
617 F.3d 88 (2010)
UNITED STATES of America, Appellee,
v.
Ellis Williams GOMEZ, Defendant-Appellant.
Docket No. 08-3829-cr.
United States Court of Appeals, Second Circuit.
Argued: May 26, 2009.
Decided: August 4, 2010.
*89 Lloyd Epstein, Epstein & Weil, New York, NY, for Defendant-Appellant.
Jocelyn E. Strauber, Assistant United States Attorney, for Lev L. Dassin, Acting United States Attorney for the Southern District of New York, New York, NY (John M. Hillebrecht, on the brief), for Appellee.
Before: LEVAL, POOLER, and B.D. PARKER, Circuit Judges.
POOLER, Circuit Judge:
Ellis Williams Gomez appeals from a judgment of conviction (Swain, J.) for conspiracy to distribute and to possess with intent to distribute 3,4-Methylenedioxy-methamphetamine ("Ecstasy") and the resulting sentence, which consisted primarily of sixty-six months imprisonment. Key testimony in the case against Gomez was provided by Detective Michael Ryan, who interviewed Gomez's co-conspirator, Fred Rivas, after Rivas was arrested for selling Ecstasy to a confidential informant. On appeal, Gomez primarily argues that the district court admitted prejudicial hearsay in permitting Detective Ryan to testify, by inference, that Rivas told Ryan that Gomez supplied Rivas with Ecstasy. The government argues that Gomez waived his hearsay objection, that any possible hearsay problem was cured by the district court's limiting instructions, and that, in any event, the evidence was admissible for the proper, non-hearsay purpose of showing context, in that it explained how Ryan came to place a call to Gomez. We hold that Ryan's testimony included highly prejudicial hearsay, and we reject the government's contentions. Moreover, while *90 we find it unnecessary to reach the issue, we note that Ryan's testimony also poses a serious Confrontation Clause issue. We vacate Gomez's conviction and remand for a new trial.

BACKGROUND
The defendants in the charged Ecstasy conspiracy were Gomez, Rivas, and Darwin Pena. Rivas and Pena were arrested after they sold 5,000 Ecstasy pills to a confidential informant. Following their arrest, Rivas cooperated with the government, and the phone calls at the center of this appeal were made as part of that cooperation. Prior to Gomez's trial, Rivas and Pena pleaded guilty.

The government's opening statement
In its opening, the government told the jury that the evidence would show that after Rivas's arrest:
[he] spoke with the agents who arrested him. And the agents instructed him to make phone calls to the person who supplied him with the five thousand pills. They told him to say he was with the guy who purchased the five thousand pills, and that Rivas had money to give to that person. So who did Rivas call? He called the defendant.

Detective Ryan's Testimony
Only three witnesses testified against Gomez at trial. The most important of the three was New York City Police Detective Michael Ryan who testified concerning the arrest and interrogation of Rivas and supplied foundation for transcripts of recorded conversations between Rivas and Gomez. Ryan testified that he arrested Rivas on June 8, 2006, and later interviewed Rivas along with United States Drug Enforcement ("DEA") Special Agent Rodolfo Cesario, who spoke Spanish, Rivas's native language. Ryan told Rivas "to call the person who had given him the 5,000 pills that we had seized that morning. We told him to call that person to tell him that you were with the person, the customer who he sold the 5,000 pills to, that he was collecting money from that person and that he needed an additional 2,000 pills of Ecstasy." The agents provided Rivas with written instructions to state: "I'm here with the guy from the five on 190th. He gave me something, he wants two more. I'll call you when I'm nearby."
Ryan testified that he dialed Gomez's number on Rivas's cell phone by locating Gomez's number on the phone, setting up a recording device for Rivas, then pressing the send button on the phone to make the call before passing the cell phone to Rivas. The defense objected on hearsay grounds to Ryan's testimony. In response to those objections, the district court instructed the jury that:
the testimony that you have just heard about instructions given to Mr. Rivas and Mr. Rivas's actions in response to those instructions is not being offered to establish and is not to be considered by you as evidence that the defendant was, in fact, Mr. Rivas's supplier. Rather, the evidence is offered for the more limited purpose of explaining how it happened that Mr. Rivas made calls to the defendant.
The government did not call Rivas as a witness at trial.

Closing arguments and charge
In closing, the government said:
[T]he crucial question and the only question that you need to decide is whether the government has proved that the defendant, Ellis Williams Gomez, was part of the agreement to distribute ecstacy. So, what was the evidence on this? Well, you heard that after Fred Rivas *91 got arrested on June 8 of 2006, he made a series of telephone calls to the defendant. You heard why those calls happened. Detective Ryan told Rivas to call the person who had given him the five thousand pills.
Defense counsel argued, in part, that even if the jury found that Gomez sold the five thousand pills to Rivaswhich counsel acknowledged would be "a fair and reasonable interpretation" of the tapesthat finding would not establish a conspiracy. In her charge, Judge Swain again instructed the jury that it could not use Ryan's testimony concerning the instructions he gave Rivas and Rivas's actions in response to those instructions to establish that Gomez was Rivas's supplier.
The jury convicted Gomez of the sole count of the indictment. In a motion for a new trial pursuant to Rule 33 of the Rules of Criminal Procedure, Gomez primarily argued that Ryan should not have been permitted to testify as to his understanding of the meaning of Rivas's statements on the recorded calls to Gomez. Gomez also argued that the jury charge on the nature of the conspiracy was erroneous, and that defense counsel was ineffective for failing to introduce Rivas's and Gomez's cell phone records at trial. Judge Swain denied Gomez's Rule 33 motion in a memorandum order of September 20, 2007. This appeal followed.

DISCUSSION

I. Prejudicial hearsay.
The Federal Rules of Evidence prohibit as hearsay the receipt of out-of-court statements offered to proved the truth of the matter asserted. See Fed. R.Evid. 801, 802. The government argues, however, that the disputed testimony was admitted for a valid, non-hearsay purposeto provide background and context for the juryand that it therefore is not hearsay. The government's claim is simply not credible. Indeed, the government in both its opening statement and its summation explicitly argued the very hearsay use of the evidence which it now contends it was not offering.[1] The bearing of Ryan's testimony on Gomez's guilt was clear and the inference that the jury would draw from itthat Gomez was Rivas's supplier in the transaction for which Rivas was arrestedwas inescapable. Ryan's testimony that he (1) told Rivas to call his supplier, (2) selected Gomez's number from Rivas's cell phone address book, (3) placed the call, and (4) then handed the telephone to Rivas, clearly informed the jury that Rivas had told Ryan that Gomez was his supplier. There was no other reasonable explanation why Ryan would have selected Gomez's number from Rivas's address book.
In Ryan v. Miller, 303 F.3d 231 (2d Cir.2002), we addressed a similar issue. *92 There, after questioning a co-conspirator, a detective told his colleague to advise the defendant of his Miranda rights. In Ryan, we rejected the State's contention that the detective' testimony was properly admissible as background information to show why the defendant was at the police station and to explain the course of the investigation that led to his arrest in part because it was not necessary to provide background, and it "was only relevant if we assume[d] its truththat [the co-defendant] accused [the defendant]which we [could] not do." Id. at 252-53. In this case, the hearsay violation is even more clear than it was in Ryan, because in Ryan, there were other reasonably plausible, non-inculpating explanations for the instruction to give a Miranda warning. For example, the defendant's co-conspirator might have told the police that the defendant had information about the crime, but not that he participated in it. Alternatively, the first detective might have told the second detective to give the warning merely because it was good police practice. Conversely, here there is no such likely non-culpatory alternative inferences to explain Ryan's selection of Gomez's number. Ryan's testimony provides background that would aid the jury in understanding why Gomez was called only if we assume that Rivas told Ryan that Gomez was his supplier, which we cannot do. If the jury inexplicably missed the obvious point of Ryan's testimony, the Assistant United States Attorney ("AUSA") drove it home with her opening and summation.
The objected-to testimony was also completely unnecessary to inform the jury of the background or to avoid confusion. Ryan's testimony that Rivas was arrested and then made several consensually recorded phone calls would have been sufficient background for the calls themselves. Although hearsay may sometimes be admitted "to prove relevant facts other than the truth of what was asserted," this is not the case "when the evidence sought to be justified in its nonhearsay use is on the unimportant issue of investigative background, and it has considerable capacity in its improper application for substantial prejudice to the defendant on the crucial issue of proof of guilt." United States v. Johnson, 529 F.3d 493, 500 (2d Cir.2008). In this case, the main import of the challenged evidence is not what it furnished as background, but rather its demonstration that Rivas identified Gomez as his supplier.[2]

II. Waiver
The government next argues Gomez waived any error related to Ryan's testimony by failing to adequately object to the scope and substance of Ryan's proposed testimony during pretrial conferences. "[W]aiver is the `intentional relinquishment or abandonment of a known right.'" United States v. Olano, 507 U.S. 725, 733, 113 S.Ct. 1770, 123 L.Ed.2d 508 (1993) (quoting Johnson v. Zerbst, 304 U.S. 458, 464, 58 S.Ct. 1019, 82 L.Ed. 1461 (1938)). A defendant who waives a claim of error may not raise it on appeal. See United States v. Polouizzi, 564 F.3d 142, 153 (2d Cir.2009).
Ryan's testimony was the subject of pretrial controversy. In an April 9, 2007, *93 pretrial conference, defense counsel objected to the admission of instructions Ryan had given Rivas for the latter's calls to Gomez. Counsel explained: "I have an objection to any testimony being elicited from detective Ryan about what he instructed Mr. Rivas to do; therefore, I also have an objection to the handwritten directions." Defense counsel later explained that:
I have no objection to detective Ryan saying that earlier in the day, for whatever reasons he provides, Mr. Rivas was arrested. He was brought down to law enforcement headquarters ... and sometime thereafter, he made a phone call that was monitored and recorded, and here is the monitored and recorded conversation.
Counsel then made clear that if the government is permitted to:
do more, not only elicit testimony but also have the opportunity to introduce this statement of instructions, it infers, implies, suggests that Mr. Rivas had conversations with law enforcement, indicating and implicating Mr. Gomez' [sic] involvement, which is a background way of getting hearsay testimony in through implication, without my having the opportunity to cross-examine Mr. Rivas.
The court overruled Gomez's objection to admitting the written instructions "and to testimony consistent with [the written instructions] as to detective Ryan having instructed Rivas to make the phone call." It explained that "[t]here is nothing that is particularly suggestive of the content of any conversation between Rivas and Ryan in these instructions ... and they are relevant in providing context for the telephone call and chronology."
Later the AUSA indicated that "Mr. Rivas was also instructed to ... make certain phonecalls [sic] to certain people or at least to a person who fell within a particular description as in the person who had supplied him with these ... five thousand pills." Defense counsel responded:
There is an objection on the same grounds that I objected to the instructions. Rivas having postarrest conversations and not being a witness in the case, the statements that he makes to Detective Ryan at that time are hearsay and inadmissible for their truth. The phone calls are in evidence. Now, according to your Honor's ruling, the written instruction that Detective Ryan gave him is going to be in evidence.
Asked by Judge Swain whether defense counsel was "objecting to the introduction into evidence of the alleged fact that ... these were calls that were to be made to an alleged supplier?", defense counsel explained:
I think that any statements Rivas made to Detective Ryan before the phonecalls [sic] are made are inadmissible because they are hearsay and because they will be taken for their truth ... proving that Gomez was, in fact, a supplier. And that's inadmissible hearsay, it would be treated as truthful by the jury and that's my fear.
The government attorney responded that she:
plan[ned] to ask the agent: Did you speak with Rivas? What instructions, if any, did you give him?
What I expect the agent to say is something along the lines of I instructed him to call the person he got the five thousand from and to ask him for moreto ask him for two more and to ask him also forto ask him if he wanted to receive his money or say he had money for him.... The only really new thing that I'm raising is the instruction *94 to call the person that you got the five from.
Defense counsel replied:
If the purported testimony by Detective Ryan is: I instructed him to call "the person" who supplied him the five thousand, I don't object.
If the instructions are based on conversations I had with Rivas, I instructed him to call Gomez, the person who supplied him with the five thousand, then I do.
The government agreed that it would not elicit testimony from Ryan that he instructed Rivas to call Gomez, whom Rivas had identified as his supplier. Thus, at this point, it appeared that everyone agreed as to the nature of the testimony Ryan could give.
However, the issue was apparently not settled to Judge Swain's satisfaction because upon her return to chambers, she instructed the government's attorneys by facsimile, with a copy to defense counsel, that the government attorney:
should address by letter to the Court the issue of why testimony to the effect that Mr. Rivas phoned Mr. Gomez in response to a directive by Detective Ryan to telephone Mr. Rivas's supplier would not constitute evidence of a "testimonial" out of court statement within the meaning of Crawford v. Washington, 541 U.S. 36 [124 S.Ct. 1354, 158 L.Ed.2d 177] (2004), and Davis v. Washington, [547 U.S. 813] 126 S.Ct. 2266 [165 L.Ed.2d 224] (2006); see also U.S. v. Basciano, 430 F.Supp.2d 87, 89-90 (E.D.N.Y.2006).
The government responded that there was no Crawford problem because it did not intend to argue based on the testimony that Rivas was instructed to call his supplier, and that Gomez was Rivas's supplier. It indicated that it would not object to an instruction that the jury could consider the applicable portion of Ryan's testimony only as "provid[ing] the necessary context for the calls between Gomez and Rivas, that is, to make clear to the jury why it is that Rivas telephoned Gomez." Ultimately, the judge drafted her own proposed instruction:
the testimony that you have just heard about instructions given to Mr. Rivas and Mr. Rivas's actions in response to those instructions is not being offered to establish and is not to be considered by you as evidence that the defendant was in fact, Mr. Rivas's supplier. Rather, the evidence is offered for the more limited purpose of explaining how it happened that Mr. Rivas made calls to the defendant.
In response to the proposed charge, defense counsel said: "[w]hile still maintaining my objection to the testimony with regard to this in any way, in light of your ruling, I have no problem with the curative instruction."
The government argues, based on these exchanges, that Gomez knowingly abandoned his objection. We disagree. Defense counsel's initial statement was a global objection to anything in Ryan's testimony that "infers, implies, suggests that Mr. Rivas had [prior] conversations with law enforcement, indicating and implicating Mr. Gomez' [sic] involvement, which is a background way of getting hearsay testimony in through implication, without my having the opportunity to cross-examine Mr. Rivas." Defense counsel later refined his objection, but never retreated from his insistence that Ryan could not properly testify that Rivas had identified Gomez as his supplier. The government's argument that the defendant waived the hearsay objection is further undermined by the fact that the government made no such contention in the district court.
*95 Nor does the government succeed in demonstrating waiver by its argument that counsel reached agreement before trial on the admission of evidence. The government's evidence went well beyond what had arguably been stipulated. The government's lawyer assured the court and defense counsel that she "would certainly caution [Ryan] from saying anything that would get close to referencing the content of [the pre-phone call] conversations that he had with Mr. Rivas." At trial, however, Ryan testified that "after the debriefing of Mr. Rivas, we placed several calls to ... Gomez." Ryan also testified that he located Gomez's phone number on Rivas's phone, something he would have done only if Rivas had identified Gomez as implicated in the scheme. The AUSA also promised Judge Swain that in soliciting Ryan's testimony about Ryan's instructions to Rivas to call "the person" who supplied Rivas with the ecstasy, "there is no one who is going to say who that person is...." Yet Ryan specifically testified that "Ellis Williams Gomez" was the person to whom he placed a call just prior to testifying that Rivas was instructed to call "the person who had given him the 5,000 pills." In short, while the defendant may have waived any objection to certain evidence, he did not waive the objection advanced on this appeal.

IV. Harmless error
Not every hearsay error requires a new trial. "Error is harmless if it is highly probable that it did not contribute to the verdict." United States v. Kaiser, 609 F.3d 556, 573 (2d Cir.2010) (internal quotation marks omitted). The following factors must be weighed in determining whether the wrongful admission of evidence constituted harmless error: "(1) the overall strength of the prosecutor's case; (2) the prosecutor's conduct with respect to the improperly admitted evidence; (3) the importance of the wrongly admitted testimony; and (4) whether such evidence was cumulative of other properly admitted evidence." United States v. Kaplan, 490 F.3d 110, 123 (2d Cir.2007). "Where the erroneously admitted evidence goes to the heart of the case against the defendant, and the other evidence against the defendant is weak, we cannot conclude that the evidence was unimportant or was not a substantial factor in the jury's verdict." United States v. Rigas, 490 F.3d 208, 222 (2d Cir.2007).
The government contends that admission of Ryan's testimony was harmless because the tapes constituted overwhelming evidence that Gomez had supplied the Ecstasy used in the sale for which Rivas had been arrested and Gomez's counsel admitted as much both in his requests to charge and in his summation to the jury. We also consider whether the court's limiting instruction was sufficient, in combination with the admissible evidence, to establish harmless error.
While the tape evidencealong with the unobjectionable portion of law enforcement personnel's testimony detailing the sale by Rivas and Pena to the confidential informant, their subsequent arrest, and the consensual recording of phone calls to Gomezwas certainly sufficient to support the jury's verdict, it is hardly overwhelming as to Gomez's involvement in the charged conspiracy. A reasonable juror would likely have been compelled to believe that Gomez and Rivas were discussing a drug deal. However, our hypothetical reasonable juror would not have been compelled to conclude that Gomez was the supplier for the earlier sale. To make the case for that proposition overwhelming, the government needed Ryan's testimony that he instructed Rivas to call the person who had supplied him the drugs for the 5,000 pill sale and that Ryan then dialed *96 Gomez's phone number. Ryan's testimony was not duplicative of the tapes because the tapes do not clearly indicate that Gomez supplied the 5,000 pills. In addition, physical evidence was lacking. Although palm prints and fingerprints were found on bags of pills from the 5,000-pill transaction, none of them matched Gomez's palm and finger prints.
We accord no importance to defense counsel's request for a charge suggesting that the jury might find that Gomez did supply the pills, or to his summation. The charge request was made before trial when competent counsel was required to anticipate that the court might allow the disputed testimony. Had the court disallowed the Ryan testimony, it is likely that the charge request would have been withdrawn. And, once Ryan's testimony had been admitted, defense counsel had to take into account the jury's likely reaction to that testimony in his summation.
Finally, the court's limiting instruction did not render admission of Ryan's testimony harmless. While we generally "presume that juries follow limiting instructions," we will not do so "where there is an overwhelming probability that the jury will be unable to follow the court's instructions and the evidence is devastating to the defense." United States v. Becker, 502 F.3d 122, 130 (2d Cir.2007) (internal quotation marks omitted). Presuming that limiting instructions were followed is "inappropriate" when the "instructions required jurors to perform mental acrobatics." Id. (internal quotation marks omitted). Here the jury was confronted with an instruction that they should not consider Ryan's testimony as proof that Gomez was Rivas's supplier despite the fact that there was no apparent reason for the testimony other than to show that Ryan dialed Gomez's number because Gomez had been identified by Rivas as Rivas's supplier. Further, the limiting instruction the court gave did not cover the most damaging portion of Ryan's testimonythat Ryan called Gomez's number after instructing Rivas that he should call the person who gave him the 5,000 pills. After Ryan's testimony, the court instructed:
Ladies and Gentlemen, the testimony that you have just heard about instructions given to Mr. Rivas and Mr. Rivas's actions in response to those instructions is not being offered to establish and is not to be considered by you as evidence that the defendant was, in fact, Rivas's supplier. Rather the evidence is offered for the more limited purpose of explaining how it happened that Mr. Rivas made calls to the defendant.[3]
Because it was Ryan who made the phone call after talking to Rivas and instructing Rivas to call his supplier, the instruction left the jury free to infer that Ryan called Gomez because Rivas identified him as his supplier. And, even apart from the portion of the objectionable testimony not covered by the instruction, it would take a certain amount of mental acrobatics for the jury to separate considering Ryan's testimony on the question of why the calls were made, which they were allowed to do, from concluding that the calls were made because Rivas had told Ryan that Gomez was his supplier. Hence, we hold that the court's instruction, even in combination with admissible evidence, did not render the hearsay violation harmless.
While we decide this case on hearsay grounds, we recognize that Gomez's claim directly implicates the Confrontation Clause and his right to confront his accusers in court. Ryan's testimony told the jury, by implication, that Rivas had accused *97 Gomez of being his drug supplier. We do not reach this overlapping claim only because the doctrine of constitutional avoidance counsels against it once the evidentiary error is established. See Ashwander v. Tennessee Valley Authority, 297 U.S. 288, 56 S.Ct. 466, 80 L.Ed. 688 (1936) (Brandeis, J. concurring); United States v. Bailey, 581 F.2d 341, 345 (3d Cir.1978). We emphasize, however, that the very concerns embedded in the Confrontation Clause are part and parcel of our harmless error analysis. In that analysis, we are bound to consider "the importance of the wrongly admitted testimony," and, as we have said, Rivas's untested accusation is significant. Kaplan, 490 F.3d at 123. The government was able to convey Rivas's statements on the central question of this criminal prosecutionthrough the testimony of a police officer, no lesswithout producing Rivas himself at trial. The improperly admitted evidence had substantial weight precisely because it went to the core of the government's case against Gomez, placing an unimpeachable accusation before the jury. Without a doubt, this is what the Confrontation Clause seeks to guard against. Viewed simply as prejudicial hearsay, we recognize the same dangers.

CONCLUSION
Because admission of Ryan's testimony constituted inadmissible hearsay and did not constitute harmless error, we vacate the conviction and remand for a new trial. In light of this disposition, we do not address the alleged sentencing error.
NOTES
[1]  We are puzzled and dismayed by the government's continued maintenance of this position both before the district court and in this court. It bears repeating that:

The United States Attorney is the representative not of an ordinary party to a controversy, but of a sovereignty whose obligation to govern impartially is as compelling as its obligation to govern at all; and whose interest, therefore, in a criminal prosecution is not that it shall win a case, but that justice shall be done. As such, he is in a peculiar and very definite sense the servant of the law, the twofold aim of which is that guilt shall not escape or innocence suffer. He may prosecute with earnestness and vigorindeed, he should do so. But, while he may strike hard blows, he is not at liberty to strike foul ones. It is as much his duty to refrain from improper methods calculated to produce a wrongful conviction as it is to use every legitimate means to bring about a just one.
Berger v. United States, 295 U.S. 78, 88, 55 S.Ct. 629, 79 L.Ed. 1314 (1935).
[2]  Although Ryan's instruction to Rivas to call his supplier by itself included no hearsay, the coincidence of that instruction with Rivas's recorded telephone call to Gomez communicates the same hearsay message to the jury that Rivas identified Gomez as his supplier. Because the government will surely use the recorded telephone conversation as evidence on retrial, and because the fact of Ryan's instruction has no importance (except when used improperly), the district court should exclude the officer's instruction.
[3]  This was also the charge given in the court's final instructions to the jury.
