From [email protected] Mon Nov 30 06:37:48 19981Return-Path: <[email protected]>2Received: from math.berkeley.edu (math.Berkeley.EDU [128.32.183.94])3by bmw.autobahn.org (8.8.7/8.8.7) with ESMTP id GAA116494for <[email protected]>; Mon, 30 Nov 1998 06:37:34 -08005Received: from cerber.mat.ub.es (cerber.mat.ub.es [161.116.4.1])6by math.berkeley.edu (8.8.7/8.8.7) with ESMTP id GAA064187for <[email protected]>; Mon, 30 Nov 1998 06:41:48 -0800 (PST)8Received: from localhost ([email protected])9by cerber.mat.ub.es (8.9.1/8.9.1) with SMTP id PAA2113610for <[email protected]>; Mon, 30 Nov 1998 15:38:25 +0100 (MET)11Date: Mon, 30 Nov 1998 15:38:25 +0100 (MET)12From: Luis Victor Dieulefait <[email protected]>13To: [email protected]14Subject: newforms15Message-ID: <[email protected]>16MIME-Version: 1.017Content-Type: TEXT/PLAIN; charset=US-ASCII18Status: RO19X-Status:2021William22Hello! I'm Luis from the University of Barcelona, we met at the23ABC workshop (Arizona). Kevin Buzzard told me you may help me with24some computations with newforms of high levels. I have visited your25site on the web with tables of hecke polynomials, but I need values for26higher levels, and I don't have linux (for the moment).27The example I have is a newform of level 8192. The number field28generated by its coefficients is given by a root of:29x^8 - 24 x^6 + 164 x^4 -240 x^2 +2 , which is a factor of the30T_3 of this level (please correct if there's some mistake)31(sorry, I forgot to mention that we are in weight 2 and32trivial nebentypus)33This form seems to have an inner twist, and it also seems from the34discussion between you and Kevin that this can be proved (I have to take35a look at Shimura's book...)36If your algorithm works well for this level, please send me the a_5 and37a_7 of this newform.3839One more thing: I've found in your table the T_2, T_3 , and T_5 for level402048. Can you send me the T_7 and T_11 for this same level?4142Up to what level can you compute these Hecke polynomials?4330000 would be too much?4445Thank you!46Luis4748From [email protected] Tue Dec 1 05:54:17 199849Return-Path: <[email protected]>50Received: from cerber.mat.ub.es (cerber.mat.ub.es [161.116.4.1])51by bmw.autobahn.org (8.8.7/8.8.7) with ESMTP id FAA0863252for <[email protected]>; Tue, 1 Dec 1998 05:54:15 -080053Received: from localhost ([email protected])54by cerber.mat.ub.es (8.9.1/8.9.1) with SMTP id OAA2922655for <[email protected]>; Tue, 1 Dec 1998 14:58:40 +0100 (MET)56Date: Tue, 1 Dec 1998 14:58:39 +0100 (MET)57From: Luis Victor Dieulefait <[email protected]>58To: [email protected]59Subject: Re: newforms60In-Reply-To: <[email protected]>61Message-ID: <[email protected]>62MIME-Version: 1.063Content-Type: TEXT/PLAIN; charset=US-ASCII64Status: RO65X-Status: A6667William68Thank you very much for your help! I will see wether or not I can manage69to obtain the information I need from the polynomials reduced mod. some70primes, but I guess that combining these reduced polynomials with the71known bounds something can be done.72I still need some more details. For level 2048 I haven't been able to73find out which are the eigenvalues I need. I'm interested in the newform74whose a_3 is a root of x^4 -20* x^2 + 98 , please send me the a_5,75a_7,...., a_17 of this form.7677Regarding level 8192, the form whose a_3 I sent you in the former mail is78new and the degree 8 polynomial whose root is the a_3 is a simple factor79of the new part of the T_3 of level 8192, and this a_3 generates the whole80number field attached to this newform. With this information, theorem813.64 of Shimura's book and the value of a_3 it follows that this form has82an inner twist given83by the mod 4 character chi . (I think there is no need to look at the84oldforms because if f is new of level 8192 and f*chi is old, then85(f*chi)*chi=f would be old, thus giving a contradiction. So this form f86has an inner twist.8788Thanks again,89Luis9091From [email protected] Wed Dec 2 05:25:54 199892Return-Path: <[email protected]>93Received: from cerber.mat.ub.es (cerber.mat.ub.es [161.116.4.1])94by bmw.autobahn.org (8.8.7/8.8.7) with ESMTP id FAA3209095for <[email protected]>; Wed, 2 Dec 1998 05:25:29 -080096Received: from localhost ([email protected])97by cerber.mat.ub.es (8.9.1/8.9.1) with SMTP id OAA0594998for <[email protected]>; Wed, 2 Dec 1998 14:29:39 +0100 (MET)99Date: Wed, 2 Dec 1998 14:29:39 +0100 (MET)100From: Luis Victor Dieulefait <[email protected]>101Reply-To: Luis Victor Dieulefait <[email protected]>102To: [email protected]103Subject: Re: newforms104In-Reply-To: <[email protected]>105Message-ID: <[email protected]>106MIME-Version: 1.0107Content-Type: TEXT/PLAIN; charset=US-ASCII108Status: RO109X-Status:110111William,112The relation between the number field corresponding to f := Q_f113and the subfield fixed by the action of the inner twists:= F_f ,114gets into the picture of the abelian variety A_f. More precisely:115THe endomorphism algebra of A_f (over Q) is a central simple algebra over116F_f which contains Q_f as a maximal conmutative subfield. Its degree over117Q is [Q_f : Q]*[Q_f : F_f] .118119This is proved in : K. Ribet: "Twists of Modular Forms and Endomorphisms120of Abelian VArieties", Math. Ann. 253, 43-62 (1980)121122Regarding the coefficient a_5 and a_7 of the form of level 8192 that are123"probably" 0, I think that, for example for the a_5, the upper bound for124the absolute value of it, 2*sqrt(5), proves that it will be 0 when you can125show this mod some primes whose product is greater that (2*sqrt(5))^8 =126160000. The two primes you took are not enough, but if you do the same for127one more prime greater than 25 then it will be enough to get a proof that128a_5 = 0. If you think this argument is correct, please do this computation129with such a prime, say 29.130131Regarding modular forms of very big level (more than 30000) I will like to132discuss with you some other time the possibility of doing computations, at133least mod some primes, maybe some previous information I have about the134number field Q_f in some cases can make the computations easier....135136BEst Regards,137Luis138PD: During december when you mail me, please send a copy to139[email protected] , I will spend some weeks there.140141From [email protected] Wed Dec 2 07:16:38 1998142Return-Path: <[email protected]>143Received: from cerber.mat.ub.es (cerber.mat.ub.es [161.116.4.1])144by bmw.autobahn.org (8.8.7/8.8.7) with ESMTP id HAA02553145for <[email protected]>; Wed, 2 Dec 1998 07:07:44 -0800146Received: from localhost ([email protected])147by cerber.mat.ub.es (8.9.1/8.9.1) with SMTP id QAA06895148for <[email protected]>; Wed, 2 Dec 1998 16:12:00 +0100 (MET)149Date: Wed, 2 Dec 1998 16:12:00 +0100 (MET)150From: Luis Victor Dieulefait <[email protected]>151To: [email protected]152Subject: Re: newforms153In-Reply-To: <[email protected]>154Message-ID: <[email protected]>155MIME-Version: 1.0156Content-Type: TEXT/PLAIN; charset=US-ASCII157Status: RO158X-Status:159160William,161As a matter of fact, I can prove already that a_5=0 for the newform of162level 8192, because we can use the fact that this form has an inner163twist given by the mod 4 character to see that a_5 belongs to F_f, the164subfield of Q_f fixed by the automorphism giving the inner twist.165The field F_f has degree 4, so that taking norms we get from the fact166that the abs. value of a_5 is bounded by 2*sqrt(5) that its norm is167smaller than 400. Then with the minimal polynomial mod p you computed168for those two primes it's enough to conclude that the norm of a_5, and169the a_5, is 0, becouse the product of the 2 primes is bigger that 400.170For the a_7, more computation would be needed (the twist is no longer171useful). Are you convinced with this proof that a_5=0 ?172Regards,173Luis174175176From [email protected] Fri Dec 4 05:59:45 1998177Return-Path: <[email protected]>178Received: from cerber.mat.ub.es (cerber.mat.ub.es [161.116.4.1])179by bmw.autobahn.org (8.8.7/8.8.7) with ESMTP id FAA06251180for <[email protected]>; Fri, 4 Dec 1998 05:58:53 -0800181Received: from localhost ([email protected])182by cerber.mat.ub.es (8.9.1/8.9.1) with SMTP id OAA22548183for <[email protected]>; Fri, 4 Dec 1998 14:48:03 +0100 (MET)184Date: Fri, 4 Dec 1998 14:48:03 +0100 (MET)185From: Luis Victor Dieulefait <[email protected]>186To: [email protected]187Subject: Re: newforms188In-Reply-To: <[email protected]>189Message-ID: <[email protected]>190MIME-Version: 1.0191Content-Type: TEXT/PLAIN; charset=US-ASCII192Status: RO193X-Status:194195William,196I have just realize that in the 2 examples of newforms of level 2048197and 8192, we can prove (using theorem 3.64 of Shimura's book) that the 2198characters: chi: (Z/4Z)* -->{+-1} and199psi: (Z/8Z)* -->{+-1} give the inner twist: chi*f = psi*f = f^(gamma),200where gamma is the "real conjugation": a_3 --> - a_3 , of Q_f. (this201automorphism has F_f = Q((a_3)^2) as its fixed field)202From this it follows that (in the 2 examples):203a_p = 0 for every p congruent to 5 or 7 (mod 8)204205(Proof: take p congruent to 5 mod 8. If a_p is not 0, chi(p)=1 implies206that a_p belongs to F_f; psi(p)= -1 implies that a_p doesn't belong to207F_f. Then a_p =0 )208209In particular, in the example of level 8192, this proves that a_3 = a_5 =210a_13 = 0 , as suggested by the computations you have done.211212Luis213214From [email protected] Tue Jan 19 07:27:02 1999215Received: from bmw.autobahn.org (bmw.autobahn.org [206.79.223.28])216by math.berkeley.edu (8.8.7/8.8.7) with ESMTP id HAA29168217for <[email protected]>; Tue, 19 Jan 1999 07:26:56 -0800 (PST)218Received: from cerber.mat.ub.es (cerber.mat.ub.es [161.116.4.1])219by bmw.autobahn.org (8.9.2/8.9.2) with ESMTP id HAA12862220for <[email protected]>; Tue, 19 Jan 1999 07:28:52 -0800 (PST)221Received: from localhost ([email protected])222by cerber.mat.ub.es (8.9.1/8.9.1) with SMTP id QAA26507223for <[email protected]>; Tue, 19 Jan 1999 16:24:57 +0100 (MET)224Date: Tue, 19 Jan 1999 16:24:57 +0100 (MET)225From: Luis Victor Dieulefait <[email protected]>226To: [email protected]227Subject: Re: newforms228In-Reply-To: <[email protected]>229Message-ID: <[email protected]>230MIME-Version: 1.0231Content-Type: TEXT/PLAIN; charset=US-ASCII232Status: RO233X-Status: A234235William,236So it seems that the examples of newforms we were working on have CM.237I think that the argument of Shimura I thought was contradictory with this238fact can only be used with odd primes in the level, but in the case of the239power of 2 level I can't say nothing about CM (I mean "a priori").240I want examples similar to the ones we were working on , but without241CM. I have a few candidates to check.242Can you send me the fourier expansion of the level 1024 newforms243corresponding to the following factor of the characteristic polynomial of244the T_3: (x^4-8*x^2+8)^2245Maybe we are lucky in this example and the forms don't have CM. The fact246that the factor appears with mult. 2 makes posible that the form has one247(and only one!) twist. With a few more coefficients I think this can be248checked.249If this example doesn't work, a good place to look at is the space250of newforms with level 3*1024 = 3072, where CM can not occur.251252In the case of level 8192, remember that there was another newform whose253corresp. number field was of degree 8 (the a_3 is in fact the square root254of an element belonging to the maximal real subfiel of the cyclotomic255field of the 16-th roots of unity). Computing its a_p modulo some random256prime, does this form also seem to have CM ??257258By the way, your site in the internet is unreachable these days, do you259have any idea of what can be the problem?260Thank you a lot! Best wishes,261Luis262PS: If you can,please send me the results of the computations this week,263because I have a congress starting this sunday and it will be great if I264have this results before leaving.265266267From [email protected] Tue Jan 19 08:08:46 1999268Received: from cerber.mat.ub.es (cerber.mat.ub.es [161.116.4.1])269by math.berkeley.edu (8.8.7/8.8.7) with ESMTP id IAA00603270for <[email protected]>; Tue, 19 Jan 1999 08:08:42 -0800 (PST)271Received: from localhost ([email protected])272by cerber.mat.ub.es (8.9.1/8.9.1) with SMTP id RAA27137273for <[email protected]>; Tue, 19 Jan 1999 17:07:50 +0100 (MET)274Date: Tue, 19 Jan 1999 17:07:49 +0100 (MET)275From: Luis Victor Dieulefait <[email protected]>276To: William Arthur Stein <[email protected]>277Subject: Re: newforms278In-Reply-To: <[email protected]>279Message-ID: <[email protected]>280MIME-Version: 1.0281Content-Type: TEXT/PLAIN; charset=US-ASCII282Status: RO283X-Status:284285William,286287In case it helps you, here's the polynomial giving the a_3 of the level2888192 newform whose coefficients I've asked you to compute:289578 -624*x^2 + 196*x^4 - 24*x^6 + x^8290I'm reaching your site (with the new address you gave me) without291problems. Thanks again, best regards,292Luis293294295296