From luisd@cerber.mat.ub.es Mon Nov 30 06:37:48 1998 Return-Path: Received: from math.berkeley.edu (math.Berkeley.EDU [128.32.183.94]) by bmw.autobahn.org (8.8.7/8.8.7) with ESMTP id GAA11649 for ; Mon, 30 Nov 1998 06:37:34 -0800 Received: from cerber.mat.ub.es (cerber.mat.ub.es [161.116.4.1]) by math.berkeley.edu (8.8.7/8.8.7) with ESMTP id GAA06418 for ; Mon, 30 Nov 1998 06:41:48 -0800 (PST) Received: from localhost (luisd@localhost) by cerber.mat.ub.es (8.9.1/8.9.1) with SMTP id PAA21136 for ; Mon, 30 Nov 1998 15:38:25 +0100 (MET) Date: Mon, 30 Nov 1998 15:38:25 +0100 (MET) From: Luis Victor Dieulefait To: was@math.berkeley.edu Subject: newforms Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: William Hello! I'm Luis from the University of Barcelona, we met at the ABC workshop (Arizona). Kevin Buzzard told me you may help me with some computations with newforms of high levels. I have visited your site on the web with tables of hecke polynomials, but I need values for higher levels, and I don't have linux (for the moment). The example I have is a newform of level 8192. The number field generated by its coefficients is given by a root of: x^8 - 24 x^6 + 164 x^4 -240 x^2 +2 , which is a factor of the T_3 of this level (please correct if there's some mistake) (sorry, I forgot to mention that we are in weight 2 and trivial nebentypus) This form seems to have an inner twist, and it also seems from the discussion between you and Kevin that this can be proved (I have to take a look at Shimura's book...) If your algorithm works well for this level, please send me the a_5 and a_7 of this newform. One more thing: I've found in your table the T_2, T_3 , and T_5 for level 2048. Can you send me the T_7 and T_11 for this same level? Up to what level can you compute these Hecke polynomials? 30000 would be too much? Thank you! Luis From luisd@cerber.mat.ub.es Tue Dec 1 05:54:17 1998 Return-Path: Received: from cerber.mat.ub.es (cerber.mat.ub.es [161.116.4.1]) by bmw.autobahn.org (8.8.7/8.8.7) with ESMTP id FAA08632 for ; Tue, 1 Dec 1998 05:54:15 -0800 Received: from localhost (luisd@localhost) by cerber.mat.ub.es (8.9.1/8.9.1) with SMTP id OAA29226 for ; Tue, 1 Dec 1998 14:58:40 +0100 (MET) Date: Tue, 1 Dec 1998 14:58:39 +0100 (MET) From: Luis Victor Dieulefait To: was@bmw.autobahn.org Subject: Re: newforms In-Reply-To: Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: A William Thank you very much for your help! I will see wether or not I can manage to obtain the information I need from the polynomials reduced mod. some primes, but I guess that combining these reduced polynomials with the known bounds something can be done. I still need some more details. For level 2048 I haven't been able to find out which are the eigenvalues I need. I'm interested in the newform whose a_3 is a root of x^4 -20* x^2 + 98 , please send me the a_5, a_7,...., a_17 of this form. Regarding level 8192, the form whose a_3 I sent you in the former mail is new and the degree 8 polynomial whose root is the a_3 is a simple factor of the new part of the T_3 of level 8192, and this a_3 generates the whole number field attached to this newform. With this information, theorem 3.64 of Shimura's book and the value of a_3 it follows that this form has an inner twist given by the mod 4 character chi . (I think there is no need to look at the oldforms because if f is new of level 8192 and f*chi is old, then (f*chi)*chi=f would be old, thus giving a contradiction. So this form f has an inner twist. Thanks again, Luis From luisd@cerber.mat.ub.es Wed Dec 2 05:25:54 1998 Return-Path: Received: from cerber.mat.ub.es (cerber.mat.ub.es [161.116.4.1]) by bmw.autobahn.org (8.8.7/8.8.7) with ESMTP id FAA32090 for ; Wed, 2 Dec 1998 05:25:29 -0800 Received: from localhost (luisd@localhost) by cerber.mat.ub.es (8.9.1/8.9.1) with SMTP id OAA05949 for ; Wed, 2 Dec 1998 14:29:39 +0100 (MET) Date: Wed, 2 Dec 1998 14:29:39 +0100 (MET) From: Luis Victor Dieulefait Reply-To: Luis Victor Dieulefait To: was@bmw.autobahn.org Subject: Re: newforms In-Reply-To: Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: William, The relation between the number field corresponding to f := Q_f and the subfield fixed by the action of the inner twists:= F_f , gets into the picture of the abelian variety A_f. More precisely: THe endomorphism algebra of A_f (over Q) is a central simple algebra over F_f which contains Q_f as a maximal conmutative subfield. Its degree over Q is [Q_f : Q]*[Q_f : F_f] . This is proved in : K. Ribet: "Twists of Modular Forms and Endomorphisms of Abelian VArieties", Math. Ann. 253, 43-62 (1980) Regarding the coefficient a_5 and a_7 of the form of level 8192 that are "probably" 0, I think that, for example for the a_5, the upper bound for the absolute value of it, 2*sqrt(5), proves that it will be 0 when you can show this mod some primes whose product is greater that (2*sqrt(5))^8 = 160000. The two primes you took are not enough, but if you do the same for one more prime greater than 25 then it will be enough to get a proof that a_5 = 0. If you think this argument is correct, please do this computation with such a prime, say 29. Regarding modular forms of very big level (more than 30000) I will like to discuss with you some other time the possibility of doing computations, at least mod some primes, maybe some previous information I have about the number field Q_f in some cases can make the computations easier.... BEst Regards, Luis PD: During december when you mail me, please send a copy to egallese@agatha.unr.edu.ar , I will spend some weeks there. From luisd@cerber.mat.ub.es Wed Dec 2 07:16:38 1998 Return-Path: Received: from cerber.mat.ub.es (cerber.mat.ub.es [161.116.4.1]) by bmw.autobahn.org (8.8.7/8.8.7) with ESMTP id HAA02553 for ; Wed, 2 Dec 1998 07:07:44 -0800 Received: from localhost (luisd@localhost) by cerber.mat.ub.es (8.9.1/8.9.1) with SMTP id QAA06895 for ; Wed, 2 Dec 1998 16:12:00 +0100 (MET) Date: Wed, 2 Dec 1998 16:12:00 +0100 (MET) From: Luis Victor Dieulefait To: was@bmw.autobahn.org Subject: Re: newforms In-Reply-To: Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: William, As a matter of fact, I can prove already that a_5=0 for the newform of level 8192, because we can use the fact that this form has an inner twist given by the mod 4 character to see that a_5 belongs to F_f, the subfield of Q_f fixed by the automorphism giving the inner twist. The field F_f has degree 4, so that taking norms we get from the fact that the abs. value of a_5 is bounded by 2*sqrt(5) that its norm is smaller than 400. Then with the minimal polynomial mod p you computed for those two primes it's enough to conclude that the norm of a_5, and the a_5, is 0, becouse the product of the 2 primes is bigger that 400. For the a_7, more computation would be needed (the twist is no longer useful). Are you convinced with this proof that a_5=0 ? Regards, Luis From luisd@cerber.mat.ub.es Fri Dec 4 05:59:45 1998 Return-Path: Received: from cerber.mat.ub.es (cerber.mat.ub.es [161.116.4.1]) by bmw.autobahn.org (8.8.7/8.8.7) with ESMTP id FAA06251 for ; Fri, 4 Dec 1998 05:58:53 -0800 Received: from localhost (luisd@localhost) by cerber.mat.ub.es (8.9.1/8.9.1) with SMTP id OAA22548 for ; Fri, 4 Dec 1998 14:48:03 +0100 (MET) Date: Fri, 4 Dec 1998 14:48:03 +0100 (MET) From: Luis Victor Dieulefait To: was@bmw.autobahn.org Subject: Re: newforms In-Reply-To: Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: William, I have just realize that in the 2 examples of newforms of level 2048 and 8192, we can prove (using theorem 3.64 of Shimura's book) that the 2 characters: chi: (Z/4Z)* -->{+-1} and psi: (Z/8Z)* -->{+-1} give the inner twist: chi*f = psi*f = f^(gamma), where gamma is the "real conjugation": a_3 --> - a_3 , of Q_f. (this automorphism has F_f = Q((a_3)^2) as its fixed field) From this it follows that (in the 2 examples): a_p = 0 for every p congruent to 5 or 7 (mod 8) (Proof: take p congruent to 5 mod 8. If a_p is not 0, chi(p)=1 implies that a_p belongs to F_f; psi(p)= -1 implies that a_p doesn't belong to F_f. Then a_p =0 ) In particular, in the example of level 8192, this proves that a_3 = a_5 = a_13 = 0 , as suggested by the computations you have done. Luis From luisd@cerber.mat.ub.es Tue Jan 19 07:27:02 1999 Received: from bmw.autobahn.org (bmw.autobahn.org [206.79.223.28]) by math.berkeley.edu (8.8.7/8.8.7) with ESMTP id HAA29168 for ; Tue, 19 Jan 1999 07:26:56 -0800 (PST) Received: from cerber.mat.ub.es (cerber.mat.ub.es [161.116.4.1]) by bmw.autobahn.org (8.9.2/8.9.2) with ESMTP id HAA12862 for ; Tue, 19 Jan 1999 07:28:52 -0800 (PST) Received: from localhost (luisd@localhost) by cerber.mat.ub.es (8.9.1/8.9.1) with SMTP id QAA26507 for ; Tue, 19 Jan 1999 16:24:57 +0100 (MET) Date: Tue, 19 Jan 1999 16:24:57 +0100 (MET) From: Luis Victor Dieulefait To: was@bmw.autobahn.org Subject: Re: newforms In-Reply-To: Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: A William, So it seems that the examples of newforms we were working on have CM. I think that the argument of Shimura I thought was contradictory with this fact can only be used with odd primes in the level, but in the case of the power of 2 level I can't say nothing about CM (I mean "a priori"). I want examples similar to the ones we were working on , but without CM. I have a few candidates to check. Can you send me the fourier expansion of the level 1024 newforms corresponding to the following factor of the characteristic polynomial of the T_3: (x^4-8*x^2+8)^2 Maybe we are lucky in this example and the forms don't have CM. The fact that the factor appears with mult. 2 makes posible that the form has one (and only one!) twist. With a few more coefficients I think this can be checked. If this example doesn't work, a good place to look at is the space of newforms with level 3*1024 = 3072, where CM can not occur. In the case of level 8192, remember that there was another newform whose corresp. number field was of degree 8 (the a_3 is in fact the square root of an element belonging to the maximal real subfiel of the cyclotomic field of the 16-th roots of unity). Computing its a_p modulo some random prime, does this form also seem to have CM ?? By the way, your site in the internet is unreachable these days, do you have any idea of what can be the problem? Thank you a lot! Best wishes, Luis PS: If you can,please send me the results of the computations this week, because I have a congress starting this sunday and it will be great if I have this results before leaving. From luisd@cerber.mat.ub.es Tue Jan 19 08:08:46 1999 Received: from cerber.mat.ub.es (cerber.mat.ub.es [161.116.4.1]) by math.berkeley.edu (8.8.7/8.8.7) with ESMTP id IAA00603 for ; Tue, 19 Jan 1999 08:08:42 -0800 (PST) Received: from localhost (luisd@localhost) by cerber.mat.ub.es (8.9.1/8.9.1) with SMTP id RAA27137 for ; Tue, 19 Jan 1999 17:07:50 +0100 (MET) Date: Tue, 19 Jan 1999 17:07:49 +0100 (MET) From: Luis Victor Dieulefait To: William Arthur Stein Subject: Re: newforms In-Reply-To: Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: William, In case it helps you, here's the polynomial giving the a_3 of the level 8192 newform whose coefficients I've asked you to compute: 578 -624*x^2 + 196*x^4 - 24*x^6 + x^8 I'm reaching your site (with the new address you gave me) without problems. Thanks again, best regards, Luis