Bill Allombert on Sat, 23 Nov 2019 19:55:07 +0100

 Re: Finding the generating funcction for a theta sequence?

```On Sat, Nov 23, 2019 at 10:51:58PM +1100, Kevin Acres wrote:
> Hi Karim,
>
> Thanks for the pointer. A longer sequence is:
> [1, -12, 116, -12, -1804, 8120, 116, -155744, 684532, -12, -13237576,
> 58212208, -1804, -1125531816, 4949148576, 8120, -95692200972,
> 420774756136, 116, -8135721271536, 35774143649208, -155744,
> -691696548706960, 3041506787016416, 684532, -58807829742387572,
> 258587980022941272, -12];
>
> But I didn't get a match on that.

This cannot be a plain theta series since there are negative
coefficients.

> It's basically newform 24.6.a.b divided by 24.4.a.a (From LMFDB).
... once you replace q^2 by q.

Indeed:
mf4=mfinit([24,4,1]); mf6=mfinit([24,6,1]);
S4=mfeigenbasis(mf4); S6=mfeigenbasis(mf6);
F = mfdiv(S6[2],S4[1]);
mfcoefs(F,100)
[1,0,-12,0,116,0,-12,0,-1804,0,8120,0,116,0,-155744,0,684532,0,-12,0,-13237576,0,58212208,0,-1804,0,-1125531816,0,4949148576,0,8120,0,-95692200972,0,420774756136,0,116,0,-8135721271536,0,35774143649208,0,-155744,0,-691696548706960,0,3041506787016416,0,684532,0,-58807829742387572,0,258587980022941272,0,-12,0,-4999823761392810080,0,21985071250116140696,0,-13237576,0,-425083492363859255680,0,1869164057145124702196,0,58212208,0,-36140468965190515554264,0,158915758460649561308608,0,-1804,0,-3072651656644273090003400,0,13510969350489254857017616,0,-1125531816,0,-261235907374978890400355400,0,1148698496348690403460496904,0,4949148576,0,-22210197226378695797930607120,0,97661996062922066757106510704,0,8120,0,-1888304199034038995783841477920,0,8303193140159697330525128080832,0,-95692200972,0,-160543047490576416258971456111276,0,705934950155802356197027251490316]

F is a weight-2 modular form of level 24.

Cheers,
Bill

```