Kevin Acres on Sun, 24 Nov 2019 03:02:01 +0100

 Re: Finding the generating funcction for a theta sequence?

```Hello Bill,

Thanks for getting back.  My end goal was to try and come up with a,
relatively simple, eta quotient that also generates this sequence.

Are you able to offer any advice in that area?

Best Regards,

Kevin.

On Sun, November 24, 2019 5:55 am, Bill Allombert wrote:
> 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,-132
> 37576,0,58212208,0,-1804,0,-1125531816,0,4949148576,0,8120,0,-95692200972
> ,0,420774756136,0,116,0,-8135721271536,0,35774143649208,0,-155744,0,-6916
> 96548706960,0,3041506787016416,0,684532,0,-58807829742387572,0,2585879800
> 22941272,0,-12,0,-4999823761392810080,0,21985071250116140696,0,-13237576,
> 0,-425083492363859255680,0,1869164057145124702196,0,58212208,0,-361404689
> 65190515554264,0,158915758460649561308608,0,-1804,0,-30726516566442730900
> 03400,0,13510969350489254857017616,0,-1125531816,0,-261235907374978890400
> 355400,0,1148698496348690403460496904,0,4949148576,0,-2221019722637869579
> 7930607120,0,97661996062922066757106510704,0,8120,0,-18883041990340389957
> 83841477920,0,8303193140159697330525128080832,0,-95692200972,0,-160543047
> 490576416258971456111276,0,705934950155802356197027251490316]
>
>
> F is a weight-2 modular form of level 24.
>
>
> Cheers,
> Bill
>
>

```