Re: Finding the generating funcction for a theta sequence?

[Karim Belabas <Karim.Belabas@math.u-bordeaux.fr>]

* Kevin Acres [2019-11-23 12:11]:
> I have sequence:
> 
> [1, -12, 116, -12, -1804, 8120, 116, -155744, 684532, -12]
> 
> that has a couple of siblings:
> 
> OEIS A186100
> [1, -12, -12, -12, -12, -72, -12, -96, -12, -12, -72, -144]
> 
> and
> OEIS A125510
> [1, 6, 6, 42, 6, 36, 42, 48, 6, 150, 36, 72]
> 
> I strongly suspect my sequence to also be a theta series, which raises my
> question - is there a way to try and derive it's generating function using
> pari/gp?

No direct support for this, but you can try to recognize them as modular forms:

? m = [1, -12, 116, -12, -1804, 8120, 116, -155744, 684532, -12];
? L = mfsearch([[1..30], 2], m);
? [ print(mfcoefs(f, 11)) | f <- L ];
[1, -12, 116, -12, -1804, 8120, 116, -155744, 684532, -12, 672816, 856096]
\\ single solution in level <= 30 and weight 2.

You may have to input more terms: there are 336 solutions in level <= 300 ...
(31 of which have integer coefficients)

Once you identify the form, in particular its level, you can look for theta
series in the corresponding modular form space [using, e.g., mffromqf and/or
a database of lattices]

Cheers,

    K.B.
--
Karim Belabas, IMB (UMR 5251)  Tel: (+33) (0)5 40 00 26 17
Universite de Bordeaux         Fax: (+33) (0)5 40 00 21 23
351, cours de la Liberation    http://www.math.u-bordeaux.fr/~kbelabas/
F-33405 Talence (France)       http://pari.math.u-bordeaux.fr/  [PARI/GP]
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