Karim Belabas on Sun, 24 Nov 2019 16:51:28 +0100

 Re: Finding the generating funcction for a theta sequence?

```* Kevin Acres [2019-11-24 03:02]:
> 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?

Starting from Bill's construction, you can easily show that it can't be
expressed as a simple eta quotient:

? mf4=mfinit([24,4,1]); mf6=mfinit([24,6,1]); S4=mfeigenbasis(mf4);
? S6=mfeigenbasis(mf6);
? F = mfdiv(S6,S4);
? mfisetaquo(F)
%4 = 0

On the other hand, it's actually a combination of Eisenstein series:

? mf = mfinit([24,2], 3);
? mftobasis(mf, F)
%6 = [6, 0, -38, -6, 272, -910/3, 208/3]~
? E = mfbasis(mf);
? for(i = 1, #E, print(mfdescribe(E[i])))
LIN([F_2(1), B(2)(F_2(1))], [1, -2])
LIN([F_2(1), B(3)(F_2(1))], [1, -3])
LIN([F_2(1), B(4)(F_2(1))], [1, -4])
LIN([F_2(1), B(6)(F_2(1))], [1, -6])
LIN([F_2(1), B(8)(F_2(1))], [1, -8])
LIN([F_2(1), B(12)(F_2(1))], [1, -12])
LIN([F_2(1), B(24)(F_2(1))], [1, -24])

So you can actually write it only in terms of F_2 (= mfeisenstein(2)). More
precisely

(-12 B(2) + 152 B(4) + 36 B(6) - 2176 B(8) + 3640 B(12) - 1664 B(24)) . F_2

(where B(n) is the q -> q^n operator). In fact, from this description, your

? mftobasis(mf, [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])])
%8 = [-76, -12, 544, -1820/3, 416/3]~
? E2 = mfbasis(mf);
? for(i = 1, #E2, print(mfdescribe(E2[i])))
LIN([F_2(1), B(2)(F_2(1))], [1, -2])
LIN([F_2(1), B(3)(F_2(1))], [1, -3])
LIN([F_2(1), B(4)(F_2(1))], [1, -4])
LIN([F_2(1), B(6)(F_2(1))], [1, -6])
LIN([F_2(1), B(12)(F_2(1))], [1, -12])

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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