Re: Fwd: Q-expansion question

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

* Kevin Acres [2018-12-28 22:33]:
> I’m trying to reproduce with pari/gp the q expansion shown at :
> http://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/137/2/1/a/
> 
> I’m using pari/gp  2.12.0 but struggling to get a match.
> 
> Any ideas?

(22:42) gp > mf = mfinit([137,2], 0);
(22:43) gp > F = mfeigenbasis(mf); #F
%2 = 2
(22:43) gp > nfs = mffields(mf)
%3 = [y^4 - y^3 - 3*y^2 + y + 1, y^7 - 12*y^5 - 10*y^4 + 24*y^3 + 31*y^2 + 11*y + 1]

The first one is obviously isomorphic to the a^4 + 3*a^3 − 4*a − 1 = 0 in LMFDB:

(22:43) gp > nfisisom(nfs[1], a^4 + 3*a^3 - 4*a - 1)
%4 = [a + 1, a^3 + 2*a^2 - 2*a - 2]

(22:43) gp > v = lift(mfcoefs(F[1],10))
%5 = [0, 1, y - 1, y^3 - 2*y^2 - 2*y + 1, y^2 - 2*y - 1, -2*y^3 + 3*y^2 + 3*y - 3, -2*y^3 + 3*y^2 + 2*y - 2, -y^3 + y^2 + 3*y - 4, y^3 - 3*y^2 - y + 3, 2*y^2 - y - 2, 3*y^3 - 6*y^2 - 4*y + 5]

\\ apply isomorphism (the other one gives a conjugate)
(22:43) gp > lift(subst(v, 'y, Mod(a + 1, a^4 + 3*a^3 - 4*a - 1)))
%6 = [0, 1, a, a^3 + a^2 - 3*a - 2, a^2 - 2, -2*a^3 - 3*a^2 + 3*a + 1, -2*a^3 - 3*a^2 + 2*a + 1, -a^3 - 2*a^2 + 2*a - 1, a^3 - 4*a, 2*a^2 + 3*a - 1, 3*a^3 + 3*a^2 - 7*a - 2]

Hope this helps,

    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/
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