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A253193
Expansion of a weight 2 Gamma0(23) cusp form in powers of q with a(1) = 1, a(2) = 0.
2
1, 0, -1, -1, 0, -2, 2, -1, 2, 2, -4, 3, 3, 2, -4, 0, 2, 0, -2, -2, -6, -2, 1, 5, -1, 0, 1, -4, -3, 2, 3, 5, 8, -2, 4, -2, 0, 0, -3, -4, -1, -2, 0, 6, 0, 0, -1, -6, 1, -4, 2, -3, -2, 2, -4, -6, 2, 0, 4, 2, -2, 6, 4, 1, 0, 6, -4, 0, -1, 0, 11, -2, 9, -2, 9, 2
OFFSET
1,6
LINKS
FORMULA
If b(n) = a(n) - (1 + sqrt(5))/2 * A232506(n) then b() is multiplicative with b(23^e) = 1, otherwise b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)).
G.f. is a period 1 Fourier series which satisfies f(-1 / (23 t)) = 23 (t/i)^2 f(t) where q = exp(2 Pi i t).
EXAMPLE
G.f. = q - q^3 - q^4 - 2*q^6 + 2*q^7 - q^8 + 2*q^9 + 2*q^10 - 4*q^11 + ...
MATHEMATICA
a[ n_] := With[ {B = QPochhammer[ q^2] QPochhammer[ q^46]}, With[ {A = QPochhammer[ q] QPochhammer[ q^23] / B}, SeriesCoefficient[ q (2 q + A) (2 q^2 + q A + A^2) B^2, {q, 0, n}]]];
PROG
(PARI) {a(n) = my(A, B); if( n<1, 0, n--; A = x * O(x^n); B = eta(x^2 + A) * eta(x^46 + A); A = eta(x + A) * eta(x^23 + A) / B; polcoeff( (2*x + A) * (2*x^2 + x*A + A^2) * B^2, n))};
(Magma) Basis( ModularForms( Gamma0(23), 2), 83)[2];
(Sage) ModularForms( Gamma0(23), 2, prec=83).0;
CROSSREFS
Cf. A232506.
Sequence in context: A255771 A334590 A115034 * A027869 A156748 A351455
KEYWORD
sign
AUTHOR
Michael Somos, Mar 24 2015
STATUS
approved