A318375 Coefficients of modular form for elliptic curve "108a1": y^2 = x^3 + 4 divided by q in powers of q^6.

1, 5, -7, -1, -5, -4, -1, 8, 18, 0, -13, 11, 17, -13, 0, -35, 5, -7, 2, 0, -11, 20, -5, -7, 0, 23, 14, 17, 36, -25, -19, 0, -25, 17, 0, 29, -20, -28, -22, 0, -31, 7, 0, -5, 0, -1, 26, 32, -17, 0, 40, -16, -13, 0, 35, -31, 29, 55, 23, 0, -18, -31, -25, -37, 0

Offset

0,2

Links

Robin Visser, Table of n, a(n) for n = 0..10000

LMFDB, Elliptic Curve 108.a2 (Cremona label 108a1).

Formula

G.f. is a period 1 Fourier series which satisfies f(-1 / (108 t)) = 108 (t/i)^2 f(t) where q = exp(2 Pi i t).

a(n) = b(6*n + 1) where b() is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)) if p>5, where b(p) = p minus number of points of elliptic curve modulo p.

Example

G.f. = 1 + 5*x - 7*x^2 - x^3 - 5*x^4 - 4*x^5 - x^6 + 8*x^7 + ...
G.f. = q + 5*q^7 - 7*q^13 - q^19 - 5*q^25 - 4*q^31 - q^37 + ...

Mathematica

a[ n_] := Module[ {x, y, p, e}, If[ n < 1, Boole[n == 0], Times @@ ( If[ # < 5, 0, {p, e} = {##}; p^(e/2) ChebyshevU[ e, -Sum[ KroneckerSymbol[ k^3 + 4, p], {k, p}] *p^(-1/2) / 2 ]] & @@@ FactorInteger@(6 n + 1) )]];

Prog

(PARI) {a(n) = if( n<0, 0, n = 6*n + 1; ellan(ellinit([0, 0, 0, 0, 4]), n)[n])};
(PARI) {a(n) = my(A, p, e, y='y); if( n<0, 0, A = factor(6*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, 0, substpol( polchebyshev(e, 2, -1/2/y * sum(k=1, p, kronecker(k^3 + 4, p))) * y^e, y^2, p))))};
(Magma) qExpansion( ModularForm( EllipticCurve( [0, 0, 0, 0, 4])), 386);
(Magma) A := Basis( CuspForms( Gamma0(108), 2), 386); A[1] + 5*A[6] - 7*A[9] - A[10];
(Sage)
def a(n):
    return EllipticCurve("108a1").an(6*n+1)  # Robin Visser, Jan 03 2024

Crossrefs

Sequence in context: A021179 A153613 A356706 * A023571 A160631 A155066

Adjacent sequences: A318372 A318373 A318374 * A318376 A318377 A318378

Keyword

sign

Author

Michael Somos, Aug 24 2018

Status

approved