A332332 Coefficients of L-series for elliptic curve "33a1": y^2 + x*y = x^3 + x^2 - 11*x.

1, 1, -1, -1, -2, -1, 4, -3, 1, -2, 1, 1, -2, 4, 2, -1, -2, 1, 0, 2, -4, 1, 8, 3, -1, -2, -1, -4, -6, 2, -8, 5, -1, -2, -8, -1, 6, 0, 2, 6, -2, -4, 0, -1, -2, 8, 8, 1, 9, -1, 2, 2, 6, -1, -2, -12, 0, -6, -4, -2, 6, -8, 4, 7, 4, -1, -4, 2, -8, -8, 0, -3, -14, 6

Offset: 1

Author

Michael Somos, Feb 23 2020

Keywords

• sign
• mult

Examples

			G.f. = x + x^2 - x^3 - x^4 - 2*x^5 - x^6 + 4*x^7 - 3*x^8 + x^9 + ...
		

Links

• Robin Visser, Table of n, a(n) for n = 1..10000
• LMFDB, Elliptic Curve with LMFDB label 33.a2 (Cremona label 33a1)

Crossrefs

Cf. A107920, A168175.

Programs

Magma

A := Basis( ModularForms( Gamma0(33), 2), 75); A[2] + A[3] - A[4] - A[5] - A[6];

PARI

{a(n) = if( n<1, 0, ellak( ellinit( [1, 1, 0, -11, 0], 1), n))};

PARI

{a(n) = my(A, t1, t3); if( n<1, 0, n--; A = x * O(x^n); t1 = eta(x + A) * eta(x^11 + A); t3 = x * eta(x^3 + A) * eta(x^33 + A); polcoeff( t1^2 + 3*t1*t3 + 3*t3^2, n))};

Sage

def a(n):
    return EllipticCurve("33a1").an(n)  # Robin Visser, Sep 30 2023

Formula

Expansion of eta(q)^2*eta(q^11)^2 + 3*eta(q)*eta(q^3)*eta(q^11)*eta(q^33) + 3*eta(q^3)^2*eta(q^33)^2 in powers of q.

G.f. is a period 1 Fourier series which satisfies f(-1 / (33 t)) = 33 (t/i)^2 f(t) where q = exp(2 Pi i t).

a(n) is multiplicative with a(3^n) = (-1)^n, a(11^n) = 1. a(2^n) = A107920(n+1). a(7^n) = A168175(n).