A007653 Coefficients of L-series for elliptic curve "37a1": y^2 + y = x^3 - x.

1, -2, -3, 2, -2, 6, -1, 0, 6, 4, -5, -6, -2, 2, 6, -4, 0, -12, 0, -4, 3, 10, 2, 0, -1, 4, -9, -2, 6, -12, -4, 8, 15, 0, 2, 12, -1, 0, 6, 0, -9, -6, 2, -10, -12, -4, -9, 12, -6, 2, 0, -4, 1, 18, 10, 0, 0, -12, 8, 12, -8, 8, -6, -8, 4, -30, 8, 0, -6, -4, 9, 0, -1, 2, 3, 0, 5, -12, 4, 8, 9, 18, -15, 6, 0, -4, -18, 0, 4, 24, 2, 4, 12, 18, 0

Offset: 1

Author

N. J. A. Sloane

Keywords

• sign
• easy
• mult

Comments

G.f. is Fourier series of a weight 2 level 37 modular cusp form.

Examples

			G.f. = q - 2*q^2 - 3*q^3 + 2*q^4 - 2*q^5 + 6*q^6 - q^7 + 6*q^9 + 4*q^10 - 5*q^11 + ...
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Robin Visser, Table of n, a(n) for n = 1..10000
• N. D. Elkies, Elliptic and modular curves over finite fields and related computational issues, in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 57.
• LMFDB, Elliptic Curve 37a1.
• Don Zagier, Modular points, modular curves, modular surfaces and modular forms, Arbeitstagung Bonn 1984: Proceedings of the meeting held by the Max-Planck-Institut für Mathematik, Bonn June 15-22, 1984. Springer Berlin Heidelberg, 1985. See Eq. (8).

Crossrefs

Cf. A000748, A045866, A045867.

Programs

Magma

A := Basis( CuspForms( Gamma0(37), 2), 72); A[1] - 2*A[2]; /* Michael Somos, Jan 02 2017 */

PARI

{a(n) = if( n<1, 0, ellak( ellinit([ 0, 0, -1, -1, 0]), n))}; /* Michael Somos, Mar 04 2011 */

PARI

{a(n) = if( n<1, 0, qfrep([ 2, 1, 0, 1; 1, 8, 1, -3; 0, 1, 10, 2; 1, -3, 2, 12 ], n, 1)[n] - qfrep([ 4, 1, 2, 1; 1, 4, 1, 0; 2, 1, 6, -2; 1, 0, -2, 20 ], n, 1)[n])}; /* Michael Somos, Apr 02 2006 */

Sage

def a(n):
    return EllipticCurve("37a1").an(n)  # Robin Visser, Aug 02 2023

Formula

a(3^n) = A000748(n).

a(n) = (A045866(n) - A045867(n)) / 2.

a(n) is multiplicative with a(p^e) = a(p) * a(p^(e-1)) - p * a(p^(e-2)) where a(p) = p+1 - number of solutions of y^2 + y = x^3 - x modulo p including the point at infinity. - Michael Somos, Mar 03 2011

G.f. is a period 1 Fourier series which satisfies f(-1 / (37 t)) = -37 (t/i)^2 f(t) where q = exp(2 Pi i t).

Extensions

More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 22 2000