A007653 Coefficients of L-series for elliptic curve "37a1": y^2 + y = x^3 - x. (Formerly M0419)

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

1,2

Comments

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

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).

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).

Example

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 + ...

Prog

(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 */
(Magma) A := Basis( CuspForms( Gamma0(37), 2), 72); A[1] - 2*A[2]; /* Michael Somos, Jan 02 2017 */
(Sage)
def a(n):
    return EllipticCurve("37a1").an(n)  # Robin Visser, Aug 02 2023

Crossrefs

Cf. A000748, A045866, A045867.

Sequence in context: A093797 A214320 A119809 * A353623 A272181 A154179

Adjacent sequences: A007650 A007651 A007652 * A007654 A007655 A007656

Keyword

sign,easy,mult

Author

N. J. A. Sloane

Extensions

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

Status

approved