A045867 -id:A045867 - cp's OEIS frontend

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

N. J. A. Sloane

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

			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 + ...

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

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).

Cf. A000748, A045866, A045867.

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

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

{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 */

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

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).

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

A045866 Theta series of quadratic form with Gram matrix [ 2, 1, 0, 1; 1, 8, 1, -3; 0, 1, 10, 2; 1, -3, 2, 12 ].

Original entry on oeis.org

1, 2, 0, 0, 6, 2, 14, 4, 10, 14, 16, 4, 12, 6, 18, 22, 18, 14, 14, 14, 24, 24, 34, 20, 40, 18, 32, 16, 36, 24, 36, 16, 50, 48, 36, 34, 74, 0, 40, 42, 60, 16, 58, 34, 44, 40, 44, 24, 96, 30, 64, 50, 64, 36, 98, 58, 80, 54, 48, 52, 124, 36, 72, 64, 74, 60, 66, 52, 80, 60, 92, 52

Offset: 0

N. J. A. Sloane

nonn

This is the 4-dimensional Elkies_B lattice.

			1 + 2*q^2 + 6*q^8 + 2*q^10 + 14*q^12 + 4*q^14 + 10*q^16 + 14*q^18 + 16*q^20 + ...

John Cannon, Table of n, a(n) for n = 0..5000
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.

Dual lattice to that in A045867.

B(x, y, z, w)=2*x^2+8*y^2+10*z^2+12*w^2+2*x*(y+w)+2*y*(z-3*w)+4*z*w;
thetaB(n, N, bx, by, bz, bw, ix, iy, iz, iw, nn)=n=2*n; bx=floor(sqrt(n)*(1+1/sqrt(6))); bz=floor(sqrt(n/7)); bw=floor(sqrt(n/6)); by=floor(sqrt(n/3)); N=vector(n/2+2); for(ix=-bx, bx, for(iy=-by, by, for(iz=-bz, bz, for(iw=-bw, bw, nn=B(ix, iy, iz, iw); if (nn<=n, N[1+nn/2]++); )))); N;
thetaB(80)

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

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

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

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A045866 Theta series of quadratic form with Gram matrix [ 2, 1, 0, 1; 1, 8, 1, -3; 0, 1, 10, 2; 1, -3, 2, 12 ].

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