A115672 Coefficients of L-series for elliptic curve "35a3": y^2 + y = x^3 + x^2 - x.

1, 0, 1, -2, -1, 0, 1, 0, -2, 0, -3, -2, 5, 0, -1, 4, 3, 0, 2, 2, 1, 0, -6, 0, 1, 0, -5, -2, 3, 0, -4, 0, -3, 0, -1, 4, 2, 0, 5, 0, -12, 0, -10, 6, 2, 0, 9, 4, 1, 0, 3, -10, 12, 0, 3, 0, 2, 0, 0, 2, 8, 0, -2, -8, -5, 0, -4, -6, -6, 0, 0, 0, 2, 0, 1, -4, -3, 0, -1, -4, 1, 0, 12, -2, -3, 0, 3, 0, -12, 0, 5, 12, -4, 0, -2, 0, -1, 0, 6, -2, 6

Offset: 1

Author

Michael Somos, Jan 29 2006

Keywords

• sign
• mult

Examples

			q + q^3 - 2*q^4 - q^5 + q^7 - 2*q^9 - 3*q^11 - 2*q^12 + 5*q^13 - q^15 + ...
		

Links

• Robin Visser, Table of n, a(n) for n = 1..10000
• LMFDB, Elliptic curve with LMFDB label 35.a2 (Cremona label 35a3)
• W. Stein, Modular Forms Database.

Crossrefs

Cf. A106852(n) = a(3^n).

Programs

PARI

{a(n)=if( n<1, 0, ellak( ellinit([ 0, 1, 1, -1, 0]), n))} /* Michael Somos, Mar 03 2011 */

PARI

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^5 + A) * eta(x^7 + A))^2 + x^2 * (eta(x + A) * eta(x^35 + A))^2, n))}

Sage

def a(n):
    return EllipticCurve("35a3").an(n)  # Robin Visser, Sep 30 2023

Formula

a(n) is multiplicative with a(5^e) = (-1)^e, a(7^e) = 1, a(p^e) = a(p) * a(p^(e-1)) - p * a(p^(e-2)) otherwise.

Expansion of (eta(q^5) * eta(q^7))^2 + (eta(q) * eta(q^35))^2 in powers of q. Expansion of a newform level 35 weight 2 and trivial character.