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A187096
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Coefficients of L-series for elliptic curve "19a3": y^2 + y = x^3 + x^2 + x.
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2
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1, 0, -2, -2, 3, 0, -1, 0, 1, 0, 3, 4, -4, 0, -6, 4, -3, 0, 1, -6, 2, 0, 0, 0, 4, 0, 4, 2, 6, 0, -4, 0, -6, 0, -3, -2, 2, 0, 8, 0, -6, 0, -1, -6, 3, 0, -3, -8, -6, 0, 6, 8, 12, 0, 9, 0, -2, 0, -6, 12, -1, 0, -1, -8, -12, 0, -4, 6, 0, 0, 6, 0, -7, 0, -8, -2, -3, 0, 8, 12, -11, 0, 12, -4, -9, 0, -12, 0, 12, 0, 4, 0, 8, 0, 3, 0, 8, 0, 3, -8, 6
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OFFSET
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1,3
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COMMENTS
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LINKS
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FORMULA
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Expansion of q * (psi(q^4) * phi(q^38) - q^2 * psi(q) * psi(q^19) + q^9 * phi(q^2) * psi(q^76))^2 in powers of q where phi(), psi() are Ramanujan theta functions.
a(n) is multiplicative with a(19^e) = 1, a(p^e) = a(p) * a(p^(e-1)) - p * a(p^(e-2)) where a(p) = p+1 minus number of points of elliptic curve modulo p including point at infinity.
G.f. is a period 1 Fourier series which satisfies f(-1 / (19 t)) = 19 (t/i)^2 f(t) where q = exp(2 Pi i t).
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EXAMPLE
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G.f. = q - 2*q^3 - 2*q^4 + 3*q^5 - q^7 + q^9 + 3*q^11 + 4*q^12 - 4*q^13 + ...
If p = 2, then the solutions to y^2 + y = x^3 + x^2 + x modulo 2 are (0,0), (0,1) and the point at infinity. Thus a(2) = 2+1-3 = 0.
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ (-EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^19] + 2 (EllipticTheta[ 3, 0, q^76] EllipticTheta[ 2, 0, q^4] + EllipticTheta[ 3, 0, q^4] EllipticTheta[ 2, 0, q^76]))^2 / 16, {q, 0, 2 n}]; (* Michael Somos, Oct 17 2016 *)
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PROG
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(PARI) {a(n) = if( n<1, 0, ellak( ellinit([ 0, 1, 1, 1, 0], 1), n))};
(PARI) {a(n) = my(phi1, psi1); if( n<1, 0, n--; phi1 = 1 + 2 * sum( k=1, sqrtint( n), x^k^2, x * O(x^n)); psi1 = sum( k=1, ( sqrtint( 8*n + 1) + 1 ) \ 2, x^((k^2 - k)/2), x * O(x^n)); polcoeff( sqr( subst( psi1 + x * O(x^(n \ 4)), x, x^4) * subst( phi1 + x * O(x^(n \ 38)), x, x^38) - x^2 * psi1 * subst( psi1 + x * O(x^(n \ 19)), x, x^19) + x^9 * subst( phi1 + x * O(x^(n \ 2)), x, x^2) * subst( psi1 + x * O(x^(n \ 76)), x, x^76)), n))};
(Sage) CuspForms( Gamma0(19), 2, prec=100).0; # Michael Somos, May 28 2013
(Magma) Basis( CuspForms( Gamma0(19), 2), 100) [1]; /* Michael Somos, May 27 2014 */
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CROSSREFS
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KEYWORD
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sign,mult
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AUTHOR
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STATUS
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approved
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