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A187846
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Coefficients of L-series for elliptic curve "21a4": y^2 + x * y = x^3 + x.
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1
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1, -1, 1, -1, -2, -1, -1, 3, 1, 2, 4, -1, -2, 1, -2, -1, -6, -1, 4, 2, -1, -4, 0, 3, -1, 2, 1, 1, -2, 2, 0, -5, 4, 6, 2, -1, 6, -4, -2, -6, 2, 1, -4, -4, -2, 0, 0, -1, 1, 1, -6, 2, 6, -1, -8, -3, 4, 2, 12, 2, -2, 0, -1, 7, 4, -4, 4, 6, 0, -2, 0, 3, -6, -6, -1, -4, -4, 2, -16, 2, 1, -2, -12, 1, 12, 4, -2, 12, -14, 2, 2, 0, 0, 0, -8, -5, 18, -1, 4, 1, 14, 6
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OFFSET
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1,5
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LINKS
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FORMULA
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Expansion of eta(q^7) * ( 3 * eta(q)^2 * eta(q^7)^2 * eta(q^9)^4 - eta(q^3)^5 * eta(q^7) * eta(q^9) * eta(q^21) + 7 * eta(q) * eta(q^3)^2 * eta(q^9) * eta(q^21)^4 + 3 * eta(q)^3 * eta(q^7) * eta(q^9)^3 * eta(q^63) - 3 * eta(q) * eta(q^3)^5 * eta(q^21) * eta(q^63) + 3 * eta(q)^4 * eta(q^9)^2 * eta(q^63)^2 ) / ( 2 * eta(q)^2 * eta(q^3) * eta(q^9) * eta(q^21) ) in powers of q.
a(n) is multiplicative with a(3^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 / (21 t)) = 21 (t/i)^2 f(t) where q = exp(2 Pi i t).
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EXAMPLE
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G.f. = q - q^2 + q^3 - q^4 - 2*q^5 - q^6 - q^7 + 3*q^8 + q^9 + 2*q^10 + ...
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PROG
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(PARI) {a(n) = if( n<1, 0, ellak( ellinit( [1, 0, 0, 1, 0], 1), n))}
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^7 + A) * ( + 3 * eta(x + A)^2 * eta(x^7 + A)^2 * eta(x^9 + A)^4 - 1 * eta(x^3 + A)^5 * eta(x^7 + A) * eta(x^9 + A) * eta(x^21 + A) + 7 * x^2 * eta(x + A) * eta(x^3 + A)^2 * eta(x^9 + A) * eta(x^21 + A)^4 + 3 * x^2 * eta(x + A)^3 * eta(x^7 + A) * eta(x^9 + A)^3 * eta(x^63 + A) - 3 * x^2 * eta(x + A) * eta(x^3 + A)^5 * eta(x^21 + A) * eta(x^63 + A) + 3 * x^4 * eta(x + A)^4 * eta(x^9 + A)^2 * eta(x^63 + A)^2 ) / ( 2 * eta(x + A)^2 * eta(x^3 + A) * eta(x^9 + A) * eta(x^21 + A) ), n))}
(Sage) CuspForms( Gamma0(21), 2, prec=100).0 # Michael Somos, May 28 2013
(Magma) Basis( CuspForms( Gamma0( 21), 2), 100) [1] /* Michael Somos, Dec 10 2013 */
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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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