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%I #25 Jan 03 2024 08:03:19
%S 1,-2,0,0,-2,4,7,0,-5,0,-11,0,-10,0,-13,0,0,4,0,0,13,-8,-16,0,7,-14,
%T -4,0,0,0,0,0,-5,10,-20,0,-19,0,0,0,-11,22,-1,0,0,0,16,0,0,20,23,0,
%U -14,0,17,0,-9,26,0,0,7,0,0,0,2,0,-17,0,0,-8,29,0,0,0,28,0,-29,0,0,0,31,-26,-14,0,0,16,0,0,0,32,1
%N Coefficient of modular form for elliptic curve "1323m1": y^2 + y = x^3 - 2 divided by q in powers of q^3.
%H Robin Visser, <a href="/A127862/b127862.txt">Table of n, a(n) for n = 0..10000</a>
%H LMFDB, <a href="http://www.lmfdb.org/EllipticCurve/Q/1323/j/1">Elliptic Curve 1323.j1 (Cremona label 1323m1)</a>.
%F a(n) = b(3*n + 1) where b(n) is multiplicative and b(3^e) = b(7^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * (-p)^(e/2) if p == 2 (mod 3), b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)) if p == 1 (mod 3) where b(p) = -Sum_{x=0..p-1} Kronecker(4*x^3 - 7, p).
%F a(4*n + 3) = a(7*n + 2) = 0.
%e G.f. = 1 - 2*x - 2*x^4 + 4*x^5 + 7*x^6 - 5*x^8 - 11*x^10 - 10*x^12 + ...
%e G.f. = q - 2*q^4 - 2*q^13 + 4*q^16 + 7*q^19 - 5*q^25 - 11*q^31 + ...
%o (PARI) {a(n) = if( n<0, 0, ellak( ellinit( [0, 0, 1, 0, -2], 1), 3*n + 1))};
%o (PARI) {a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 3*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3 || p==7, 0, a0 = 1; a1 = y = -sum( x=0, p-1, kronecker( 4*x^3 - 7, p)); for( i=2, e, x = y*a1 - p*a0; a0 = a1; a1 = x); a1)))};
%o (Magma) qExpansion( ModularForm( EllipticCurve( [0, 0, 1, 0, -2])), 274); /* _Michael Somos_, Sep 07 2018 */
%o (Sage)
%o def a(n):
%o return EllipticCurve("1323m1").an(3*n+1) # _Robin Visser_, Jan 03 2024
%K sign
%O 0,2
%A _Michael Somos_, Feb 03 2007
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