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%I M0419 #37 Aug 02 2023 11:45:59
%S 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,
%T -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,
%U 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
%N Coefficients of L-series for elliptic curve "37a1": y^2 + y = x^3 - x.
%C G.f. is Fourier series of a weight 2 level 37 modular cusp form.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Robin Visser, <a href="/A007653/b007653.txt">Table of n, a(n) for n = 1..10000</a>
%H N. D. Elkies, <a href="http://www.math.harvard.edu/~elkies/modular.pdf">Elliptic and modular curves over finite fields and related computational issues</a>, in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 57.
%H LMFDB, <a href="http://www.lmfdb.org/EllipticCurve/Q/37/a/1">Elliptic Curve 37a1</a>.
%H Don Zagier, <a href="https://pub.math.leidenuniv.nl/~vonkjb/seminars/grosszagier/Zag1.pdf">Modular points, modular curves, modular surfaces and modular forms</a>, 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).
%F a(3^n) = A000748(n).
%F a(n) = (A045866(n) - A045867(n)) / 2.
%F 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
%F 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).
%e 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 + ...
%o (PARI) {a(n) = if( n<1, 0, ellak( ellinit([ 0, 0, -1, -1, 0]), n))}; /* _Michael Somos_, Mar 04 2011 */
%o (PARI) {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 */
%o (Magma) A := Basis( CuspForms( Gamma0(37), 2), 72); A[1] - 2*A[2]; /* _Michael Somos_, Jan 02 2017 */
%o (Sage)
%o def a(n):
%o return EllipticCurve("37a1").an(n) # _Robin Visser_, Aug 02 2023
%Y Cf. A000748, A045866, A045867.
%K sign,easy,mult
%O 1,2
%A _N. J. A. Sloane_
%E More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 22 2000
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