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 A007653 Coefficients of L-series for elliptic curve "37a1": y^2 + y = x^3 - x. (Formerly M0419) 2
 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, -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, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS G.f. is Fourier series of a weight 2 level 37 modular cusp form. REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS N. D. Elkies, Elliptic and modular curves over finite fields and related computational issues, in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 57. LMFDB, Elliptic Curve 37a1. FORMULA A000748(n) = a(3^n). a(n) = (A045866(n) - A045867(n)) / 2. 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 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). EXAMPLE 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 + ... PROG (PARI) {a(n) = if( n<1, 0, ellak( ellinit([ 0, 0, -1, -1, 0]), n))}; /* Michael Somos, Mar 04 2011 */ (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 */ (MAGMA) A := Basis( CuspForms( Gamma0(37), 2), 72); A[1] - 2*A[2]; /* Michael Somos, Jan 02 2017 */ CROSSREFS Cf. A000748, A045866, A045867. Sequence in context: A093797 A214320 A119809 * A272181 A154179 A300862 Adjacent sequences:  A007650 A007651 A007652 * A007654 A007655 A007656 KEYWORD sign,easy,mult AUTHOR EXTENSIONS More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 22 2000 STATUS approved

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Last modified April 12 22:36 EDT 2021. Contains 342933 sequences. (Running on oeis4.)