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Search: id:A007653
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| A007653 |
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Coefficients of Dirichlet expansion of L-series for elliptic curve y^2-y=x^3-x (curve "37a1").
(Formerly M0419)
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+0
3
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| 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; listen)
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OFFSET
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1,2
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. D. Elkies, Elliptic and modular curves..., in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 57.
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LINKS
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W. Stein, Modular Forms Database.
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FORMULA
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A000748(n)=a(3^n).
a(n) is multiplicative with a(p^e)=a(p)a(p^(e-1))-p*a(p^(e-2)).
Also for p prime, a(p) = p - number of solutions of y^2-y=x^3-x mod p. - Michael Somos Jul 27 2006
G.f. is Fourier series of a weight 2 level 37 modular cusp form. f(-1/ (37 t)) = -37 (t/i)^2 f(t) where q = exp(2 pi i t).
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EXAMPLE
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q - 2*q^2 - 3*q^3 + 2*q^4 - 2*q^5 + 6*q^6 - q^7 + 6*q^9 + 4*q^10 - ...
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PROGRAM
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(PARI) a(n)=ellak(ellinit([0, 0, -1, -1, 0]), n)
(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 */
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CROSSREFS
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a(n)=(A045866(n)-A045867(n))/2.
Sequence in context: A128651 A093797 A119809 this_sequence A154179 A151863 A134142
Adjacent sequences: A007650 A007651 A007652 this_sequence A007654 A007655 A007656
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KEYWORD
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sign,easy,mult
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 22 2000
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