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A006571
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Expansion of (eta(q) * eta(q^11))^2 in powers of q.
(Formerly M0092)
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12
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1, -2, -1, 2, 1, 2, -2, 0, -2, -2, 1, -2, 4, 4, -1, -4, -2, 4, 0, 2, 2, -2, -1, 0, -4, -8, 5, -4, 0, 2, 7, 8, -1, 4, -2, -4, 3, 0, -4, 0, -8, -4, -6, 2, -2, 2, 8, 4, -3, 8, 2, 8, -6, -10, 1, 0, 0, 0, 5, -2, 12, -14, 4, -8, 4, 2, -7, -4, 1, 4, -3, 0, 4, -6, 4, 0, -2, 8, -10, -4, 1, 16, -6, 4, -2, 12, 0, 0, 15, 4, -8, -2, -7, -16, 0, -8, -7, 6, -2, -8
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OFFSET
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1,2
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COMMENTS
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Number 23 of the 74 eta-quotients listed in Table I of Martin 1996.
Unique cusp form of weight 2 for congruence group Gamma_1(11). - Michael Somos, Aug 11 2011
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REFERENCES
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Barry Cipra, What's Happening in the Mathematical Sciences, Vol. 5, Amer. Math. Soc., 2002; see p. 5.
H. Darmon, A proof of the full Shimura-Taniyama-Weil conjecture is announced, Notices Amer. Math. Soc., Dec. 1999, pp. 1397-1401.
F. Diamond, Congruences between modular forms: raising the level and dropping Euler factors, in Elliptic curves and modular forms (Washington, DC, 1996). Proc. Nat. Acad. Sci. U.S.A. 94 (1997), 11143-11146.
N. D. Elkies, Elliptic and modular curves..., in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 42.
Shimura, Goro; A reciprocity law in non-solvable extensions. J. Reine Angew. Math. 221 1966 209-220.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, p. 412.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. Wiles, Modular forms, elliptic curves and Fermat's last theorem, pp. 243-245 of Proc. Intern. Congr. Math. (Zurich), Vol. 1, 1994.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1002
J. Cowles, Some congruence properties of three well-known sequences: Two notes, J. Num. Theory 12(1) (1980) 84.
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FORMULA
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a(n) = A000594(n) (mod 11). [Cowles]. - R. J. Mathar, Feb 13 2007
Euler transform of period 11 sequence [ -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -4, ...] . - Michael Somos, Feb 12 2006
a(n) is multiplicative with a(11^e) = 1, a(p^e) = a(p) * a(p^(e-1)) - p * a(p^(e-2)) for p != 11. - Michael Somos, Feb 12 2006
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u*w*(u + 4*v + 4*w) - v^3 . - Michael Somos, Mar 21 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11 (t/i)^2 f(t) where q = exp(2 pi i t).
Convolution square of A030200.
Coefficients of L-series for elliptic curve "11a3": y^2 + y = x^3 - x^2. - Michael Somos, May 23 2008
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EXAMPLE
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q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 - 2*q^9 - 2*q^10 + q^11 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ q (QPochhammer[ q, q] QPochhammer[ q^11, q^11])^2, {q, 0, n}] (* Michael Somos, Aug 11 2011 *)
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PROG
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(PARI) {a(n) = if( n<1, 0, ellak( ellinit( [0, -1, 1, 0, 0], 1), n))}
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^11 + A))^2, n))}
(PARI) {a(n) = local(A, p, e, x, y, a0, a1); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==11, 1, a0=1; a1 = y = -sum(x=0, p-1, kronecker( 4*x^3-4*x^2+1, p)); for( i=2, e, x = y*a1 - p*a0; a0=a1; a1=x); a1))))} /* Michael Somos, Aug 13 2006 */
(SAGE) CuspForms( Gamma1(11), 2, prec = 100). 0 # Michael Somos, Aug 11 2011
(MAGMA) [ Coefficient(qEigenform(EllipticCurve([0, -1, 1, 0, 0]), n+1), n) : n in [1..100] ]; /* Klaus Brockhaus, Jan 29 2007 */
(MAGMA) [ Coefficient(Basis(ModularForms(Gamma0(11), 2))[2], n) : n in [1..100] ]; /* Klaus Brockhaus, Jan 31 2007 */
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CROSSREFS
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Cf. A002070 (terms with prime indices), A030200.
Sequence in context: A002107 A208845 A133099 * A100889 A206828 A094781
Adjacent sequences: A006568 A006569 A006570 * A006572 A006573 A006574
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KEYWORD
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sign,easy,nice,mult
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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