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A006571
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Expansion of q*Product_{k>=1} (1-q^k)^2*(1-q^(11*k))^2.
(Formerly M0092)
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20
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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
For some elliptic curves with p-defects given by this sequence, and for more references, see A272196. See also the Michael Somos formula from May 23 2008 below. - Wolfdieter Lang, Apr 25 2016
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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.
M. du Sautoy, Review of "Love and Math: The Heart of Hidden Reality" by Edward Frenkel, Nature, 502 (Oct 03 2013), p. 36.
N. D. Elkies, Elliptic and modular curves..., in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 42.
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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Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
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FORMULA
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Expansion of (eta(q) * eta(q^11))^2 in powers of q.
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(q) satisfies 0 = f(A(q), A(q^2), A(q^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).
Coefficients of L-series for elliptic curve "11a3": y^2 + y = x^3 - x^2. - Michael Somos, May 23 2008
a(prime(n)) = prime(n) - A272196(n), n >= 3.
a(2) = -2 is not 2 - A272196(1) = 0. Modularity pattern of some elliptic curves. - Wolfdieter Lang, Apr 25 2016
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EXAMPLE
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G.f.: 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] QPochhammer[ q^11])^2, {q, 0, n}]; (* Michael Somos, Aug 11 2011 *)
a[ n_] := SeriesCoefficient[ q (Product[ (1 - q^k), {k, 11, n, 11}] Product[ 1 - q^k, {k, n}])^2, {q, 0, n}]; (* Michael Somos, May 27 2014 *)
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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) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^11 + A))^2, n))};
(PARI) {a(n) = my(A, p, e, x, y, a0, a1); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; 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 = 101).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 */
(Magma) Basis( CuspForms( Gamma1(11), 2), 101) [1]; /* Michael Somos, Jul 14 2014 */
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CROSSREFS
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KEYWORD
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sign,easy,nice,mult
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AUTHOR
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STATUS
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approved
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