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A129522 Expansion of unique weight 3 level 11 multiplicative cusp form in powers of q. 4

%I

%S 1,0,-5,4,-1,0,0,0,16,0,-11,-20,0,0,5,16,0,0,0,-4,0,0,35,0,-24,0,-35,

%T 0,0,0,-37,0,55,0,0,64,-25,0,0,0,0,0,0,-44,-16,0,50,-80,49,0,0,0,-70,

%U 0,11,0,0,0,107,20,0,0,0,64,0,0,35,0,-175,0,-133,0,0

%N Expansion of unique weight 3 level 11 multiplicative cusp form in powers of q.

%C This is a member of an infinite family of odd weight level 11 multiplicative modular forms. g_1 = A035179, g_3 = A129522, g_5 = A065099, g_7 = A138661.

%F Expansion of (F(q)^2 + 4*F(q^2)^2 + 8*F(q^4)^2) * F(q)^2 / F(q^2) in powers of q where F(q) := eta(q) * eta(q^11) is the g.f. of A030200.

%F a(n) is multiplicative with a(11^e) = (-11)^e, a(p^e) = (1+(-1)^e)/2*p^e if p == 2, 6, 7, 8, 10 (mod 11), a(p^e) = a(p)*a(p^(e-1)) - p^2*a(p^(e-2)) if p == 1, 3, 4, 5, 9 (mod 11) where a(p) = y^2 - 2*p and 4*p = y^2 + 11*x^2.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t).

%F G.f.: (1/2) * Sum_{u, v in Z} (u*u - 3*v*v) * x^(u*u + u*v + 3*v*v). - _Michael Somos_, Jun 14 2007

%F Convolution of A006571 and A028609. - _Michael Somos_, Aug 14 2012

%F a(4*n + 2) = 0. - _Michael Somos_, Nov 11 2015

%e G.f. = q - 5*q^3 + 4*q^4 - q^5 + 16*q^9 - 11*q^11 - 20*q^12 + 5*q^15 + 16*q^16 + ...

%t a[ n_] := Module[ {A, B}, B = QPochhammer[ q] QPochhammer[ q^11]; A = B / (q QPochhammer[ q^3] QPochhammer[ q^33]); SeriesCoefficient[ q B^3 (1 + 3 / A) Sqrt[ q (A + 1 + 3 / A)], {q, 0, n}]]; (* _Michael Somos_, Mar 26 2015 *)

%o (PARI) {a(n) = my(A, B); if( n<1, 0, n--; A = x * O(x^n); B = eta(x + A) * eta(x^11 + A); A = B /( x * eta(x^3 + A) * eta(x^33 + A)); A = B^3 * (1 + 3/A) * sqrt(x * (A + 1 + 3/A)); polcoeff(A, n))};

%o (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, (-11)^e, kronecker( -11, p)==-1, if( e%2, 0, p^e), for( x=1, sqrtint(4*p\11), if( issquare(4*p - 11*x^2, &y), break)); y = y^2 - p*2; a0=1; a1=y; for( i=2, e, x = y*a1 - p^2*a0; a0=a1; a1=x); a1)))}; /* _Michael Somos_, Jun 06 2007 */

%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); A = eta(x + A) * eta(x^11 + A); polcoeff( A^2 / subst(A + x * O(x^(n\2)), x, x^2) * (A^2 + 4*x * subst(A + x * O(x^(n\2)), x, x^2)^2 + 8 * x^3 * subst(A + x * O(x^(n\4)), x, x^4)^2), n))}; /* _Michael Somos_, Jun 06 2007 */

%o (MAGMA) A := Basis( CuspForms( Gamma1(11), 3), 73); A[1] - 5*A[3] + 4*A[4] - A[5]; /* _Michael Somos_, Mar 26 2015 */

%Y Cf. A006571, A028609, A030200, A035179, A065099, A138661.

%K sign,mult

%O 1,3

%A _Michael Somos_, Apr 19 2007, Jun 06 2007

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Last modified May 19 14:45 EDT 2019. Contains 323395 sequences. (Running on oeis4.)