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Search: id:A129522
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| A129522 |
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Expansion of unique weight 3 level 11 cusp form in powers of q. |
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2
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| 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, 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, 0, 11, 0, 0, 0, 107, 20, 0, 0, 0, 64, 0, 0, 35, 0, -175, 0, -133, 0, 0, 0, 120, 0, 0, 0, 0, -16, 31, 0, 0, 0, 0, 0, 0, 0, -97, 0
(list; graph; listen)
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OFFSET
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1,3
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FORMULA
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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).
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.
G.f. is Fourier series of a weight 3 level 11 modular form. f(-1/ (11 t)) = sqrt(11)^3 (t/i)^3 f(t) where q = exp(2 pi i t).
G.f.: (1/2)* Sum_{u,v} (u*u -3*v*v)* x^(u*u +u*v +3*v*v). - Michael Somos Jun 14 2007
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EXAMPLE
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q - 5*q^3 + 4*q^4 - q^5 + 16*q^9 - 11*q^11 - 20*q^12 + 5*q^15 + 16*q^16 + ...
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PROGRAM
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(PARI) {a(n)= local(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))}
(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, (-11)^e, if(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 */
(PARI) {a(n)= local(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 */
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CROSSREFS
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Adjacent sequences: A129519 A129520 A129521 this_sequence A129523 A129524 A129525
Sequence in context: A021653 A020847 A128355 this_sequence A133842 A115637 A124602
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KEYWORD
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sign,mult
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AUTHOR
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Michael Somos, Apr 19 2007, Jun 06 2007
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