Search: id:A129522
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%I A129522
%S A129522 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,
%T A129522 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,
%U A129522 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
%V A129522 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,
%W A129522 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,
%X A129522 0,-175,0,-133,0,0,0,120,0,0,0,0,-16,31,0,0,0,0,0,0,0,-97,0
%N A129522 Expansion of unique weight 3 level 11 cusp form in powers of q.
%F A129522 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).
%F A129522 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 A129522 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).
%F A129522 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
%e A129522 q - 5*q^3 + 4*q^4 - q^5 + 16*q^9 - 11*q^11 - 20*q^12 + 5*q^15 + 16*q^16 
               + ...
%o A129522 (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))}
%o A129522 (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 */
%o A129522 (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 */
%Y A129522 Sequence in context: A021653 A020847 A128355 this_sequence A133842 A115637 
               A124602
%Y A129522 Adjacent sequences: A129519 A129520 A129521 this_sequence A129523 A129524 
               A129525
%K A129522 sign,mult
%O A129522 1,3
%A A129522 Michael Somos, Apr 19 2007, Jun 06 2007

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