OFFSET
1,3
COMMENTS
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. - Michael Somos, Jun 07 2015
LINKS
K. Ono, On the Circular Summation of the Eleventh Powers of Ramanujan's Theta Function, Journal of Number Theory, Volume 76, Issue 1, May 1999, Pages 62-65.
G. Shimura, On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields, Nagoya Math. J. 43 (1971) p. 205.
FORMULA
G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11^(5/2) (t/i)^5 f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 08 2007
a(n) is multiplicative with a(11^e) = 121^e, a(p^e) = (1 + (-1)^e) / 2 * p^(2*e) if p == 2, 6, 7, 8, 10 (mod 11), a(p^e) = a(p) * a(p^(e-1)) - p^4 * a(p^(e-2)) if p == 1, 3, 4, 5, 9 (mod 11) where a(p) = y^4 - 4 * p*y^2 + 2 * p^2 and 4*p = y^2 + 11 * x^2. - Michael Somos, Jun 08 2007
EXAMPLE
G.f. = q + 7*q^3 + 16*q^4 - 49*q^5 - 32*q^9 + 121*q^11 + 112*q^12 - 343*q^15 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ With[{F1 = (QPochhammer[ q] QPochhammer[ q^11])^2, F2 = (QPochhammer[ q^2] QPochhammer[ q^22])^2, F3 = (QPochhammer[ q^2] QPochhammer[ q^22])^3, F4 = (QPochhammer[ q^4] QPochhammer[ q^44])^2}, (F1^4 + 8 q F1^3 F2 + 32 q^2 F1^2 F2^2 + 88 q^3 F1 F2^3 + 64 q^4 F2^4 + 96 q^6 F4 F2^3 + 128 q^5 F1 F4 (F2^2 + q^2 F2 F4 + q^4 F4^2)) / F3], {q, 0, n}]; (* Michael Somos, Jun 07 2015 *)
PROG
(PARI) { B(N, a, x, y, x2, y2)= a=vector(N); for (x=0, floor(sqrt(4*N)), for (y=0, floor(sqrt(4*N/11)), x2=x*x; y2=y*y; n=(x2+11*y2); if (n%4==0 && n<=4*N && n>0, w=(2*x2*x2-132*x2*y2+242*y2*y2)/32; a[n/4]+=w; if (x*y !=0, a[n/4]+=w)))); a }
(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, 121^e, kronecker( -11, p)==-1, if( e%2, 0, p^(2*e)), for( x=1, sqrtint(4*p\11), if( issquare(4*p - 11*x^2, &y), break)); y = y^4 - 4 * p*y^2 + 2 * p^2; a0=1; a1=y; for( i=2, e, x=y*a1 - p^4*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Jun 08 2007 */
(PARI) {a(n) = my(A, F1, F2, F4); if( n<1, 0, n--; A = x * O(x^n); F1 = (eta(x + A) * eta(x^11 + A))^2; F2 = (eta(x^2 + A) * eta(x^22 + A))^2; F4 = (eta(x^4 + A) * eta(x^44 + A))^2; polcoeff( (F1^4 + 8 * x * F1^3*F2 + 32 * x^2 * F1^2*F2^2 + 88 * x^3 * F1*F2^3 + 64 * x^4 * F2^4 + 96 * x^6 * F4*F2^3 + 128 * x^5 * F1*F4 * (F2^2 + x^2 * F2*F4 + x^4 * F4^2)) / (eta(x^2 + A) * eta(x^22 + A))^3, n))}; /* Michael Somos, Jun 08 2007 */
(PARI) {a(n) = if( n<1, 0, n*=4; sum( y=0, sqrtint(n\11), if( issquare( n - 11 * y^2), if( (n > 11*y^2) && y, 2, 1) * (n^2 - 88 * n*y^2 + 968 * y^4) / 16)))}; /* Michael Somos, Jun 08 2007 */
(Magma) A := Basis( CuspForms( Gamma1(11), 5), 71); A[1] + 7*A[3] + 16*A[4] - 49*A[5] - 32*A[9] + 121*A[11] + 112*A[12] - 343*A[15]; /* Michael Somos, Aug 26 2015 */
CROSSREFS
KEYWORD
sign,mult
AUTHOR
Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Nov 20 2001
STATUS
approved