A065099 - OEIS

%I #24 Sep 08 2022 08:45:04

%S 1,0,7,16,-49,0,0,0,-32,0,121,112,0,0,-343,256,0,0,0,-784,0,0,167,0,

%T 1776,0,-791,0,0,0,-553,0,847,0,0,-512,-2113,0,0,0,0,0,0,1936,1568,0,

%U -1918,1792,2401,0,0,0,-718,0,-5929,0,0,0,4487,-5488,0,0,0,4096

%N Weight 5 level 11 cusp form with complex multiplication by Q(sqrt(11)) and trivial character.

%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. - _Michael Somos_, Jun 07 2015

%H K. Ono, <a href="https://doi.org/10.1006/jnth.1998.2354">On the Circular Summation of the Eleventh Powers of Ramanujan's Theta Function</a>, Journal of Number Theory, Volume 76, Issue 1, May 1999, Pages 62-65.

%H G. Shimura, <a href="https://projecteuclid.org/euclid.nmj/1118798376">On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields</a>, Nagoya Math. J. 43 (1971) p. 205.

%F 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

%F 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

%e G.f. = q + 7*q^3 + 16*q^4 - 49*q^5 - 32*q^9 + 121*q^11 + 112*q^12 - 343*q^15 + ...

%t 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 *)

%o (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 }

%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, 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 */

%o (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 */

%o (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 */

%o (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 */

%Y Cf. A035179, A129522, A138661.

%K sign,mult

%O 1,3

%A Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Nov 20 2001