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A065099 Weight 5 level 11 cusp form with complex multiplication by Q(sqrt(11)) and trivial character. 4
1, 0, 7, 16, -49, 0, 0, 0, -32, 0, 121, 112, 0, 0, -343, 256, 0, 0, 0, -784, 0, 0, 167, 0, 1776, 0, -791, 0, 0, 0, -553, 0, 847, 0, 0, -512, -2113, 0, 0, 0, 0, 0, 0, 1936, 1568, 0, -1918, 1792, 2401, 0, 0, 0, -718, 0, -5929, 0, 0, 0, 4487, -5488, 0, 0, 0, 4096 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..64.

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

Cf. A035179, A129522, A138661.

Sequence in context: A154141 A173661 A152530 * A001345 A225128 A056613

Adjacent sequences:  A065096 A065097 A065098 * A065100 A065101 A065102

KEYWORD

sign,mult

AUTHOR

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

STATUS

approved

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Last modified May 24 06:53 EDT 2019. Contains 323529 sequences. (Running on oeis4.)