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Search: id:A065099
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| A065099 |
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Weight 5 level 11 cusp form with complex multiplication by Q(sqrt(11)) and trivial character. |
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+0
2
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| 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, 0, 0, -7753, 0, 1169, 0
(list; graph; listen)
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OFFSET
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1,3
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REFERENCES
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K. Ono, On the Circular Summation of the Eleventh Powers of Ramanujan's Theta Function, J. Number Theory 76, 62-65 (1999).
G. Shimura, On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields, Nagoya Math. J. 43 (1971) p. 205
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LINKS
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W. Stein,Modular Forms Database.
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FORMULA
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Newform number 1 of degree 1 in Full modular forms space of level 11, weight 5, and character [5]. [MFD]
G.f. is Fourier series of a weight 5 level 11 modular form. f(-1/ (11 t)) = sqrt(11)^5 (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
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EXAMPLE
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q + 7*q^3 + 16*q^4 - 49*q^5 - 32*q^9 + 121*q^11 + 112*q^12 - 343*q^15 + ...
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PROGRAM
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(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)= 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, 121^e, if(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)= local(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 */
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CROSSREFS
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Adjacent sequences: A065096 A065097 A065098 this_sequence A065100 A065101 A065102
Sequence in context: A055553 A066009 A037241 this_sequence A001345 A056613 A029498
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KEYWORD
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sign,mult
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AUTHOR
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Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Nov 20 2001
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