/* This code can be loaded, or copied and paste using cpaste, into Sage. It will load the data associated to the HMF, including the field, level, and Hecke and Atkin-Lehner eigenvalue data. */ P. = PolynomialRing(QQ) g = P([6, 0, -6, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([9, 3, -w^2 + 3]) primes_array = [ [2, 2, -w^2 + w + 2],\ [3, 3, w^2 - w - 3],\ [11, 11, -w^2 + w + 1],\ [11, 11, -w^2 - w + 1],\ [13, 13, w^3 - 4*w + 1],\ [13, 13, -w^3 + 4*w + 1],\ [25, 5, -w^2 - 2*w + 1],\ [25, 5, w^2 - 2*w - 1],\ [37, 37, w^3 - 3*w - 1],\ [37, 37, w^3 - 3*w + 1],\ [59, 59, w^2 - w - 5],\ [59, 59, -w^2 - w + 5],\ [61, 61, -w^3 + w^2 + 4*w - 7],\ [61, 61, w^3 - 3*w^2 - 6*w + 11],\ [73, 73, 2*w^2 - w - 5],\ [73, 73, 2*w - 1],\ [73, 73, -2*w - 1],\ [73, 73, 2*w^2 + w - 5],\ [83, 83, 2*w^3 + w^2 - 9*w - 7],\ [83, 83, -2*w^3 + w^2 + 7*w + 1],\ [107, 107, w^3 + w^2 - 4*w - 1],\ [107, 107, -w^3 + w^2 + 4*w - 1],\ [109, 109, -w^3 + 2*w^2 + 5*w - 11],\ [109, 109, w^3 + 2*w^2 - 5*w - 11],\ [121, 11, 2*w^2 - 5],\ [131, 131, 3*w^3 - 2*w^2 - 14*w + 11],\ [131, 131, 3*w^3 - 12*w + 5],\ [157, 157, -3*w^3 + 2*w^2 + 13*w - 11],\ [157, 157, -w^3 + 2*w^2 + 7*w - 5],\ [167, 167, w^2 + 2*w - 5],\ [167, 167, -w^3 + w^2 + w + 1],\ [167, 167, 3*w^3 - w^2 - 9*w - 5],\ [167, 167, w^2 - 2*w - 5],\ [169, 13, -w^2 + 7],\ [179, 179, -w^3 + w^2 + 6*w - 1],\ [179, 179, w^3 + w^2 - 6*w - 1],\ [181, 181, -w^3 + 5*w - 5],\ [181, 181, w^3 - 5*w - 5],\ [191, 191, -2*w^3 + 2*w^2 + 6*w - 1],\ [191, 191, -2*w^2 + 2*w + 7],\ [191, 191, -4*w^2 + 4*w + 11],\ [191, 191, -2*w^3 + 8*w + 5],\ [227, 227, -w^3 + 2*w^2 + 5*w - 5],\ [227, 227, w^3 + 2*w^2 - 5*w - 5],\ [229, 229, -w^3 + 3*w + 5],\ [229, 229, -w^3 - 4*w^2 + 7*w + 13],\ [239, 239, -w^3 + w^2 + 5*w - 1],\ [239, 239, 2*w^3 - 4*w^2 - 12*w + 13],\ [239, 239, 2*w^3 - 2*w^2 - 10*w + 7],\ [239, 239, w^3 + w^2 - 5*w - 1],\ [241, 241, 2*w^3 - w^2 - 8*w + 5],\ [241, 241, -3*w^2 + 4*w + 7],\ [241, 241, -2*w^3 + w^2 + 6*w + 1],\ [241, 241, -2*w^3 - w^2 + 8*w + 5],\ [251, 251, -w^3 + 2*w - 5],\ [251, 251, 3*w^3 - 2*w^2 - 12*w + 13],\ [263, 263, 2*w^3 - 2*w^2 - 8*w + 11],\ [263, 263, -w^3 + w^2 + 3*w - 7],\ [263, 263, 3*w^3 - w^2 - 11*w + 11],\ [263, 263, -2*w^3 + 6*w - 7],\ [277, 277, -3*w^2 - w + 11],\ [277, 277, 3*w^2 - w - 11],\ [311, 311, -w^3 + 5*w^2 + 7*w - 19],\ [311, 311, 2*w^3 - 3*w^2 - 8*w + 17],\ [311, 311, -2*w^3 + 6*w - 5],\ [311, 311, -2*w^3 - 2*w^2 + 8*w + 5],\ [337, 337, -2*w^3 + 8*w + 1],\ [337, 337, -w^3 + 3*w^2 + 5*w - 11],\ [337, 337, w^3 + 3*w^2 - 5*w - 11],\ [337, 337, 2*w^3 - 8*w + 1],\ [347, 347, w^3 - 3*w^2 - 6*w + 13],\ [347, 347, -w^3 - 3*w^2 + 6*w + 13],\ [349, 349, -w^3 + 4*w - 5],\ [349, 349, w^3 - 4*w - 5],\ [361, 19, 4*w^3 - 2*w^2 - 17*w + 13],\ [361, 19, 2*w^3 - 2*w^2 - 11*w + 7],\ [373, 373, -2*w^3 - 5*w^2 + 13*w + 17],\ [373, 373, 2*w^3 - w^2 - 7*w + 1],\ [383, 383, 4*w^3 - w^2 - 16*w + 13],\ [383, 383, w^2 + 2*w - 7],\ [383, 383, w^2 - 2*w - 7],\ [383, 383, 2*w^3 + 3*w^2 - 6*w - 5],\ [397, 397, -w^3 + 2*w^2 + 7*w - 7],\ [397, 397, w^3 + 2*w^2 - 7*w - 7],\ [419, 419, 5*w^2 - 5*w - 11],\ [419, 419, 3*w^2 - 3*w - 5],\ [421, 421, w^3 + 4*w^2 - 5*w - 17],\ [421, 421, -w^3 + 4*w^2 + 5*w - 17],\ [433, 433, w^3 - w^2 - 7*w + 1],\ [433, 433, 3*w - 1],\ [433, 433, -3*w - 1],\ [433, 433, 5*w^3 - 3*w^2 - 21*w + 19],\ [443, 443, w^3 - 4*w^2 - 6*w + 17],\ [443, 443, -3*w^3 + 6*w^2 + 16*w - 25],\ [467, 467, -w^3 + w^2 + 2*w - 5],\ [467, 467, w^3 + w^2 - 2*w - 5],\ [491, 491, 3*w^3 - w^2 - 14*w + 7],\ [491, 491, -3*w^3 - w^2 + 14*w + 7],\ [529, 23, 3*w^2 - 11],\ [529, 23, 3*w^2 - 7],\ [541, 541, 3*w^3 + 2*w^2 - 10*w - 1],\ [541, 541, 3*w^3 - 4*w^2 - 16*w + 17],\ [563, 563, w^3 + 2*w^2 - 2*w - 7],\ [563, 563, -w^3 + 2*w^2 + 2*w - 7],\ [587, 587, -2*w^3 + w^2 + 5*w - 5],\ [587, 587, 2*w^3 + w^2 - 5*w - 5],\ [613, 613, 3*w^3 - 13*w - 1],\ [613, 613, 3*w^3 - 13*w + 1],\ [659, 659, w^3 + 3*w^2 - 4*w - 5],\ [659, 659, -w^3 + 3*w^2 + 4*w - 5],\ [661, 661, w^3 + 4*w^2 - 9*w - 13],\ [661, 661, -3*w^3 + 4*w^2 + 7*w - 5],\ [673, 673, -2*w^3 + 7*w + 1],\ [673, 673, w^3 + w^2 - 7*w - 5],\ [673, 673, -w^3 + w^2 + 7*w - 5],\ [673, 673, 2*w^3 - 7*w + 1],\ [683, 683, -2*w^3 - w^2 + 9*w + 1],\ [683, 683, 2*w^3 - w^2 - 9*w + 1],\ [709, 709, -3*w^3 + 2*w^2 + 8*w + 1],\ [709, 709, -w^3 + 4*w^2 - 2*w - 7],\ [733, 733, 2*w^3 - 5*w^2 - 11*w + 19],\ [733, 733, -2*w^3 + w^2 + 7*w - 11],\ [743, 743, 2*w^3 + 2*w^2 - 11*w - 5],\ [743, 743, 2*w^3 - w^2 - 8*w - 1],\ [743, 743, -2*w^3 - w^2 + 8*w - 1],\ [743, 743, -2*w^3 + 2*w^2 + 11*w - 5],\ [757, 757, 2*w^3 + 3*w^2 - 3*w + 1],\ [757, 757, -2*w^3 + 5*w^2 + 11*w - 17],\ [827, 827, -5*w^3 + 4*w^2 + 21*w - 23],\ [827, 827, -w^3 + w - 7],\ [829, 829, 2*w^3 - 3*w^2 - 9*w + 17],\ [829, 829, -2*w^3 - 3*w^2 + 9*w + 17],\ [841, 29, 2*w^2 - w - 13],\ [841, 29, 2*w^2 + w - 13],\ [853, 853, -w^3 + 6*w - 7],\ [853, 853, -2*w^3 + w^2 + 11*w - 5],\ [863, 863, w^3 - 5*w^2 - 9*w + 13],\ [863, 863, 2*w^3 + 4*w^2 - 5*w - 11],\ [863, 863, -2*w^3 + 4*w^2 + 5*w - 11],\ [863, 863, w^3 - 3*w^2 - 7*w + 7],\ [877, 877, -w^3 - 3*w^2 + 4*w + 17],\ [877, 877, w^3 - 3*w^2 - 4*w + 17],\ [887, 887, -2*w^3 - w^2 + 10*w + 1],\ [887, 887, 2*w^3 + 2*w^2 - 9*w - 5],\ [887, 887, -2*w^3 + 2*w^2 + 9*w - 5],\ [887, 887, 2*w^3 - w^2 - 10*w + 1],\ [911, 911, -2*w^3 + 11*w - 5],\ [911, 911, -w^3 + w^2 + w - 5],\ [911, 911, w^3 + w^2 - w - 5],\ [911, 911, 2*w^3 - 11*w - 5],\ [937, 937, -4*w^3 - 2*w^2 + 16*w + 17],\ [937, 937, 3*w^3 - w^2 - 11*w - 1],\ [937, 937, -3*w^3 - 5*w^2 + 17*w + 19],\ [937, 937, -4*w^2 + 2*w + 13],\ [947, 947, -3*w^3 + 2*w^2 + 10*w + 1],\ [947, 947, 3*w^3 - 12*w - 7],\ [971, 971, w^3 + w^2 - 2*w - 7],\ [971, 971, -w^3 + w^2 + 2*w - 7],\ [983, 983, -w^3 - 3*w^2 + 7*w + 13],\ [983, 983, 2*w^3 - 2*w^2 - 7*w + 1],\ [983, 983, -2*w^3 - 2*w^2 + 7*w + 1],\ [983, 983, w^3 - 7*w^2 + 3*w + 17],\ [997, 997, -w^3 + 2*w^2 + 4*w - 13],\ [997, 997, 3*w^3 - 8*w^2 - 18*w + 29]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^8 - 16*x^6 + 82*x^4 - 152*x^2 + 72 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, 0, -1/4*e^7 + 7/2*e^5 - 27/2*e^3 + 13*e, -1/4*e^7 + 7/2*e^5 - 27/2*e^3 + 13*e, 1/4*e^6 - 7/2*e^4 + 25/2*e^2 - 7, 1/4*e^6 - 7/2*e^4 + 25/2*e^2 - 7, -3/4*e^6 + 19/2*e^4 - 63/2*e^2 + 23, -3/4*e^6 + 19/2*e^4 - 63/2*e^2 + 23, -2*e^2 + 8, -2*e^2 + 8, 1/2*e^7 - 6*e^5 + 17*e^3 - 4*e, 1/2*e^7 - 6*e^5 + 17*e^3 - 4*e, -1/2*e^6 + 5*e^4 - 9*e^2 - 4, -1/2*e^6 + 5*e^4 - 9*e^2 - 4, 1/4*e^6 - 5/2*e^4 + 5/2*e^2 + 11, 3/4*e^6 - 19/2*e^4 + 63/2*e^2 - 19, 3/4*e^6 - 19/2*e^4 + 63/2*e^2 - 19, 1/4*e^6 - 5/2*e^4 + 5/2*e^2 + 11, -1/4*e^7 + 5/2*e^5 - 3/2*e^3 - 17*e, -1/4*e^7 + 5/2*e^5 - 3/2*e^3 - 17*e, e^5 - 10*e^3 + 18*e, e^5 - 10*e^3 + 18*e, -1/4*e^6 + 3/2*e^4 + 15/2*e^2 - 25, -1/4*e^6 + 3/2*e^4 + 15/2*e^2 - 25, -e^6 + 12*e^4 - 38*e^2 + 38, e^7 - 13*e^5 + 44*e^3 - 34*e, e^7 - 13*e^5 + 44*e^3 - 34*e, e^6 - 12*e^4 + 36*e^2 - 16, e^6 - 12*e^4 + 36*e^2 - 16, -e^5 + 12*e^3 - 34*e, 1/2*e^7 - 8*e^5 + 37*e^3 - 40*e, 1/2*e^7 - 8*e^5 + 37*e^3 - 40*e, -e^5 + 12*e^3 - 34*e, -e^6 + 12*e^4 - 30*e^2 + 8, -4*e^3 + 24*e, -4*e^3 + 24*e, 5/4*e^6 - 31/2*e^4 + 93/2*e^2 - 19, 5/4*e^6 - 31/2*e^4 + 93/2*e^2 - 19, -e^5 + 10*e^3 - 26*e, -1/2*e^7 + 8*e^5 - 39*e^3 + 52*e, -1/2*e^7 + 8*e^5 - 39*e^3 + 52*e, -e^5 + 10*e^3 - 26*e, 1/4*e^7 - 5/2*e^5 + 3/2*e^3 + 25*e, 1/4*e^7 - 5/2*e^5 + 3/2*e^3 + 25*e, -9/4*e^6 + 59/2*e^4 - 209/2*e^2 + 83, -9/4*e^6 + 59/2*e^4 - 209/2*e^2 + 83, e^5 - 12*e^3 + 26*e, 1/2*e^7 - 8*e^5 + 37*e^3 - 40*e, 1/2*e^7 - 8*e^5 + 37*e^3 - 40*e, e^5 - 12*e^3 + 26*e, -1/4*e^6 + 5/2*e^4 + 3/2*e^2 - 31, 1/4*e^6 - 13/2*e^4 + 77/2*e^2 - 49, 1/4*e^6 - 13/2*e^4 + 77/2*e^2 - 49, -1/4*e^6 + 5/2*e^4 + 3/2*e^2 - 31, -3/4*e^7 + 19/2*e^5 - 65/2*e^3 + 33*e, -3/4*e^7 + 19/2*e^5 - 65/2*e^3 + 33*e, -1/2*e^7 + 6*e^5 - 19*e^3 + 16*e, 3/2*e^7 - 20*e^5 + 75*e^3 - 80*e, 3/2*e^7 - 20*e^5 + 75*e^3 - 80*e, -1/2*e^7 + 6*e^5 - 19*e^3 + 16*e, 3/4*e^6 - 17/2*e^4 + 43/2*e^2 - 13, 3/4*e^6 - 17/2*e^4 + 43/2*e^2 - 13, -3/2*e^7 + 19*e^5 - 61*e^3 + 38*e, 3/2*e^7 - 19*e^5 + 65*e^3 - 58*e, 3/2*e^7 - 19*e^5 + 65*e^3 - 58*e, -3/2*e^7 + 19*e^5 - 61*e^3 + 38*e, 3/4*e^6 - 19/2*e^4 + 47/2*e^2 + 5, 7/4*e^6 - 43/2*e^4 + 131/2*e^2 - 43, 7/4*e^6 - 43/2*e^4 + 131/2*e^2 - 43, 3/4*e^6 - 19/2*e^4 + 47/2*e^2 + 5, 1/4*e^7 - 11/2*e^5 + 67/2*e^3 - 49*e, 1/4*e^7 - 11/2*e^5 + 67/2*e^3 - 49*e, -3/2*e^6 + 21*e^4 - 79*e^2 + 68, -3/2*e^6 + 21*e^4 - 79*e^2 + 68, -2*e^6 + 28*e^4 - 104*e^2 + 80, -2*e^6 + 28*e^4 - 104*e^2 + 80, -1/2*e^6 + 5*e^4 - 5*e^2 - 16, -1/2*e^6 + 5*e^4 - 5*e^2 - 16, 1/2*e^7 - 7*e^5 + 31*e^3 - 46*e, -e^7 + 12*e^5 - 34*e^3 + 8*e, -e^7 + 12*e^5 - 34*e^3 + 8*e, 1/2*e^7 - 7*e^5 + 31*e^3 - 46*e, 3/2*e^6 - 17*e^4 + 47*e^2 - 28, 3/2*e^6 - 17*e^4 + 47*e^2 - 28, -3/4*e^7 + 21/2*e^5 - 89/2*e^3 + 71*e, -3/4*e^7 + 21/2*e^5 - 89/2*e^3 + 71*e, -9/4*e^6 + 55/2*e^4 - 169/2*e^2 + 35, -9/4*e^6 + 55/2*e^4 - 169/2*e^2 + 35, -e^6 + 12*e^4 - 34*e^2 + 8, 2*e^4 - 24*e^2 + 44, 2*e^4 - 24*e^2 + 44, -e^6 + 12*e^4 - 34*e^2 + 8, 7/4*e^7 - 43/2*e^5 + 133/2*e^3 - 41*e, 7/4*e^7 - 43/2*e^5 + 133/2*e^3 - 41*e, 5/4*e^7 - 29/2*e^5 + 79/2*e^3 - 15*e, 5/4*e^7 - 29/2*e^5 + 79/2*e^3 - 15*e, 2*e^7 - 27*e^5 + 102*e^3 - 94*e, 2*e^7 - 27*e^5 + 102*e^3 - 94*e, e^6 - 16*e^4 + 74*e^2 - 58, -e^6 + 16*e^4 - 70*e^2 + 86, 1/4*e^6 + 1/2*e^4 - 39/2*e^2 + 29, 1/4*e^6 + 1/2*e^4 - 39/2*e^2 + 29, 1/4*e^7 - 11/2*e^5 + 75/2*e^3 - 73*e, 1/4*e^7 - 11/2*e^5 + 75/2*e^3 - 73*e, 1/2*e^7 - 7*e^5 + 27*e^3 - 38*e, 1/2*e^7 - 7*e^5 + 27*e^3 - 38*e, -1/2*e^6 + 7*e^4 - 17*e^2 - 16, -1/2*e^6 + 7*e^4 - 17*e^2 - 16, 1/2*e^7 - 7*e^5 + 27*e^3 - 38*e, 1/2*e^7 - 7*e^5 + 27*e^3 - 38*e, 2*e^4 - 10*e^2 - 4, 2*e^4 - 10*e^2 - 4, e^6 - 14*e^4 + 50*e^2 - 22, -3*e^6 + 38*e^4 - 126*e^2 + 86, -3*e^6 + 38*e^4 - 126*e^2 + 86, e^6 - 14*e^4 + 50*e^2 - 22, 1/4*e^7 - 7/2*e^5 + 35/2*e^3 - 45*e, 1/4*e^7 - 7/2*e^5 + 35/2*e^3 - 45*e, 1/4*e^6 - 7/2*e^4 + 41/2*e^2 - 55, 1/4*e^6 - 7/2*e^4 + 41/2*e^2 - 55, 7/4*e^6 - 41/2*e^4 + 119/2*e^2 - 25, 7/4*e^6 - 41/2*e^4 + 119/2*e^2 - 25, -e^7 + 12*e^5 - 34*e^3 + 8*e, -3/2*e^7 + 19*e^5 - 65*e^3 + 62*e, -3/2*e^7 + 19*e^5 - 65*e^3 + 62*e, -e^7 + 12*e^5 - 34*e^3 + 8*e, 5/4*e^6 - 31/2*e^4 + 101/2*e^2 - 55, 5/4*e^6 - 31/2*e^4 + 101/2*e^2 - 55, 1/2*e^7 - 9*e^5 + 51*e^3 - 90*e, 1/2*e^7 - 9*e^5 + 51*e^3 - 90*e, 7/4*e^6 - 45/2*e^4 + 135/2*e^2 - 1, 7/4*e^6 - 45/2*e^4 + 135/2*e^2 - 1, -9/4*e^6 + 53/2*e^4 - 157/2*e^2 + 65, -9/4*e^6 + 53/2*e^4 - 157/2*e^2 + 65, -3*e^6 + 40*e^4 - 144*e^2 + 104, -3*e^6 + 40*e^4 - 144*e^2 + 104, 2*e^7 - 25*e^5 + 80*e^3 - 42*e, 1/2*e^7 - 8*e^5 + 37*e^3 - 48*e, 1/2*e^7 - 8*e^5 + 37*e^3 - 48*e, 2*e^7 - 25*e^5 + 80*e^3 - 42*e, 5/2*e^6 - 31*e^4 + 101*e^2 - 88, 5/2*e^6 - 31*e^4 + 101*e^2 - 88, -e^7 + 11*e^5 - 20*e^3 - 34*e, -e^7 + 13*e^5 - 38*e^3 + 2*e, -e^7 + 13*e^5 - 38*e^3 + 2*e, -e^7 + 11*e^5 - 20*e^3 - 34*e, e^7 - 14*e^5 + 58*e^3 - 68*e, -3/2*e^7 + 19*e^5 - 57*e^3 + 22*e, -3/2*e^7 + 19*e^5 - 57*e^3 + 22*e, e^7 - 14*e^5 + 58*e^3 - 68*e, -3*e^6 + 40*e^4 - 134*e^2 + 80, 3*e^6 - 36*e^4 + 114*e^2 - 88, 3*e^6 - 36*e^4 + 114*e^2 - 88, -3*e^6 + 40*e^4 - 134*e^2 + 80, e^7 - 13*e^5 + 48*e^3 - 50*e, e^7 - 13*e^5 + 48*e^3 - 50*e, -5/4*e^7 + 39/2*e^5 - 175/2*e^3 + 93*e, -5/4*e^7 + 39/2*e^5 - 175/2*e^3 + 93*e, e^7 - 11*e^5 + 28*e^3 - 14*e, e^7 - 13*e^5 + 46*e^3 - 50*e, e^7 - 13*e^5 + 46*e^3 - 50*e, e^7 - 11*e^5 + 28*e^3 - 14*e, 1/2*e^6 - 5*e^4 + e^2 + 44, 1/2*e^6 - 5*e^4 + e^2 + 44] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([3,3,w^2-w-3])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]