/* 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([29, 3, -12, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([20, 10, 1/2*w^3 - w^2 - 4*w + 11/2]) primes_array = [ [4, 2, 1/2*w^3 - 2*w^2 - 2*w + 15/2],\ [4, 2, -w^2 + w + 8],\ [5, 5, -1/4*w^3 + 1/2*w^2 + 1/2*w - 9/4],\ [5, 5, 1/2*w^3 - w^2 - 4*w + 9/2],\ [19, 19, 1/4*w^3 - 3/2*w^2 - 1/2*w + 21/4],\ [19, 19, 1/4*w^3 - 3/2*w^2 - 1/2*w + 41/4],\ [29, 29, w],\ [29, 29, 1/4*w^3 - 1/2*w^2 - 1/2*w + 1/4],\ [29, 29, 1/4*w^3 - 1/2*w^2 - 5/2*w + 9/4],\ [29, 29, -1/2*w^3 + w^2 + 4*w - 5/2],\ [41, 41, 3/4*w^3 - 5/2*w^2 - 7/2*w + 47/4],\ [41, 41, 1/4*w^3 + 1/2*w^2 - 5/2*w - 15/4],\ [61, 61, -3/2*w^3 + 3*w^2 + 11*w - 21/2],\ [61, 61, w^3 - 2*w^2 - 4*w + 6],\ [79, 79, 3/4*w^3 - 5/2*w^2 - 7/2*w + 27/4],\ [79, 79, 1/4*w^3 + 1/2*w^2 - 5/2*w - 35/4],\ [81, 3, -3],\ [89, 89, 7/4*w^3 - 11/2*w^2 - 21/2*w + 119/4],\ [89, 89, 1/4*w^3 + 3/2*w^2 - 3/2*w - 23/4],\ [109, 109, -1/2*w^3 + 3*w^2 + 2*w - 41/2],\ [109, 109, -5/4*w^3 + 7/2*w^2 + 9/2*w - 45/4],\ [121, 11, -3/4*w^3 + 3/2*w^2 + 9/2*w - 23/4],\ [121, 11, -1/4*w^3 + 1/2*w^2 + 3/2*w - 21/4],\ [131, 131, -1/2*w^3 + w^2 + 5*w - 3/2],\ [131, 131, 3/4*w^3 - 1/2*w^2 - 7/2*w - 1/4],\ [139, 139, -5/4*w^3 + 11/2*w^2 + 9/2*w - 73/4],\ [139, 139, -1/4*w^3 + 7/2*w^2 - 3/2*w - 113/4],\ [149, 149, -1/4*w^3 + 3/2*w^2 + 1/2*w - 49/4],\ [149, 149, 1/4*w^3 - 3/2*w^2 - 1/2*w + 13/4],\ [151, 151, 1/2*w^3 - 3*w^2 - w + 23/2],\ [151, 151, -1/4*w^3 + 3/2*w^2 + 3/2*w - 45/4],\ [151, 151, 3/4*w^3 - 3/2*w^2 - 7/2*w + 11/4],\ [151, 151, -2*w^3 + 6*w^2 + 7*w - 20],\ [169, 13, -1/4*w^3 + 1/2*w^2 + 1/2*w - 25/4],\ [169, 13, 1/2*w^3 - w^2 - 4*w + 17/2],\ [179, 179, -1/4*w^3 + 1/2*w^2 + 7/2*w - 21/4],\ [179, 179, 1/4*w^3 - 1/2*w^2 - 7/2*w - 3/4],\ [181, 181, 1/2*w^3 - 3*w - 7/2],\ [181, 181, -5/4*w^3 + 7/2*w^2 + 15/2*w - 57/4],\ [191, 191, -3/4*w^3 + 5/2*w^2 + 9/2*w - 39/4],\ [191, 191, w^2 - 8],\ [199, 199, -3/4*w^3 + 7/2*w^2 + 5/2*w - 43/4],\ [199, 199, -1/4*w^3 + 5/2*w^2 - 1/2*w - 81/4],\ [199, 199, -3/4*w^3 + 7/2*w^2 + 5/2*w - 75/4],\ [199, 199, -1/4*w^3 + 5/2*w^2 - 1/2*w - 49/4],\ [229, 229, -1/2*w^3 + 2*w^2 + w - 21/2],\ [229, 229, -3/4*w^3 + 5/2*w^2 + 7/2*w - 55/4],\ [239, 239, 3/2*w^3 - 4*w^2 - 9*w + 35/2],\ [239, 239, 3/2*w^3 - 5*w^2 - 7*w + 45/2],\ [241, 241, -5/4*w^3 + 9/2*w^2 + 13/2*w - 93/4],\ [241, 241, -1/4*w^3 - 1/2*w^2 - 1/2*w + 3/4],\ [251, 251, -w^3 + 3*w^2 + 8*w - 13],\ [251, 251, -7/4*w^3 + 13/2*w^2 + 13/2*w - 99/4],\ [269, 269, -9/4*w^3 + 13/2*w^2 + 15/2*w - 85/4],\ [269, 269, -3/4*w^3 + 5/2*w^2 + 3/2*w - 47/4],\ [271, 271, 5/4*w^3 - 5/2*w^2 - 21/2*w + 53/4],\ [271, 271, 5/4*w^3 - 1/2*w^2 - 23/2*w - 23/4],\ [271, 271, 7/4*w^3 - 11/2*w^2 - 13/2*w + 83/4],\ [271, 271, -1/2*w^3 + w^2 - 13/2],\ [281, 281, 3/4*w^3 - 1/2*w^2 - 13/2*w - 1/4],\ [281, 281, -w^3 + 3*w^2 + 4*w - 13],\ [311, 311, -1/4*w^3 + 1/2*w^2 - 1/2*w - 5/4],\ [311, 311, 7/4*w^3 - 13/2*w^2 - 19/2*w + 155/4],\ [311, 311, -1/4*w^3 + 7/2*w^2 + 1/2*w - 49/4],\ [311, 311, -3/4*w^3 + 3/2*w^2 + 13/2*w - 23/4],\ [331, 331, -3/4*w^3 + 1/2*w^2 + 15/2*w + 5/4],\ [331, 331, -3/4*w^3 + 5/2*w^2 + 3/2*w - 39/4],\ [349, 349, -1/4*w^3 + 1/2*w^2 + 7/2*w - 17/4],\ [349, 349, 1/4*w^3 - 1/2*w^2 - 7/2*w + 1/4],\ [359, 359, -5/4*w^3 - 1/2*w^2 + 25/2*w + 55/4],\ [359, 359, w^2 - 2*w - 2],\ [359, 359, -9/4*w^3 + 15/2*w^2 + 17/2*w - 113/4],\ [359, 359, -1/4*w^3 + 3/2*w^2 - 1/2*w - 45/4],\ [361, 19, -w^3 + 2*w^2 + 6*w - 8],\ [379, 379, -5/4*w^3 + 5/2*w^2 + 17/2*w - 37/4],\ [379, 379, -5/4*w^3 + 7/2*w^2 + 9/2*w - 41/4],\ [379, 379, 5/4*w^3 - 3/2*w^2 - 21/2*w - 3/4],\ [379, 379, -w^3 + 2*w^2 + 5*w - 7],\ [389, 389, 9/4*w^3 - 17/2*w^2 - 15/2*w + 117/4],\ [389, 389, -3/4*w^3 - 5/2*w^2 + 21/2*w + 113/4],\ [401, 401, -3/4*w^3 - 7/2*w^2 + 23/2*w + 149/4],\ [401, 401, 11/4*w^3 - 21/2*w^2 - 19/2*w + 143/4],\ [409, 409, -3/4*w^3 + 5/2*w^2 + 9/2*w - 31/4],\ [409, 409, w^2 - 10],\ [421, 421, 5/4*w^3 - 5/2*w^2 - 19/2*w + 49/4],\ [421, 421, -1/4*w^3 + 5/2*w^2 - 1/2*w - 41/4],\ [421, 421, 5/4*w^3 - 5/2*w^2 - 7/2*w + 41/4],\ [421, 421, 9/4*w^3 - 9/2*w^2 - 35/2*w + 77/4],\ [431, 431, 7/4*w^3 - 9/2*w^2 - 11/2*w + 59/4],\ [431, 431, 5/4*w^3 - 9/2*w^2 - 13/2*w + 85/4],\ [431, 431, 5/4*w^3 - 5/2*w^2 - 11/2*w + 33/4],\ [431, 431, -2*w^2 + w + 12],\ [439, 439, -1/2*w^3 + w^2 + 5*w - 7/2],\ [439, 439, 2*w - 1],\ [461, 461, -7/4*w^3 + 13/2*w^2 + 21/2*w - 163/4],\ [461, 461, 3/2*w^3 - 5*w^2 - 4*w + 31/2],\ [491, 491, 5/4*w^3 - 3/2*w^2 - 17/2*w + 21/4],\ [491, 491, -7/4*w^3 + 9/2*w^2 + 19/2*w - 83/4],\ [509, 509, -3/2*w^3 + 4*w^2 + 7*w - 27/2],\ [509, 509, -3/2*w^3 + 3*w^2 + 10*w - 27/2],\ [521, 521, -5/4*w^3 + 3/2*w^2 + 11/2*w - 17/4],\ [521, 521, -5/2*w^3 + 6*w^2 + 17*w - 53/2],\ [521, 521, -1/2*w^3 + 3*w^2 + w - 31/2],\ [541, 541, 9/4*w^3 - 7/2*w^2 - 35/2*w + 37/4],\ [541, 541, w^3 - 2*w^2 - 8*w + 12],\ [541, 541, -1/4*w^3 - 3/2*w^2 + 7/2*w + 35/4],\ [541, 541, -3/4*w^3 + 9/2*w^2 + 5/2*w - 119/4],\ [569, 569, -w^3 + 3*w^2 + 6*w - 7],\ [569, 569, -7/4*w^3 + 9/2*w^2 + 23/2*w - 75/4],\ [599, 599, 1/4*w^3 + 5/2*w^2 - 9/2*w - 95/4],\ [599, 599, -7/4*w^3 + 13/2*w^2 + 15/2*w - 91/4],\ [619, 619, 9/4*w^3 - 13/2*w^2 - 19/2*w + 97/4],\ [619, 619, -7/4*w^3 + 3/2*w^2 + 29/2*w + 9/4],\ [619, 619, -1/4*w^3 + 3/2*w^2 - 3/2*w - 45/4],\ [619, 619, -1/4*w^3 - 1/2*w^2 + 9/2*w - 1/4],\ [631, 631, -9/4*w^3 + 13/2*w^2 + 15/2*w - 93/4],\ [631, 631, -13/4*w^3 + 21/2*w^2 + 39/2*w - 233/4],\ [631, 631, 7/4*w^3 - 9/2*w^2 - 15/2*w + 67/4],\ [631, 631, 1/4*w^3 + 7/2*w^2 - 3/2*w - 51/4],\ [641, 641, -1/4*w^3 + 5/2*w^2 + 1/2*w - 57/4],\ [641, 641, -w^3 + 4*w^2 + 5*w - 19],\ [661, 661, -2*w^3 + 20*w + 19],\ [661, 661, 5/4*w^3 - 11/2*w^2 - 13/2*w + 149/4],\ [691, 691, -5/4*w^3 + 3/2*w^2 + 13/2*w - 9/4],\ [691, 691, -7/4*w^3 + 11/2*w^2 + 23/2*w - 135/4],\ [739, 739, -2*w^3 + 7*w^2 + 10*w - 34],\ [739, 739, 1/4*w^3 + 5/2*w^2 - 7/2*w - 59/4],\ [751, 751, -1/2*w^3 + 3*w^2 + 2*w - 33/2],\ [751, 751, -3/4*w^3 + 7/2*w^2 + 7/2*w - 67/4],\ [761, 761, -w^3 + 4*w^2 + 5*w - 27],\ [761, 761, 7/4*w^3 - 9/2*w^2 - 15/2*w + 63/4],\ [769, 769, -5/4*w^3 + 1/2*w^2 + 25/2*w + 19/4],\ [769, 769, 3/2*w^3 - 5*w^2 - 4*w + 39/2],\ [769, 769, -3*w^3 + 10*w^2 + 11*w - 37],\ [769, 769, -7/4*w^3 + 5/2*w^2 + 27/2*w - 31/4],\ [811, 811, -w^3 + 3*w^2 + 6*w - 9],\ [811, 811, -3/4*w^3 + 7/2*w^2 + 5/2*w - 39/4],\ [811, 811, -1/4*w^3 + 5/2*w^2 - 1/2*w - 85/4],\ [811, 811, -5/4*w^3 + 7/2*w^2 + 17/2*w - 57/4],\ [821, 821, -3/4*w^3 + 3/2*w^2 + 3/2*w - 35/4],\ [821, 821, -5/4*w^3 + 11/2*w^2 + 9/2*w - 121/4],\ [829, 829, 7/4*w^3 - 13/2*w^2 - 19/2*w + 115/4],\ [829, 829, 1/4*w^3 - 7/2*w^2 - 1/2*w + 89/4],\ [839, 839, -3/4*w^3 + 5/2*w^2 + 3/2*w - 51/4],\ [839, 839, -3/4*w^3 + 1/2*w^2 + 15/2*w - 7/4],\ [911, 911, -3/4*w^3 + 1/2*w^2 + 11/2*w - 15/4],\ [911, 911, -3/4*w^3 + 7/2*w^2 + 3/2*w - 75/4],\ [929, 929, 9/4*w^3 - 9/2*w^2 - 35/2*w + 73/4],\ [929, 929, -3/4*w^3 + 7/2*w^2 + 9/2*w - 51/4],\ [929, 929, -3/4*w^3 + 7/2*w^2 + 9/2*w - 91/4],\ [929, 929, 5/2*w^3 - 4*w^2 - 19*w + 25/2],\ [961, 31, -5/4*w^3 + 5/2*w^2 + 15/2*w - 37/4],\ [961, 31, -1/2*w^3 + w^2 + 3*w - 19/2]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^4 + 4*x^3 - 8*x - 3 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, -1, 1, -e^2 - 2*e + 1, -e^2 - 4*e - 1, e^3 + 3*e^2 - e - 5, -2*e^3 - 6*e^2 + 4*e + 10, 2*e^2 + 4*e - 4, 2*e^3 + 5*e^2 - 8*e - 11, -2*e^3 - 6*e^2 + 6*e + 12, -e^3 - 2*e^2 + 5*e - 2, e^3 + 3*e^2 + e - 3, 3*e^3 + 9*e^2 - 11*e - 17, -3*e^3 - 8*e^2 + 11*e + 12, -2*e^3 - 7*e^2 + 2*e + 3, 2*e^3 + 8*e^2 - 4*e - 14, 2*e^3 + 7*e^2 - 4*e - 13, -e^3 + 9*e - 8, -e^3 - 3*e^2 + 3*e + 3, 2*e^3 + 6*e^2 - 8*e - 10, 3*e^3 + 9*e^2 - 7*e - 11, 3*e^3 + 9*e^2 - 3*e - 9, -2*e^3 - 6*e^2 + 10*e + 10, 2*e^3 + 6*e^2 - 12*e - 16, 6*e^3 + 17*e^2 - 10*e - 17, -3*e^3 - 11*e^2 + 3*e + 19, -e^3 - 5*e^2 - 5*e + 3, -2*e^3 - 6*e^2 + 6*e + 2, e^3 + 4*e^2 + 7*e, 3*e^3 + 9*e^2 - 7*e - 7, -3*e^3 - 12*e^2 + 5*e + 20, 2*e^3 + 9*e^2 - 2*e - 19, -3*e^3 - 10*e^2 + 5*e + 2, 4*e^2 + 4*e - 22, -5*e^3 - 11*e^2 + 21*e + 23, e^3 - e^2 - 13*e + 1, -e^3 + 7*e - 8, 3*e^3 + 11*e^2 - 7*e - 17, -4*e^3 - 10*e^2 + 8*e + 8, -5*e^3 - 13*e^2 + 23*e + 27, 6*e^3 + 18*e^2 - 14*e - 24, 2*e^3 + 10*e^2 + 4*e - 6, 2*e^3 + 3*e^2 - 10*e - 3, e^3 - e^2 - 3*e + 17, -2*e^3 - 6*e^2 + 10, 7*e^3 + 22*e^2 - 15*e - 30, -e^3 - e^2 + 11*e + 3, -e^3 - 3*e^2 - e - 5, 7*e^3 + 23*e^2 - 7*e - 35, -e^3 + 5*e^2 + 19*e - 7, -4*e^2 - 12*e + 4, -e^3 - 3*e^2 + 5*e - 1, -2*e^3 - 8*e^2 - 4*e + 6, -e^3 - e^2 + 11*e - 9, 3*e^3 + 11*e^2 + 3*e - 11, 4*e^3 + 9*e^2 - 16*e - 13, -2*e^3 - 10*e^2 - 6*e + 12, -3*e^3 - 3*e^2 + 21*e + 9, 5*e^3 + 16*e^2 - 15*e - 28, 4*e^3 + 6*e^2 - 22*e - 10, 0, -6*e^2 - 6*e + 20, -3*e^3 - 11*e^2 + 5*e + 27, 6*e^3 + 20*e^2 - 8*e - 36, -8*e^3 - 24*e^2 + 22*e + 30, -5*e^3 - 15*e^2 + 3*e + 9, -3*e^3 - 6*e^2 + 13*e + 10, -2*e^3 - 7*e^2 - 4*e + 3, 9*e^3 + 29*e^2 - 21*e - 47, -8*e^2 - 8*e + 28, -2*e^2 - 8*e + 12, -5*e^3 - 15*e^2 + 11*e + 33, -e^3 + 4*e^2 + 11*e - 24, -12*e^3 - 37*e^2 + 18*e + 45, 2*e^3 + 6*e^2 - 10*e - 2, 4*e^2 + 6*e - 12, -e^3 - 4*e^2 - e - 8, 6*e^3 + 20*e^2 - 14*e - 32, -4*e^3 - 14*e^2 - 8*e + 12, -6*e^3 - 20*e^2 + 16*e + 28, -10*e^3 - 30*e^2 + 20*e + 52, 2*e^3 + 8*e^2 - 8*e - 38, 3*e^3 + 13*e^2 - 9*e - 39, -9*e^3 - 29*e^2 + 17*e + 49, -8*e^3 - 22*e^2 + 20*e + 16, 4*e^2 + 10*e + 4, 6*e^3 + 16*e^2 - 16*e - 14, -3*e^3 - 7*e^2 + 21*e + 17, 5*e^3 + 14*e^2 - 19*e - 42, e^3 - e^2 - 17*e + 1, -3*e^3 - 9*e^2 + 17*e + 17, -4*e^3 - 8*e^2 + 8*e - 4, -3*e^3 - 12*e^2 - 3*e + 28, -2*e^3 - 6*e^2 + 6*e + 18, -7*e^3 - 23*e^2 + 5*e + 15, -e^3 - 5*e^2 - 5*e - 1, 7*e^3 + 19*e^2 - 15*e - 29, -8*e^3 - 29*e^2 + 8*e + 43, 2*e^3 + 4*e^2 - 16*e + 10, -4*e^3 - 14*e^2 - 2*e + 10, -6*e^3 - 20*e^2 + 10*e + 22, -4*e^3 - 8*e^2 + 14*e - 8, -3*e^3 - 9*e^2 + 21*e + 25, e^3 + 3*e^2 - 5*e - 23, 4*e^3 + 18*e^2 + 6*e - 34, -3*e^3 - 13*e^2 + e - 5, -5*e^3 - 21*e^2 - e + 29, -4*e^3 - 13*e^2 + 10*e + 19, 8*e^3 + 16*e^2 - 36*e - 36, -14*e^3 - 36*e^2 + 38*e + 48, -4*e^3 - 14*e^2 + 8*e + 40, -3*e^3 - 12*e^2 - 3*e + 4, 5*e^3 + 19*e^2 + 9*e - 29, 12*e^3 + 32*e^2 - 32*e - 44, -6*e^3 - 13*e^2 + 14*e - 5, 8*e^3 + 18*e^2 - 34*e - 24, 3*e^3 + 5*e^2 - 7*e - 7, 15*e^3 + 42*e^2 - 41*e - 70, -7*e^3 - 29*e^2 + e + 27, 3*e^3 + 3*e^2 - 21*e - 15, -2*e^3 - 8*e^2 - 4*e - 6, 4*e^3 + 14*e^2 - 16, 10*e^3 + 30*e^2 - 24*e - 30, 12*e^3 + 38*e^2 - 18*e - 58, -4*e^3 - 6*e^2 + 14*e - 12, -17*e^3 - 51*e^2 + 29*e + 59, -2*e^3 - 12*e^2 - 18*e + 16, e^3 + 6*e^2 - 11*e - 22, 10*e^3 + 32*e^2 - 10*e - 48, -4*e^3 - 6*e^2 + 34*e + 14, -12*e^3 - 42*e^2 + 22*e + 54, 5*e^3 + 7*e^2 - 19*e + 3, -2*e^3 - 9*e^2 + 4*e + 13, 6*e^3 + 26*e^2 - 2*e - 40, 5*e^3 + 8*e^2 - 37*e - 12, -8*e^3 - 22*e^2 + 32*e + 52, 12*e^3 + 40*e^2 - 26*e - 50, 3*e^3 + 7*e^2 - 3*e + 9, 6*e^3 + 22*e^2 - 16*e - 32, -2*e^3 - 15*e^2 - 20*e + 41, 7*e^3 + 21*e^2 - 19*e - 39, -8*e^3 - 26*e^2 + 16*e + 34, -3*e^3 + 27*e + 20, 5*e^3 + 25*e^2 + 19*e - 47, -4*e^3 - 6*e^2 + 6*e - 24, -2*e^3 - 10*e^2 + 2*e + 2, 7*e^3 + 23*e^2 - 7*e - 45, 4*e^3 + 10*e^2 - 16*e + 24, -8*e^3 - 20*e^2 + 34*e + 62, 3*e^3 + 4*e^2 - 15*e + 24, e^3 + 4*e^2 - 5*e + 20, -7*e^3 - 21*e^2 + 15*e + 47, -8*e^3 - 28*e^2 + 46] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([4, 2, -w^2 + w + 8])] = 1 AL_eigenvalues[ZF.ideal([5, 5, -1/4*w^3 + 1/2*w^2 + 1/2*w - 9/4])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]