/* 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([1, -4, -1, 9, -2, -3, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([25, 5, -w^3 + 2*w^2 + 2*w - 3]) primes_array = [ [3, 3, -w^5 + 2*w^4 + 4*w^3 - 5*w^2 - 5*w + 1],\ [7, 7, -w^5 + 3*w^4 + 3*w^3 - 10*w^2 - 3*w + 5],\ [25, 5, -w^3 + 2*w^2 + 2*w - 3],\ [37, 37, w^3 - w^2 - 4*w + 2],\ [37, 37, w^3 - 2*w^2 - 3*w + 2],\ [49, 7, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 5*w - 3],\ [49, 7, -w^5 + 3*w^4 + 3*w^3 - 9*w^2 - 3*w + 4],\ [53, 53, w^4 - 2*w^3 - 3*w^2 + 3*w + 1],\ [53, 53, w^4 - 2*w^3 - 3*w^2 + 5*w],\ [61, 61, -3*w^5 + 6*w^4 + 12*w^3 - 16*w^2 - 12*w + 5],\ [61, 61, 2*w^5 - 5*w^4 - 7*w^3 + 16*w^2 + 7*w - 7],\ [61, 61, -2*w^5 + 5*w^4 + 7*w^3 - 15*w^2 - 8*w + 6],\ [61, 61, -w^3 + 2*w^2 + 4*w - 3],\ [64, 2, -2],\ [67, 67, w^5 - 3*w^4 - 3*w^3 + 10*w^2 + 2*w - 3],\ [67, 67, -w^5 + 2*w^4 + 4*w^3 - 6*w^2 - 4*w + 5],\ [71, 71, w^5 - 2*w^4 - 4*w^3 + 7*w^2 + 2*w - 4],\ [71, 71, -w^5 + 3*w^4 + 2*w^3 - 7*w^2 - w],\ [81, 3, -w^4 + 2*w^3 + 2*w^2 - 3*w + 2],\ [101, 101, w^5 - 3*w^4 - 3*w^3 + 9*w^2 + 4*w - 3],\ [101, 101, -2*w^5 + 5*w^4 + 6*w^3 - 13*w^2 - 4*w + 3],\ [103, 103, -2*w^5 + 5*w^4 + 5*w^3 - 12*w^2 - 2*w + 3],\ [103, 103, 2*w^5 - 5*w^4 - 5*w^3 + 13*w^2 + w - 3],\ [107, 107, w^5 - 2*w^4 - 4*w^3 + 5*w^2 + 4*w - 3],\ [107, 107, w^5 - 3*w^4 - 2*w^3 + 9*w^2 - w - 1],\ [113, 113, -2*w^5 + 4*w^4 + 9*w^3 - 12*w^2 - 10*w + 6],\ [113, 113, -3*w^5 + 9*w^4 + 7*w^3 - 27*w^2 - 2*w + 12],\ [113, 113, -2*w^5 + 6*w^4 + 5*w^3 - 17*w^2 - 4*w + 6],\ [113, 113, -2*w^5 + 5*w^4 + 5*w^3 - 12*w^2 + 2],\ [149, 149, 3*w^5 - 8*w^4 - 8*w^3 + 22*w^2 + 4*w - 7],\ [149, 149, 3*w^5 - 7*w^4 - 10*w^3 + 20*w^2 + 7*w - 6],\ [151, 151, -w^5 + 4*w^4 - 12*w^2 + 3*w + 3],\ [151, 151, -w^5 + w^4 + 6*w^3 - 3*w^2 - 6*w + 3],\ [163, 163, 2*w^5 - 5*w^4 - 5*w^3 + 14*w^2 - 4],\ [163, 163, 2*w^5 - 5*w^4 - 5*w^3 + 11*w^2 + 3*w - 2],\ [167, 167, -w^4 + 3*w^3 - 6*w + 2],\ [167, 167, -2*w^5 + 6*w^4 + 5*w^3 - 19*w^2 - w + 7],\ [193, 193, 4*w^5 - 10*w^4 - 11*w^3 + 27*w^2 + 4*w - 7],\ [193, 193, 4*w^5 - 10*w^4 - 11*w^3 + 26*w^2 + 5*w - 7],\ [197, 197, w^5 - 4*w^4 - 2*w^3 + 15*w^2 + 3*w - 8],\ [197, 197, -2*w^5 + 5*w^4 + 7*w^3 - 16*w^2 - 4*w + 8],\ [197, 197, -w^3 + w^2 + w - 2],\ [197, 197, 2*w^5 - 5*w^4 - 5*w^3 + 13*w^2 + 2*w - 4],\ [227, 227, -w^4 + 3*w^3 + 2*w^2 - 6*w - 3],\ [227, 227, w^5 - 3*w^4 - 3*w^3 + 11*w^2 + 2*w - 4],\ [241, 241, -2*w^5 + 6*w^4 + 5*w^3 - 20*w^2 - w + 9],\ [241, 241, -w^4 + w^3 + 4*w^2 - 3*w - 1],\ [241, 241, -4*w^5 + 11*w^4 + 10*w^3 - 31*w^2 - 2*w + 11],\ [241, 241, -2*w^5 + 4*w^4 + 9*w^3 - 11*w^2 - 12*w + 3],\ [257, 257, -w^5 + 2*w^4 + 4*w^3 - 5*w^2 - 3*w - 1],\ [257, 257, w^5 - 3*w^4 - 2*w^3 + 9*w^2 - 2*w - 4],\ [263, 263, -3*w^5 + 8*w^4 + 8*w^3 - 21*w^2 - 6*w + 5],\ [263, 263, -3*w^5 + 7*w^4 + 10*w^3 - 21*w^2 - 7*w + 9],\ [271, 271, 2*w^5 - 6*w^4 - 6*w^3 + 21*w^2 + 6*w - 11],\ [271, 271, w^5 - 3*w^4 - 4*w^3 + 12*w^2 + 6*w - 7],\ [277, 277, 4*w^5 - 9*w^4 - 15*w^3 + 27*w^2 + 13*w - 9],\ [277, 277, -4*w^5 + 11*w^4 + 11*w^3 - 32*w^2 - 6*w + 11],\ [281, 281, -w^5 + 3*w^4 + 3*w^3 - 11*w^2 - 3*w + 5],\ [281, 281, -w^5 + 2*w^4 + 5*w^3 - 6*w^2 - 9*w + 4],\ [311, 311, -w^4 + w^3 + 3*w^2 + w + 1],\ [311, 311, -2*w^5 + 5*w^4 + 6*w^3 - 13*w^2 - 5*w + 5],\ [353, 353, w^5 - 4*w^4 - w^3 + 14*w^2 - 3*w - 6],\ [353, 353, w^5 - w^4 - 7*w^3 + 3*w^2 + 11*w - 1],\ [359, 359, -4*w^5 + 10*w^4 + 11*w^3 - 28*w^2 - 4*w + 11],\ [359, 359, -2*w^2 + w + 5],\ [361, 19, -3*w^5 + 9*w^4 + 6*w^3 - 25*w^2 + 8],\ [361, 19, 3*w^5 - 6*w^4 - 12*w^3 + 17*w^2 + 11*w - 5],\ [361, 19, -w^4 + 2*w^3 + 4*w^2 - 5*w - 2],\ [367, 367, 2*w^5 - 5*w^4 - 7*w^3 + 15*w^2 + 6*w - 8],\ [367, 367, w^5 - 3*w^4 - 2*w^3 + 10*w^2 - 7],\ [383, 383, -2*w^5 + 6*w^4 + 6*w^3 - 20*w^2 - 5*w + 8],\ [383, 383, 4*w^5 - 10*w^4 - 11*w^3 + 28*w^2 + 3*w - 12],\ [419, 419, -w^4 + 5*w^2 + 3*w - 3],\ [419, 419, -2*w^5 + 6*w^4 + 6*w^3 - 20*w^2 - 6*w + 11],\ [443, 443, w^5 - 3*w^4 - w^3 + 8*w^2 - 3*w - 4],\ [443, 443, -w^5 + 2*w^4 + 3*w^3 - 3*w^2 - 3*w - 2],\ [461, 461, -w^5 + 2*w^4 + 4*w^3 - 4*w^2 - 4*w - 3],\ [461, 461, w^5 - 3*w^4 - 2*w^3 + 10*w^2 - 3*w - 6],\ [463, 463, -3*w^5 + 7*w^4 + 10*w^3 - 21*w^2 - 5*w + 8],\ [463, 463, -3*w^5 + 8*w^4 + 8*w^3 - 21*w^2 - 4*w + 4],\ [499, 499, w^5 - 2*w^4 - 2*w^3 + 2*w^2 - w + 2],\ [499, 499, w^5 - 4*w^4 + 11*w^2 - 4*w - 5],\ [509, 509, -2*w^5 + 5*w^4 + 8*w^3 - 18*w^2 - 8*w + 10],\ [509, 509, w^5 - w^4 - 6*w^3 + 4*w^2 + 7*w - 3],\ [541, 541, -2*w^5 + 6*w^4 + 6*w^3 - 20*w^2 - 5*w + 9],\ [541, 541, -2*w^5 + 4*w^4 + 10*w^3 - 14*w^2 - 13*w + 6],\ [557, 557, -w^5 + 2*w^4 + 6*w^3 - 9*w^2 - 9*w + 4],\ [557, 557, w^5 - 3*w^4 - 4*w^3 + 11*w^2 + 6*w - 7],\ [577, 577, 2*w^5 - 5*w^4 - 6*w^3 + 16*w^2 + 2*w - 9],\ [577, 577, -3*w^5 + 7*w^4 + 10*w^3 - 19*w^2 - 9*w + 8],\ [587, 587, -w^4 + 3*w^3 + w^2 - 6*w - 1],\ [587, 587, 2*w^5 - 4*w^4 - 9*w^3 + 14*w^2 + 8*w - 7],\ [587, 587, -w^5 + 4*w^4 - 12*w^2 + 6*w + 3],\ [587, 587, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 5*w - 7],\ [587, 587, -2*w^5 + 6*w^4 + 5*w^3 - 17*w^2 - 3*w + 4],\ [587, 587, -w^4 + w^3 + 4*w^2 - w - 4],\ [625, 5, -2*w^5 + 6*w^4 + 3*w^3 - 16*w^2 + 3*w + 6],\ [631, 631, 3*w^5 - 8*w^4 - 8*w^3 + 22*w^2 + 5*w - 4],\ [631, 631, -3*w^5 + 7*w^4 + 10*w^3 - 20*w^2 - 8*w + 10],\ [641, 641, -3*w^5 + 7*w^4 + 11*w^3 - 21*w^2 - 10*w + 10],\ [641, 641, 3*w^5 - 8*w^4 - 9*w^3 + 24*w^2 + 6*w - 6],\ [643, 643, 2*w^5 - 5*w^4 - 6*w^3 + 14*w^2 + 6*w - 5],\ [643, 643, 3*w^5 - 6*w^4 - 13*w^3 + 16*w^2 + 15*w - 3],\ [643, 643, 3*w^5 - 8*w^4 - 8*w^3 + 24*w^2 + 5*w - 11],\ [643, 643, 2*w^5 - 5*w^4 - 6*w^3 + 14*w^2 + 6*w - 6],\ [653, 653, -w^5 + 2*w^4 + 6*w^3 - 8*w^2 - 9*w + 4],\ [653, 653, 2*w^5 - 6*w^4 - 5*w^3 + 19*w^2 + 2*w - 6],\ [653, 653, -2*w^5 + 4*w^4 + 9*w^3 - 12*w^2 - 11*w + 6],\ [653, 653, -w^5 + 3*w^4 + 4*w^3 - 12*w^2 - 4*w + 6],\ [659, 659, 2*w^5 - 4*w^4 - 8*w^3 + 12*w^2 + 8*w - 7],\ [659, 659, -2*w^5 + 6*w^4 + 4*w^3 - 16*w^2 - 2*w + 3],\ [661, 661, 2*w^5 - 5*w^4 - 6*w^3 + 13*w^2 + 6*w - 1],\ [661, 661, -w^5 + w^4 + 5*w^3 - w^2 - 6*w - 1],\ [691, 691, -w^5 + 2*w^4 + 6*w^3 - 9*w^2 - 8*w + 6],\ [691, 691, 4*w^5 - 11*w^4 - 11*w^3 + 32*w^2 + 7*w - 11],\ [733, 733, -4*w^5 + 10*w^4 + 13*w^3 - 30*w^2 - 8*w + 14],\ [733, 733, -2*w^5 + 6*w^4 + 4*w^3 - 17*w^2 - w + 5],\ [739, 739, -w^2 + 5],\ [739, 739, w^2 - 2*w - 4],\ [757, 757, -w^5 + 3*w^4 + 2*w^3 - 8*w^2 - 2*w],\ [757, 757, -w^4 + w^3 + 5*w^2 - w - 6],\ [821, 821, -3*w^5 + 7*w^4 + 9*w^3 - 17*w^2 - 7*w + 2],\ [821, 821, -2*w^5 + 3*w^4 + 10*w^3 - 9*w^2 - 11*w + 4],\ [821, 821, -w^4 + 2*w^3 + 3*w^2 - 6*w - 2],\ [821, 821, -3*w^5 + 8*w^4 + 7*w^3 - 22*w^2 - w + 9],\ [823, 823, 3*w^5 - 7*w^4 - 10*w^3 + 19*w^2 + 10*w - 7],\ [823, 823, 3*w^5 - 8*w^4 - 8*w^3 + 23*w^2 + 5*w - 8],\ [827, 827, w^5 - w^4 - 7*w^3 + 4*w^2 + 9*w - 2],\ [827, 827, -w^5 + 4*w^4 + w^3 - 13*w^2 + 3*w + 4],\ [853, 853, -5*w^5 + 13*w^4 + 12*w^3 - 33*w^2 - 2*w + 9],\ [853, 853, 2*w^4 - 6*w^3 - 4*w^2 + 16*w - 1],\ [857, 857, 3*w^5 - 6*w^4 - 11*w^3 + 16*w^2 + 8*w - 8],\ [857, 857, 2*w^5 - 7*w^4 - 3*w^3 + 21*w^2 - 2*w - 6],\ [859, 859, -3*w^5 + 7*w^4 + 11*w^3 - 20*w^2 - 12*w + 8],\ [859, 859, -3*w^5 + 7*w^4 + 12*w^3 - 23*w^2 - 12*w + 9],\ [859, 859, 3*w^5 - 8*w^4 - 10*w^3 + 25*w^2 + 9*w - 10],\ [859, 859, -2*w^5 + 7*w^4 + 2*w^3 - 20*w^2 + 5*w + 7],\ [863, 863, w^3 - 6*w],\ [863, 863, 3*w^5 - 7*w^4 - 11*w^3 + 21*w^2 + 9*w - 8],\ [863, 863, -3*w^5 + 8*w^4 + 9*w^3 - 24*w^2 - 5*w + 7],\ [863, 863, -w^3 + 3*w^2 + 3*w - 5],\ [877, 877, 2*w^5 - 6*w^4 - 4*w^3 + 15*w^2 + 2*w - 1],\ [877, 877, -2*w^5 + 4*w^4 + 8*w^3 - 13*w^2 - 6*w + 8],\ [881, 881, -4*w^5 + 10*w^4 + 13*w^3 - 29*w^2 - 9*w + 8],\ [881, 881, -4*w^5 + 10*w^4 + 13*w^3 - 30*w^2 - 8*w + 11],\ [883, 883, -4*w^5 + 9*w^4 + 16*w^3 - 30*w^2 - 14*w + 12],\ [883, 883, w^5 - w^4 - 8*w^3 + 7*w^2 + 10*w - 5],\ [907, 907, -2*w^5 + 4*w^4 + 9*w^3 - 14*w^2 - 9*w + 7],\ [907, 907, -w^4 + 3*w^3 + w^2 - 7*w],\ [907, 907, w^5 - 4*w^4 - w^3 + 12*w^2 - 2*w - 2],\ [907, 907, -w^4 + 3*w^3 + 3*w^2 - 9*w - 1],\ [907, 907, -w^4 + w^3 + 4*w^2 - 4],\ [907, 907, -2*w^5 + 6*w^4 + 5*w^3 - 17*w^2 - 4*w + 5],\ [919, 919, -4*w^5 + 10*w^4 + 12*w^3 - 28*w^2 - 6*w + 9],\ [919, 919, 4*w^5 - 10*w^4 - 12*w^3 + 28*w^2 + 6*w - 7],\ [929, 929, -w^5 + 4*w^4 + 2*w^3 - 14*w^2 - 2*w + 8],\ [929, 929, -2*w^5 + 5*w^4 + 5*w^3 - 14*w^2 + w + 5],\ [937, 937, -w^5 + 2*w^4 + 6*w^3 - 9*w^2 - 8*w + 5],\ [937, 937, w^5 - 3*w^4 - 4*w^3 + 11*w^2 + 5*w - 5],\ [941, 941, -w^5 + 2*w^4 + 4*w^3 - 5*w^2 - 5*w + 5],\ [941, 941, -w^5 + 3*w^4 + 2*w^3 - 9*w^2],\ [947, 947, 3*w^5 - 6*w^4 - 12*w^3 + 16*w^2 + 12*w - 6],\ [947, 947, -2*w^5 + 5*w^4 + 6*w^3 - 13*w^2 - 5*w + 6],\ [947, 947, 2*w^4 - 4*w^3 - 5*w^2 + 6*w + 1],\ [947, 947, -3*w^5 + 9*w^4 + 6*w^3 - 26*w^2 + w + 7],\ [953, 953, -3*w^5 + 6*w^4 + 12*w^3 - 18*w^2 - 10*w + 10],\ [953, 953, 2*w^5 - 7*w^4 - 3*w^3 + 22*w^2 - w - 7],\ [971, 971, -w^5 + w^4 + 5*w^3 - 7*w - 3],\ [971, 971, -w^5 + 4*w^4 - w^3 - 11*w^2 + 7*w + 5],\ [997, 997, -4*w^5 + 9*w^4 + 14*w^3 - 25*w^2 - 12*w + 10],\ [997, 997, -3*w^5 + 8*w^4 + 9*w^3 - 23*w^2 - 6*w + 5],\ [997, 997, -2*w^4 + 4*w^3 + 6*w^2 - 9*w - 1],\ [997, 997, 4*w^5 - 11*w^4 - 10*w^3 + 31*w^2 + 4*w - 8],\ [1021, 1021, 3*w^5 - 7*w^4 - 11*w^3 + 23*w^2 + 10*w - 15],\ [1021, 1021, -4*w^5 + 11*w^4 + 10*w^3 - 32*w^2 - 4*w + 12],\ [1031, 1031, -2*w^5 + 6*w^4 + 4*w^3 - 17*w^2 - 2*w + 6],\ [1031, 1031, 2*w^5 - 4*w^4 - 8*w^3 + 11*w^2 + 10*w - 5],\ [1039, 1039, -w^4 + 7*w^2 + 2*w - 6],\ [1039, 1039, w^5 - 4*w^4 + w^3 + 10*w^2 - 7*w - 4],\ [1039, 1039, w^5 - 2*w^4 - 3*w^3 + 5*w^2 + 2*w - 4],\ [1039, 1039, -3*w^5 + 8*w^4 + 7*w^3 - 21*w^2 + w + 7],\ [1051, 1051, -w^4 + 3*w^3 + 2*w^2 - 5*w - 2],\ [1051, 1051, w^4 - 2*w^3 - 2*w^2 + 4*w - 4],\ [1061, 1061, 4*w^5 - 10*w^4 - 11*w^3 + 26*w^2 + 4*w - 6],\ [1061, 1061, 4*w^5 - 10*w^4 - 11*w^3 + 27*w^2 + 3*w - 7],\ [1087, 1087, 2*w^5 - 5*w^4 - 6*w^3 + 14*w^2 + w - 4],\ [1087, 1087, 2*w^5 - 5*w^4 - 6*w^3 + 14*w^2 + w - 2],\ [1097, 1097, -2*w^5 + 6*w^4 + 4*w^3 - 17*w^2 - w + 4],\ [1097, 1097, -2*w^5 + 4*w^4 + 8*w^3 - 11*w^2 - 9*w + 6],\ [1103, 1103, 4*w^5 - 10*w^4 - 13*w^3 + 30*w^2 + 8*w - 10],\ [1103, 1103, 4*w^5 - 10*w^4 - 13*w^3 + 29*w^2 + 9*w - 9],\ [1129, 1129, 3*w^5 - 9*w^4 - 4*w^3 + 24*w^2 - 8*w - 7],\ [1129, 1129, 3*w^5 - 6*w^4 - 10*w^3 + 12*w^2 + 7*w + 1],\ [1181, 1181, 4*w^5 - 10*w^4 - 14*w^3 + 32*w^2 + 9*w - 13],\ [1181, 1181, 3*w^5 - 7*w^4 - 12*w^3 + 22*w^2 + 13*w - 7],\ [1181, 1181, -4*w^5 + 10*w^4 + 12*w^3 - 29*w^2 - 5*w + 9],\ [1181, 1181, 4*w^5 - 10*w^4 - 12*w^3 + 27*w^2 + 7*w - 7],\ [1181, 1181, 3*w^5 - 8*w^4 - 10*w^3 + 26*w^2 + 8*w - 12],\ [1181, 1181, -5*w^5 + 13*w^4 + 16*w^3 - 40*w^2 - 11*w + 19],\ [1187, 1187, 2*w^5 - 7*w^4 - 4*w^3 + 22*w^2 + w - 11],\ [1187, 1187, 3*w^5 - 7*w^4 - 10*w^3 + 21*w^2 + 7*w - 12],\ [1187, 1187, 2*w^5 - 4*w^4 - 7*w^3 + 11*w^2 + 4*w - 7],\ [1187, 1187, 6*w^5 - 14*w^4 - 22*w^3 + 41*w^2 + 21*w - 14],\ [1217, 1217, 3*w^5 - 6*w^4 - 12*w^3 + 17*w^2 + 12*w - 7],\ [1217, 1217, 3*w^5 - 9*w^4 - 6*w^3 + 25*w^2 + w - 7],\ [1229, 1229, -2*w^5 + 5*w^4 + 8*w^3 - 17*w^2 - 8*w + 8],\ [1229, 1229, -2*w^5 + 5*w^4 + 8*w^3 - 17*w^2 - 8*w + 6],\ [1231, 1231, -6*w^5 + 16*w^4 + 17*w^3 - 47*w^2 - 8*w + 18],\ [1231, 1231, -6*w^5 + 14*w^4 + 21*w^3 - 40*w^2 - 17*w + 10],\ [1237, 1237, -3*w^5 + 8*w^4 + 8*w^3 - 22*w^2 - 2*w + 6],\ [1237, 1237, 3*w^5 - 7*w^4 - 10*w^3 + 20*w^2 + 5*w - 5],\ [1279, 1279, 3*w^5 - 8*w^4 - 8*w^3 + 23*w^2 + w - 6],\ [1279, 1279, 2*w^5 - 5*w^4 - 7*w^3 + 16*w^2 + 8*w - 9],\ [1291, 1291, -2*w^5 + 4*w^4 + 9*w^3 - 13*w^2 - 9*w + 8],\ [1291, 1291, -2*w^5 + 6*w^4 + 5*w^3 - 18*w^2 - 2*w + 3],\ [1297, 1297, w^5 - 2*w^4 - 4*w^3 + 7*w^2 + w - 6],\ [1297, 1297, -w^5 + 3*w^4 + 2*w^3 - 7*w^2 - 3],\ [1303, 1303, -w^4 + w^3 + 6*w^2 - 3*w - 3],\ [1303, 1303, -w^4 + 3*w^3 + 3*w^2 - 8*w],\ [1319, 1319, -2*w^5 + 5*w^4 + 7*w^3 - 16*w^2 - 8*w + 7],\ [1319, 1319, 2*w^5 - 5*w^4 - 7*w^3 + 15*w^2 + 9*w - 7],\ [1369, 37, 2*w^5 - 6*w^4 - 5*w^3 + 19*w^2 + 4*w - 13],\ [1369, 37, 2*w^5 - 4*w^4 - 9*w^3 + 12*w^2 + 13*w - 1],\ [1373, 1373, w^5 - 3*w^4 - 4*w^3 + 10*w^2 + 6*w - 6],\ [1373, 1373, 2*w^5 - 5*w^4 - 8*w^3 + 17*w^2 + 9*w - 8],\ [1373, 1373, 2*w^5 - 5*w^4 - 8*w^3 + 17*w^2 + 9*w - 7],\ [1373, 1373, 4*w^5 - 10*w^4 - 13*w^3 + 31*w^2 + 8*w - 17],\ [1381, 1381, -w^4 + 3*w^3 + 2*w^2 - 5*w - 1],\ [1381, 1381, w^5 - 2*w^4 - 5*w^3 + 5*w^2 + 9*w + 1],\ [1423, 1423, -3*w^5 + 8*w^4 + 9*w^3 - 24*w^2 - 5*w + 6],\ [1423, 1423, 3*w^5 - 7*w^4 - 11*w^3 + 21*w^2 + 9*w - 9],\ [1427, 1427, -w^5 + 3*w^4 + 2*w^3 - 8*w^2 - 2],\ [1427, 1427, w^5 - 2*w^4 - 4*w^3 + 6*w^2 + 3*w - 6],\ [1433, 1433, w^5 - w^4 - 7*w^3 + 4*w^2 + 9*w - 1],\ [1433, 1433, w^5 - 4*w^4 - w^3 + 13*w^2 - 3*w - 5],\ [1439, 1439, 3*w^5 - 7*w^4 - 9*w^3 + 17*w^2 + 5*w - 4],\ [1439, 1439, 4*w^5 - 11*w^4 - 11*w^3 + 32*w^2 + 6*w - 9],\ [1451, 1451, -w^5 + w^4 + 6*w^3 - w^2 - 9*w - 2],\ [1451, 1451, -w^5 + 4*w^4 - 13*w^2 + 6*w + 6],\ [1471, 1471, 5*w^5 - 14*w^4 - 15*w^3 + 46*w^2 + 11*w - 23],\ [1471, 1471, -5*w^5 + 14*w^4 + 11*w^3 - 38*w^2 + w + 11],\ [1489, 1489, 4*w^5 - 10*w^4 - 13*w^3 + 30*w^2 + 10*w - 9],\ [1489, 1489, 4*w^5 - 10*w^4 - 13*w^3 + 29*w^2 + 11*w - 12],\ [1499, 1499, w^5 - 3*w^4 - w^3 + 7*w^2 - 4*w - 3],\ [1499, 1499, -w^5 + 2*w^4 + 3*w^3 - 4*w^2 - 3],\ [1511, 1511, 4*w^5 - 12*w^4 - 9*w^3 + 35*w^2 + 2*w - 13],\ [1511, 1511, -2*w^5 + 6*w^4 + 3*w^3 - 18*w^2 + 4*w + 10],\ [1549, 1549, -2*w^5 + 6*w^4 + 4*w^3 - 16*w^2 + 3*w + 1],\ [1549, 1549, 2*w^5 - 4*w^4 - 8*w^3 + 12*w^2 + 3*w - 4],\ [1579, 1579, 3*w^5 - 9*w^4 - 6*w^3 + 24*w^2 - w - 6],\ [1579, 1579, 5*w^5 - 12*w^4 - 15*w^3 + 32*w^2 + 7*w - 13],\ [1607, 1607, -w^5 + 4*w^4 - 3*w^3 - 7*w^2 + 12*w - 2],\ [1607, 1607, -w^5 + w^4 + 3*w^3 + 2*w^2 - 3],\ [1613, 1613, -w^4 + 3*w^3 + w^2 - 6*w - 2],\ [1613, 1613, -w^4 + w^3 + 4*w^2 - w - 5],\ [1619, 1619, 4*w^5 - 10*w^4 - 12*w^3 + 29*w^2 + 6*w - 8],\ [1619, 1619, -2*w^5 + 6*w^4 + 3*w^3 - 17*w^2 + 5*w + 6],\ [1627, 1627, 4*w^5 - 11*w^4 - 9*w^3 + 29*w^2 - w - 4],\ [1627, 1627, 4*w^5 - 11*w^4 - 12*w^3 + 35*w^2 + 10*w - 18],\ [1637, 1637, w^5 - 4*w^4 + w^3 + 11*w^2 - 7*w - 4],\ [1637, 1637, w^5 - w^4 - 5*w^3 + 7*w + 2],\ [1657, 1657, 4*w^5 - 10*w^4 - 14*w^3 + 31*w^2 + 15*w - 15],\ [1657, 1657, -4*w^5 + 10*w^4 + 14*w^3 - 31*w^2 - 15*w + 11],\ [1681, 41, 2*w^4 - 4*w^3 - 7*w^2 + 9*w + 3],\ [1693, 1693, w^5 - 4*w^4 - w^3 + 15*w^2 - w - 10],\ [1693, 1693, -5*w^5 + 12*w^4 + 17*w^3 - 35*w^2 - 16*w + 11],\ [1697, 1697, -7*w^5 + 17*w^4 + 21*w^3 - 48*w^2 - 9*w + 19],\ [1697, 1697, 2*w^5 - 5*w^4 - 5*w^3 + 13*w^2 - w - 7],\ [1709, 1709, -3*w^5 + 9*w^4 + 7*w^3 - 27*w^2 - 3*w + 14],\ [1709, 1709, -4*w^5 + 10*w^4 + 11*w^3 - 30*w^2 - 3*w + 14],\ [1721, 1721, -w^5 + 9*w^3 - 17*w + 2],\ [1721, 1721, -w^5 + 5*w^4 - w^3 - 17*w^2 + 5*w + 7],\ [1723, 1723, -w^5 + 2*w^4 + 2*w^3 - 2*w^2 + w - 3],\ [1723, 1723, w^5 - 3*w^4 + 6*w^2 - 6*w - 1],\ [1759, 1759, w^4 - 4*w^3 - w^2 + 8*w - 1],\ [1759, 1759, -3*w^5 + 9*w^4 + 7*w^3 - 27*w^2 - 4*w + 14],\ [1777, 1777, 3*w^5 - 9*w^4 - 7*w^3 + 29*w^2 - 13],\ [1777, 1777, 3*w^5 - 8*w^4 - 8*w^3 + 26*w^2 + 3*w - 13],\ [1777, 1777, -3*w^5 + 9*w^4 + 6*w^3 - 25*w^2 + 11],\ [1777, 1777, 3*w^5 - 6*w^4 - 13*w^3 + 16*w^2 + 16*w - 3],\ [1783, 1783, 3*w^5 - 7*w^4 - 10*w^3 + 19*w^2 + 10*w - 8],\ [1783, 1783, -6*w^5 + 15*w^4 + 17*w^3 - 42*w^2 - 7*w + 15],\ [1783, 1783, 7*w^5 - 18*w^4 - 20*w^3 + 52*w^2 + 8*w - 20],\ [1783, 1783, -3*w^5 + 8*w^4 + 8*w^3 - 23*w^2 - 5*w + 7],\ [1801, 1801, -2*w^5 + 6*w^4 + 3*w^3 - 15*w^2 + 4*w + 4],\ [1801, 1801, -2*w^5 + 4*w^4 + 7*w^3 - 10*w^2 - 3*w],\ [1811, 1811, -2*w^5 + 4*w^4 + 8*w^3 - 12*w^2 - 9*w + 7],\ [1811, 1811, 3*w^5 - 10*w^4 - 7*w^3 + 33*w^2 + 3*w - 14],\ [1823, 1823, w^5 - 2*w^4 - 5*w^3 + 7*w^2 + 9*w - 5],\ [1823, 1823, -w^5 + 3*w^4 + 3*w^3 - 10*w^2 - 5*w + 5],\ [1847, 1847, w^5 - 3*w^4 - 2*w^3 + 8*w^2 + w - 6],\ [1847, 1847, -w^5 + 2*w^4 + 4*w^3 - 6*w^2 - 4*w - 1],\ [1849, 43, -2*w^4 + 4*w^3 + 6*w^2 - 8*w - 3],\ [1901, 1901, -5*w^5 + 13*w^4 + 13*w^3 - 37*w^2 - 3*w + 13],\ [1901, 1901, -5*w^5 + 12*w^4 + 15*w^3 - 30*w^2 - 11*w + 6],\ [1913, 1913, 3*w^5 - 9*w^4 - 7*w^3 + 29*w^2 + w - 13],\ [1913, 1913, -w^5 + 4*w^4 + w^3 - 12*w^2 + 3*w + 1],\ [1913, 1913, 2*w^5 - 3*w^4 - 10*w^3 + 7*w^2 + 14*w + 1],\ [1913, 1913, -2*w^4 + 2*w^3 + 9*w^2 + w - 5],\ [1931, 1931, -3*w^5 + 6*w^4 + 13*w^3 - 16*w^2 - 17*w + 5],\ [1931, 1931, 4*w^5 - 9*w^4 - 15*w^3 + 26*w^2 + 17*w - 10],\ [1949, 1949, -3*w^5 + 8*w^4 + 9*w^3 - 23*w^2 - 10*w + 10],\ [1949, 1949, 2*w^5 - 7*w^4 - 3*w^3 + 20*w^2 - w - 8],\ [1973, 1973, -2*w^5 + 6*w^4 + 5*w^3 - 17*w^2 - 3*w + 9],\ [1973, 1973, -5*w^5 + 13*w^4 + 13*w^3 - 34*w^2 - 7*w + 11],\ [1979, 1979, 6*w^5 - 15*w^4 - 18*w^3 + 43*w^2 + 8*w - 13],\ [1979, 1979, -2*w^5 + 3*w^4 + 11*w^3 - 11*w^2 - 12*w + 7],\ [1987, 1987, -5*w^5 + 13*w^4 + 13*w^3 - 35*w^2 - 3*w + 7],\ [1987, 1987, -w^5 + 2*w^4 + 6*w^3 - 6*w^2 - 11*w]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^2 + 4*x - 4 K. = NumberField(heckePol) hecke_eigenvalues_array = [0, 0, 1, -e, e, -2*e - 10, 2*e + 10, e, -e, -2*e - 6, e, -e, -2*e - 6, 3*e + 11, e - 6, -e + 6, 4*e + 4, 4*e + 4, -4*e - 14, -6*e - 14, 6*e + 14, -12, -12, 2*e, 2*e, -e + 12, -2*e - 10, -2*e - 10, e - 12, -e, e, -e + 18, e - 18, 2*e - 12, -2*e + 12, 3*e + 22, -3*e - 22, -8*e - 14, 8*e + 14, -10, 4*e + 10, -4*e - 10, -10, -6*e - 8, -6*e - 8, -4*e - 14, -5*e - 20, 5*e + 20, -4*e - 14, -2*e + 2, -2*e + 2, -6*e - 16, -6*e - 16, -5*e - 10, 5*e + 10, 6*e + 22, 6*e + 22, 7*e + 20, -7*e - 20, 2*e - 12, 2*e - 12, -3*e, 3*e, -6*e - 20, -6*e - 20, -8*e - 18, -8*e - 18, 34, 8*e + 16, 8*e + 16, 4*e + 8, 4*e + 8, 5*e + 26, -5*e - 26, -2*e - 36, -2*e - 36, -4*e + 6, -4*e + 6, 9*e + 18, -9*e - 18, 8*e + 12, 8*e + 12, -4*e - 2, -4*e - 2, 10*e + 14, 10*e + 14, 4*e + 10, -4*e - 10, -6*e - 18, -6*e - 18, 6*e + 8, 10*e + 8, 2*e + 32, -2*e - 32, 10*e + 8, -6*e - 8, 4*e - 30, -e + 22, e - 22, -6*e - 6, -6*e - 6, -2*e + 8, 12, 12, -2*e + 8, 2*e + 10, 6*e - 14, 6*e - 14, 2*e + 10, 6*e - 4, -6*e + 4, 8*e + 14, -8*e - 14, 4*e - 4, 4*e - 4, 3*e + 32, -3*e - 32, 9*e + 30, -9*e - 30, 10*e + 6, -10*e - 6, -12*e - 22, -14*e - 30, -14*e - 30, -12*e - 22, -11*e - 6, 11*e + 6, e + 38, -e - 38, 14*e + 38, -14*e - 38, -e, e, 12*e + 8, 2*e + 24, 2*e + 24, 12*e + 8, 5*e - 30, 2*e + 32, -2*e - 32, -5*e + 30, 14*e + 34, 14*e + 34, 2*e - 26, 2*e - 26, -7*e - 2, 7*e + 2, 2*e - 40, -5*e - 22, 3*e + 10, -3*e - 10, 5*e + 22, 2*e - 40, -6*e - 32, -6*e - 32, 11*e + 8, -11*e - 8, -2*e + 2, -2*e + 2, 38, 38, 4*e + 16, -12*e - 28, -12*e - 28, 4*e + 16, -2*e - 22, -2*e - 22, 7*e + 18, -7*e - 18, -2*e - 6, 6*e + 46, -6*e - 46, 2*e + 6, 15*e + 40, -15*e - 40, -3*e + 14, 3*e - 14, -2*e - 44, -14*e - 28, 14*e + 28, 2*e + 44, 7*e + 54, -7*e - 54, -4*e + 30, 4*e - 30, -4*e - 24, -4*e - 24, -4*e + 2, 4*e - 2, -16*e - 48, -16*e - 48, 7*e + 32, -7*e - 32, -14*e - 38, 8*e + 38, -16*e - 14, 16*e + 14, 8*e + 38, 14*e + 38, -5*e - 38, -3*e - 6, 3*e + 6, 5*e + 38, -3*e + 36, 3*e - 36, -2*e - 22, -2*e - 22, -2*e - 28, -2*e - 28, 22, 22, -8*e - 20, -8*e - 20, -e + 50, e - 50, -15*e - 28, 15*e + 28, 4*e + 32, 4*e + 32, -14*e - 28, -14*e - 28, 6*e + 50, -6*e - 50, 6*e + 26, 14*e + 58, 14*e + 58, -6*e - 26, 7*e + 52, -7*e - 52, -3*e + 22, 3*e - 22, 16*e + 28, 16*e + 28, 10*e - 10, -10*e + 10, -14*e - 52, 14*e + 52, -14*e - 24, -14*e - 24, -6*e + 12, 6*e - 12, -6*e - 2, -6*e - 2, -4*e + 4, 4*e - 4, 6*e + 20, 6*e + 20, 14*e + 42, 14*e + 42, 13*e + 50, -13*e - 50, 2*e - 60, 2*e - 60, -26, 26, -17*e - 10, 17*e + 10, 4*e + 4, 4*e + 4, 4*e - 30, -4*e + 30, 14*e + 34, 14*e + 34, -24*e - 54, 4*e - 2, 4*e - 2, 10*e + 22, -10*e - 22, 23*e + 56, -23*e - 56, -20*e - 54, -20*e - 54, -4*e - 28, -4*e - 28, 9*e + 26, -9*e - 26, 22*e + 62, -8*e - 46, 8*e + 46, 22*e + 62, -23*e - 58, 10*e + 4, 10*e + 4, 23*e + 58, 18*e + 10, -18*e - 10, -12*e - 16, 12*e + 16, 24*e + 56, 24*e + 56, -10*e - 60, -10*e - 60, 24*e + 50, -23*e - 48, 23*e + 48, -12*e - 38, 5*e + 20, -5*e - 20, -12*e - 38, 6*e - 40, 6*e - 40, 8*e - 18, 8*e - 18, 24*e + 50, 24*e + 50, 6*e + 16, 6*e + 16, 18*e + 60, -18*e - 60] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([25, 5, -w^3 + 2*w^2 + 2*w - 3])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]