/* 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, -6, 3, 10, -3, -3, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([51, 51, -w^5 + 3*w^4 + 2*w^3 - 7*w^2 - 3*w]) primes_array = [ [3, 3, w - 1],\ [17, 17, -w^2 + 2*w + 1],\ [17, 17, -w^3 + w^2 + 4*w],\ [19, 19, w^5 - 3*w^4 - 2*w^3 + 8*w^2 + w - 4],\ [19, 19, w^2 - w - 1],\ [37, 37, w^4 - 2*w^3 - 3*w^2 + 3*w + 2],\ [37, 37, w^4 - 4*w^3 + w^2 + 7*w - 3],\ [53, 53, 2*w^5 - 6*w^4 - 6*w^3 + 18*w^2 + 9*w - 6],\ [53, 53, -w^5 + 3*w^4 + 3*w^3 - 9*w^2 - 3*w + 2],\ [64, 2, -2],\ [71, 71, -w^5 + 4*w^4 - w^3 - 8*w^2 + 3*w - 1],\ [71, 71, 2*w^5 - 5*w^4 - 8*w^3 + 15*w^2 + 12*w - 6],\ [71, 71, 2*w^5 - 6*w^4 - 6*w^3 + 18*w^2 + 9*w - 7],\ [73, 73, -2*w^5 + 6*w^4 + 5*w^3 - 16*w^2 - 7*w + 3],\ [73, 73, -2*w^5 + 6*w^4 + 5*w^3 - 17*w^2 - 6*w + 6],\ [89, 89, w^5 - 2*w^4 - 5*w^3 + 6*w^2 + 8*w - 4],\ [89, 89, w^5 - w^4 - 9*w^3 + 7*w^2 + 16*w - 6],\ [89, 89, 2*w^5 - 6*w^4 - 4*w^3 + 15*w^2 + 3*w - 3],\ [89, 89, w^5 - 2*w^4 - 6*w^3 + 8*w^2 + 10*w - 6],\ [107, 107, w^5 - 2*w^4 - 7*w^3 + 10*w^2 + 12*w - 6],\ [107, 107, 2*w^5 - 5*w^4 - 7*w^3 + 14*w^2 + 9*w - 5],\ [107, 107, w^5 - 3*w^4 - 3*w^3 + 9*w^2 + 6*w - 3],\ [107, 107, w^4 - 2*w^3 - 3*w^2 + 4*w + 1],\ [109, 109, 2*w^5 - 5*w^4 - 8*w^3 + 15*w^2 + 13*w - 6],\ [109, 109, -w^4 + 4*w^3 - w^2 - 9*w + 3],\ [127, 127, w^4 - 3*w^3 + 5*w - 4],\ [127, 127, -w^5 + 3*w^4 + w^3 - 6*w^2 + w + 3],\ [127, 127, 2*w^5 - 5*w^4 - 8*w^3 + 16*w^2 + 11*w - 6],\ [127, 127, w^5 - 4*w^4 + 2*w^3 + 7*w^2 - 7*w],\ [163, 163, -w^5 + 2*w^4 + 6*w^3 - 8*w^2 - 9*w + 5],\ [163, 163, -w^5 + 3*w^4 + 3*w^3 - 10*w^2 - 2*w + 5],\ [163, 163, -w^5 + 3*w^4 + 4*w^3 - 10*w^2 - 8*w + 4],\ [163, 163, w^5 - 4*w^4 + 10*w^2 - w - 4],\ [179, 179, -w^5 + 3*w^4 + 3*w^3 - 7*w^2 - 7*w - 1],\ [179, 179, 2*w^5 - 6*w^4 - 5*w^3 + 18*w^2 + 5*w - 7],\ [179, 179, -2*w^5 + 5*w^4 + 10*w^3 - 19*w^2 - 18*w + 10],\ [179, 179, -2*w^5 + 5*w^4 + 9*w^3 - 17*w^2 - 14*w + 9],\ [197, 197, 2*w^5 - 6*w^4 - 5*w^3 + 18*w^2 + 3*w - 7],\ [197, 197, -w^5 + 4*w^4 - w^3 - 7*w^2 + 3*w - 2],\ [197, 197, w^5 - w^4 - 7*w^3 + 4*w^2 + 10*w - 3],\ [197, 197, -2*w^5 + 8*w^4 - 2*w^3 - 17*w^2 + 10*w + 1],\ [197, 197, -w^4 + 3*w^3 + 2*w^2 - 6*w - 2],\ [197, 197, w^5 - 4*w^4 + w^3 + 9*w^2 - 4*w - 2],\ [199, 199, -w^5 + 2*w^4 + 6*w^3 - 9*w^2 - 10*w + 4],\ [199, 199, -w^4 + 3*w^3 + 2*w^2 - 9*w + 1],\ [233, 233, 2*w^5 - 4*w^4 - 12*w^3 + 17*w^2 + 19*w - 9],\ [233, 233, -w^5 + w^4 + 8*w^3 - 6*w^2 - 13*w + 6],\ [233, 233, 2*w^5 - 6*w^4 - 6*w^3 + 19*w^2 + 9*w - 7],\ [233, 233, w^5 - 4*w^4 + 11*w^2 - 4*w - 2],\ [251, 251, -2*w^5 + 5*w^4 + 10*w^3 - 18*w^2 - 18*w + 7],\ [251, 251, w^3 - w^2 - 3*w - 2],\ [251, 251, w^5 - 3*w^4 - 3*w^3 + 10*w^2 + 3*w - 3],\ [251, 251, -w^5 + 4*w^4 - 11*w^2 + 4*w + 3],\ [269, 269, w^5 - 3*w^4 - w^3 + 6*w^2 - 2*w - 3],\ [269, 269, 2*w^5 - 7*w^4 - 3*w^3 + 20*w^2 + 3*w - 8],\ [271, 271, w^5 - 3*w^4 - 3*w^3 + 11*w^2 + 2*w - 7],\ [271, 271, -w^4 + 4*w^3 - w^2 - 10*w + 6],\ [289, 17, -w^5 + 3*w^4 + w^3 - 5*w^2 - 2],\ [289, 17, w^5 - 3*w^4 - 4*w^3 + 11*w^2 + 6*w - 4],\ [307, 307, 2*w^5 - 5*w^4 - 10*w^3 + 19*w^2 + 16*w - 8],\ [307, 307, w^5 - 4*w^4 + 2*w^3 + 7*w^2 - 6*w - 1],\ [307, 307, -w^5 + 2*w^4 + 4*w^3 - 6*w^2 - 3*w + 2],\ [307, 307, w^5 - 2*w^4 - 6*w^3 + 10*w^2 + 7*w - 9],\ [361, 19, 2*w^5 - 6*w^4 - 6*w^3 + 18*w^2 + 8*w - 5],\ [361, 19, w^5 - 3*w^4 - w^3 + 5*w^2 + 3],\ [379, 379, w^5 - 2*w^4 - 5*w^3 + 6*w^2 + 9*w - 5],\ [379, 379, -2*w + 3],\ [397, 397, -w^5 + 5*w^4 - 6*w^3 - 6*w^2 + 14*w - 5],\ [397, 397, -w^5 + 3*w^4 + w^3 - 4*w^2 - w - 5],\ [397, 397, 3*w^5 - 9*w^4 - 7*w^3 + 23*w^2 + 9*w - 6],\ [397, 397, -w^4 + 5*w^3 - 2*w^2 - 10*w + 1],\ [431, 431, -3*w^5 + 8*w^4 + 11*w^3 - 24*w^2 - 18*w + 9],\ [431, 431, 2*w^5 - 6*w^4 - 3*w^3 + 13*w^2 + w - 3],\ [431, 431, -3*w^5 + 8*w^4 + 10*w^3 - 23*w^2 - 14*w + 8],\ [431, 431, -2*w^5 + 6*w^4 + 4*w^3 - 14*w^2 - 5*w],\ [433, 433, -3*w^5 + 9*w^4 + 8*w^3 - 25*w^2 - 11*w + 9],\ [433, 433, -2*w^5 + 6*w^4 + 4*w^3 - 16*w^2 - 3*w + 7],\ [433, 433, 2*w^5 - 5*w^4 - 7*w^3 + 13*w^2 + 11*w - 4],\ [433, 433, -w^4 + 4*w^3 + w^2 - 10*w - 2],\ [449, 449, -w^5 + 3*w^4 + 2*w^3 - 9*w^2 + w + 3],\ [449, 449, -w^4 + 4*w^3 - 2*w^2 - 5*w + 5],\ [449, 449, -w^5 + 4*w^4 - w^3 - 8*w^2 + 3*w - 2],\ [449, 449, -w^3 + 5*w + 3],\ [449, 449, w^3 - 5*w],\ [449, 449, w^5 - 3*w^4 - 2*w^3 + 9*w^2 - w - 6],\ [467, 467, -2*w^4 + 7*w^3 - 16*w + 7],\ [467, 467, -2*w^4 + 6*w^3 + w^2 - 10*w + 3],\ [487, 487, -w^5 + 2*w^4 + 6*w^3 - 10*w^2 - 8*w + 9],\ [487, 487, 2*w^5 - 6*w^4 - 5*w^3 + 15*w^2 + 7*w - 3],\ [487, 487, 2*w^5 - 5*w^4 - 7*w^3 + 15*w^2 + 8*w - 6],\ [487, 487, -w^5 + 2*w^4 + 6*w^3 - 8*w^2 - 8*w + 1],\ [503, 503, 2*w^5 - 7*w^4 - 3*w^3 + 19*w^2 + 3*w - 6],\ [503, 503, 2*w^5 - 6*w^4 - 5*w^3 + 18*w^2 + 4*w - 6],\ [503, 503, 2*w^5 - 5*w^4 - 9*w^3 + 17*w^2 + 13*w - 7],\ [503, 503, -w^5 + 4*w^4 - 10*w^2 + 3*w - 1],\ [521, 521, w^5 - 4*w^4 + 11*w^2 - w - 6],\ [521, 521, w^5 - w^4 - 9*w^3 + 8*w^2 + 15*w - 6],\ [521, 521, -w^5 + 4*w^4 - 2*w^3 - 8*w^2 + 9*w],\ [521, 521, -2*w^5 + 4*w^4 + 11*w^3 - 15*w^2 - 17*w + 9],\ [523, 523, 3*w^5 - 8*w^4 - 10*w^3 + 24*w^2 + 12*w - 10],\ [523, 523, -w^4 + 3*w^3 - w^2 - 4*w + 5],\ [523, 523, -2*w^5 + 4*w^4 + 11*w^3 - 14*w^2 - 17*w + 7],\ [523, 523, -w^5 + 5*w^4 - 4*w^3 - 9*w^2 + 9*w + 1],\ [577, 577, -3*w^5 + 8*w^4 + 10*w^3 - 22*w^2 - 14*w + 7],\ [577, 577, -2*w^5 + 7*w^4 + w^3 - 15*w^2 + 2*w + 2],\ [593, 593, 2*w^5 - 5*w^4 - 9*w^3 + 17*w^2 + 15*w - 10],\ [593, 593, w^3 - w^2 - 2*w - 2],\ [593, 593, -3*w^5 + 9*w^4 + 9*w^3 - 27*w^2 - 14*w + 10],\ [593, 593, 2*w^5 - 6*w^4 - 6*w^3 + 19*w^2 + 8*w - 7],\ [613, 613, w^5 - 2*w^4 - 7*w^3 + 9*w^2 + 12*w - 5],\ [613, 613, w^5 - w^4 - 9*w^3 + 8*w^2 + 13*w - 7],\ [613, 613, w^5 - 5*w^4 + 3*w^3 + 11*w^2 - 6*w - 2],\ [613, 613, -w^4 + 3*w^3 + w^2 - 5*w + 4],\ [613, 613, w^5 - w^4 - 9*w^3 + 7*w^2 + 14*w - 4],\ [613, 613, w^5 - 2*w^4 - 7*w^3 + 11*w^2 + 12*w - 8],\ [631, 631, 2*w^5 - 7*w^4 + 15*w^2 - 6*w - 2],\ [631, 631, -3*w^5 + 8*w^4 + 10*w^3 - 23*w^2 - 12*w + 10],\ [631, 631, w^5 - 3*w^4 + w^3 + 3*w^2 - 6*w + 3],\ [631, 631, -3*w^5 + 8*w^4 + 13*w^3 - 29*w^2 - 21*w + 15],\ [683, 683, -w^5 + 4*w^4 - w^3 - 8*w^2 + 2*w - 1],\ [683, 683, w^5 - 2*w^4 - 6*w^3 + 7*w^2 + 11*w - 4],\ [701, 701, -w^5 + 3*w^4 + w^3 - 7*w^2 + 3*w + 2],\ [701, 701, 2*w^2 - 3*w - 4],\ [719, 719, 2*w^5 - 5*w^4 - 8*w^3 + 16*w^2 + 13*w - 8],\ [719, 719, w^5 - 5*w^4 + 4*w^3 + 10*w^2 - 10*w - 1],\ [719, 719, w^5 - w^4 - 9*w^3 + 6*w^2 + 16*w - 2],\ [719, 719, 2*w^5 - 5*w^4 - 9*w^3 + 18*w^2 + 15*w - 10],\ [739, 739, -2*w^5 + 9*w^4 - 6*w^3 - 15*w^2 + 16*w - 7],\ [739, 739, 4*w^5 - 9*w^4 - 19*w^3 + 30*w^2 + 28*w - 14],\ [757, 757, 2*w^5 - 5*w^4 - 9*w^3 + 16*w^2 + 17*w - 7],\ [757, 757, 2*w^5 - 5*w^4 - 7*w^3 + 15*w^2 + 6*w - 6],\ [757, 757, 2*w^5 - 6*w^4 - 7*w^3 + 20*w^2 + 11*w - 6],\ [757, 757, -2*w^4 + 5*w^3 + 4*w^2 - 10*w - 1],\ [773, 773, 2*w^5 - 8*w^4 + 3*w^3 + 16*w^2 - 11*w],\ [773, 773, w^5 - 2*w^4 - 6*w^3 + 8*w^2 + 12*w - 6],\ [809, 809, 2*w^5 - 6*w^4 - 3*w^3 + 13*w^2 + 2*w - 3],\ [809, 809, -2*w^3 + 5*w^2 + 4*w - 6],\ [811, 811, -w^4 + 3*w^3 + 3*w^2 - 7*w - 3],\ [811, 811, 2*w^5 - 7*w^4 - w^3 + 17*w^2 - 3*w - 6],\ [827, 827, w^5 - 4*w^4 + w^3 + 10*w^2 - 5*w - 4],\ [827, 827, w^5 - 4*w^4 + w^3 + 10*w^2 - 5*w - 5],\ [829, 829, 2*w^5 - 4*w^4 - 11*w^3 + 15*w^2 + 18*w - 9],\ [829, 829, 2*w^5 - 6*w^4 - 7*w^3 + 19*w^2 + 13*w - 7],\ [829, 829, -2*w^5 + 9*w^4 - 5*w^3 - 18*w^2 + 15*w],\ [829, 829, 3*w^5 - 9*w^4 - 9*w^3 + 28*w^2 + 11*w - 11],\ [829, 829, w^3 - 2*w^2 - 3*w - 1],\ [829, 829, -2*w^5 + 4*w^4 + 12*w^3 - 17*w^2 - 20*w + 10],\ [863, 863, -w^4 + 4*w^3 + w^2 - 11*w],\ [863, 863, w^4 - 4*w^3 - w^2 + 11*w + 1],\ [919, 919, -w^5 + 5*w^4 - 3*w^3 - 11*w^2 + 4*w + 1],\ [919, 919, -w^5 + 4*w^4 - w^3 - 8*w^2 + 6*w - 4],\ [919, 919, -w^4 + 5*w^3 - w^2 - 12*w + 1],\ [919, 919, -2*w^5 + 5*w^4 + 7*w^3 - 13*w^2 - 11*w + 6],\ [937, 937, 2*w^5 - 6*w^4 - 3*w^3 + 12*w^2 + 3*w],\ [937, 937, 2*w^5 - 4*w^4 - 10*w^3 + 12*w^2 + 16*w - 5],\ [937, 937, w^5 - 4*w^4 + 8*w^2 + w + 1],\ [937, 937, 2*w^4 - 8*w^3 + 2*w^2 + 16*w - 5],\ [953, 953, -w^5 + 4*w^4 - 11*w^2 + 3*w + 7],\ [953, 953, w^5 - 2*w^4 - 7*w^3 + 9*w^2 + 13*w - 7],\ [953, 953, -w^5 + 3*w^4 + w^3 - 7*w^2 + 4*w + 1],\ [953, 953, 2*w^5 - 8*w^4 + 3*w^3 + 16*w^2 - 11*w - 1],\ [971, 971, w^4 - w^3 - 5*w^2 + 2*w + 4],\ [971, 971, -w^4 + w^3 + 4*w^2 - w + 1],\ [971, 971, -2*w^5 + 5*w^4 + 10*w^3 - 20*w^2 - 15*w + 9],\ [971, 971, w^4 - 2*w^3 - 4*w^2 + 6*w + 1],\ [991, 991, -w - 3],\ [991, 991, -2*w^5 + 7*w^4 + 3*w^3 - 19*w^2 - 5*w + 6],\ [991, 991, -w^5 + 3*w^4 + 3*w^3 - 9*w^2 - 5*w + 7],\ [991, 991, 2*w^5 - 7*w^4 - 2*w^3 + 16*w^2 + 2*w],\ [1009, 1009, 2*w^4 - 6*w^3 - 2*w^2 + 11*w],\ [1009, 1009, -w^5 + w^4 + 9*w^3 - 7*w^2 - 15*w + 6],\ [1061, 1061, -w^5 + 3*w^4 + 3*w^3 - 10*w^2 - 4*w + 2],\ [1061, 1061, w^5 - 3*w^4 - 2*w^3 + 8*w^2 + 2*w - 7],\ [1063, 1063, 2*w^5 - 5*w^4 - 10*w^3 + 18*w^2 + 18*w - 9],\ [1063, 1063, -2*w^5 + 5*w^4 + 8*w^3 - 15*w^2 - 14*w + 8],\ [1063, 1063, -w^5 + 4*w^4 - 11*w^2 + 4*w + 5],\ [1063, 1063, w^5 - 2*w^4 - 7*w^3 + 10*w^2 + 14*w - 5],\ [1117, 1117, -2*w^5 + 6*w^4 + 5*w^3 - 17*w^2 - 7*w + 5],\ [1117, 1117, -w^5 + 2*w^4 + 3*w^3 - 2*w^2 - 3*w - 3],\ [1117, 1117, w^5 - 3*w^4 - 3*w^3 + 10*w^2 + 5*w - 8],\ [1117, 1117, w^5 - 4*w^4 + 3*w^3 + 4*w^2 - 7*w + 4],\ [1151, 1151, -w^4 + 4*w^3 - 10*w - 1],\ [1151, 1151, w^5 - 3*w^4 - w^3 + 8*w^2 - 3*w - 3],\ [1151, 1151, 3*w^5 - 8*w^4 - 8*w^3 + 19*w^2 + 8*w - 3],\ [1151, 1151, w^5 - 2*w^4 - 6*w^3 + 7*w^2 + 12*w - 5],\ [1171, 1171, w^5 - 2*w^4 - 5*w^3 + 5*w^2 + 7*w + 1],\ [1171, 1171, 3*w^5 - 10*w^4 - 4*w^3 + 25*w^2 + 3*w - 7],\ [1171, 1171, -w^5 + 5*w^4 - 3*w^3 - 11*w^2 + 8*w + 3],\ [1171, 1171, -2*w^5 + 5*w^4 + 7*w^3 - 16*w^2 - 7*w + 6],\ [1187, 1187, -3*w^5 + 8*w^4 + 12*w^3 - 27*w^2 - 18*w + 11],\ [1187, 1187, w^4 - 2*w^3 - 2*w^2 + 4*w - 3],\ [1259, 1259, 2*w^5 - 6*w^4 - 5*w^3 + 16*w^2 + 9*w - 5],\ [1259, 1259, w^5 - 3*w^4 - 4*w^3 + 11*w^2 + 9*w - 7],\ [1277, 1277, -3*w^5 + 11*w^4 + w^3 - 25*w^2 + 2*w + 3],\ [1277, 1277, -w^4 + w^3 + 5*w^2 - w - 6],\ [1277, 1277, 2*w^5 - 6*w^4 - 4*w^3 + 13*w^2 + 5*w + 1],\ [1277, 1277, w^4 - 5*w^3 + 3*w^2 + 9*w - 7],\ [1277, 1277, -w^5 + 3*w^4 + 2*w^3 - 7*w^2 - 4*w - 1],\ [1277, 1277, 3*w^5 - 9*w^4 - 6*w^3 + 22*w^2 + 5*w - 4],\ [1279, 1279, -2*w^5 + 4*w^4 + 12*w^3 - 17*w^2 - 18*w + 11],\ [1279, 1279, -3*w^5 + 11*w^4 - 23*w^2 + 5*w + 2],\ [1279, 1279, w^3 + w^2 - 5*w - 2],\ [1279, 1279, -2*w^5 + 6*w^4 + 7*w^3 - 21*w^2 - 9*w + 9],\ [1279, 1279, -w^5 + 3*w^4 + 5*w^3 - 12*w^2 - 9*w + 4],\ [1279, 1279, -2*w^3 + 5*w^2 + 3*w - 8],\ [1297, 1297, -2*w^5 + 4*w^4 + 11*w^3 - 14*w^2 - 19*w + 7],\ [1297, 1297, -w^5 + w^4 + 10*w^3 - 9*w^2 - 19*w + 7],\ [1331, 11, -3*w^5 + 7*w^4 + 16*w^3 - 26*w^2 - 28*w + 9],\ [1331, 11, w^3 - w^2 - w - 3],\ [1367, 1367, 2*w^5 - 6*w^4 - 4*w^3 + 13*w^2 + 5*w],\ [1367, 1367, w^3 - 6*w],\ [1367, 1367, 3*w^5 - 9*w^4 - 6*w^3 + 22*w^2 + 5*w - 5],\ [1367, 1367, w^3 - 6*w - 2],\ [1369, 37, 2*w^5 - 6*w^4 - 7*w^3 + 20*w^2 + 10*w - 8],\ [1369, 37, 3*w^5 - 9*w^4 - 8*w^3 + 25*w^2 + 10*w - 8],\ [1423, 1423, 3*w^5 - 10*w^4 - 5*w^3 + 26*w^2 + 6*w - 3],\ [1423, 1423, -2*w^4 + 7*w^3 - w^2 - 13*w + 4],\ [1459, 1459, -w^5 + 4*w^4 - 3*w^3 - 4*w^2 + 8*w - 5],\ [1459, 1459, 2*w^5 - 5*w^4 - 6*w^3 + 11*w^2 + 8*w],\ [1459, 1459, -2*w^5 + 7*w^4 + 4*w^3 - 22*w^2 - 2*w + 7],\ [1459, 1459, -3*w^5 + 8*w^4 + 14*w^3 - 29*w^2 - 24*w + 11],\ [1493, 1493, -3*w^5 + 9*w^4 + 9*w^3 - 25*w^2 - 16*w + 6],\ [1493, 1493, -2*w^5 + 3*w^4 + 15*w^3 - 14*w^2 - 27*w + 9],\ [1493, 1493, w^5 - 4*w^4 + 2*w^3 + 5*w^2 - 5*w + 3],\ [1493, 1493, -3*w^5 + 8*w^4 + 9*w^3 - 21*w^2 - 11*w + 7],\ [1531, 1531, -w^5 + 3*w^4 + 2*w^3 - 8*w^2 + w + 4],\ [1531, 1531, 2*w^5 - 6*w^4 - 6*w^3 + 17*w^2 + 11*w - 7],\ [1549, 1549, w^5 - 4*w^4 - 3*w^3 + 14*w^2 + 7*w - 4],\ [1549, 1549, -4*w^5 + 10*w^4 + 18*w^3 - 34*w^2 - 28*w + 15],\ [1567, 1567, -3*w^5 + 6*w^4 + 17*w^3 - 23*w^2 - 25*w + 9],\ [1567, 1567, -w^5 + 3*w^4 + 6*w^3 - 13*w^2 - 12*w + 4],\ [1583, 1583, w^5 - 2*w^4 - 8*w^3 + 11*w^2 + 15*w - 7],\ [1583, 1583, w^5 - 4*w^4 - w^3 + 13*w^2 - w - 7],\ [1601, 1601, -w^4 + 4*w^3 - 3*w^2 - 6*w + 7],\ [1601, 1601, 3*w^5 - 8*w^4 - 9*w^3 + 22*w^2 + 10*w - 8],\ [1619, 1619, 2*w^5 - 5*w^4 - 11*w^3 + 19*w^2 + 21*w - 6],\ [1619, 1619, w^5 - 2*w^4 - 8*w^3 + 13*w^2 + 12*w - 12],\ [1619, 1619, -4*w^5 + 9*w^4 + 20*w^3 - 32*w^2 - 31*w + 13],\ [1619, 1619, w^5 - 2*w^4 - 6*w^3 + 10*w^2 + 10*w - 12],\ [1637, 1637, -w^5 + 2*w^4 + 4*w^3 - 4*w^2 - 5*w - 3],\ [1637, 1637, w^5 - 4*w^4 + 2*w^3 + 6*w^2 - 5*w + 4],\ [1637, 1637, 4*w^5 - 10*w^4 - 17*w^3 + 31*w^2 + 29*w - 12],\ [1637, 1637, -w^5 + 5*w^4 - 5*w^3 - 8*w^2 + 15*w - 1],\ [1657, 1657, w^4 - 2*w^3 - w^2 + w - 4],\ [1657, 1657, 2*w^5 - 5*w^4 - 8*w^3 + 17*w^2 + 9*w - 7],\ [1693, 1693, w^5 - 2*w^4 - 9*w^3 + 14*w^2 + 15*w - 8],\ [1693, 1693, w^5 - 3*w^4 - 3*w^3 + 9*w^2 + w],\ [1709, 1709, 2*w^5 - 7*w^4 - 3*w^3 + 18*w^2 + 4*w - 3],\ [1709, 1709, 3*w^5 - 8*w^4 - 11*w^3 + 25*w^2 + 14*w - 9],\ [1709, 1709, -w^5 + w^4 + 11*w^3 - 11*w^2 - 20*w + 9],\ [1709, 1709, -w^5 + w^4 + 7*w^3 - 3*w^2 - 12*w + 1],\ [1747, 1747, -w^5 + 5*w^4 - 3*w^3 - 9*w^2 + 7*w - 4],\ [1747, 1747, -3*w^5 + 11*w^4 - 24*w^2 + 5*w + 3],\ [1783, 1783, -w^4 + 3*w^3 + 2*w^2 - 5*w - 4],\ [1783, 1783, -2*w^5 + 7*w^4 + 2*w^3 - 18*w^2 - w + 4],\ [1871, 1871, -w^5 + 3*w^4 + 4*w^3 - 11*w^2 - 9*w + 6],\ [1871, 1871, 2*w^5 - 8*w^4 + w^3 + 17*w^2 - 3*w - 2],\ [1871, 1871, 2*w^5 - 6*w^4 - 5*w^3 + 16*w^2 + 9*w - 6],\ [1871, 1871, -3*w^5 + 11*w^4 - w^3 - 23*w^2 + 12*w - 3],\ [1873, 1873, -w^5 + w^4 + 8*w^3 - 5*w^2 - 11*w + 4],\ [1873, 1873, 3*w^5 - 6*w^4 - 15*w^3 + 19*w^2 + 22*w - 9],\ [1889, 1889, 2*w^5 - 5*w^4 - 8*w^3 + 14*w^2 + 13*w - 5],\ [1889, 1889, -2*w^5 + 5*w^4 + 8*w^3 - 15*w^2 - 12*w + 8],\ [1889, 1889, 2*w^5 - 7*w^4 - w^3 + 16*w^2 - 3*w - 5],\ [1889, 1889, w^5 - 4*w^4 + w^3 + 8*w^2 - 4*w - 3],\ [1907, 1907, -3*w^5 + 6*w^4 + 18*w^3 - 25*w^2 - 28*w + 12],\ [1907, 1907, 2*w^5 - 5*w^4 - 7*w^3 + 14*w^2 + 8*w - 1],\ [1979, 1979, 2*w^4 - 9*w^3 + 5*w^2 + 18*w - 11],\ [1979, 1979, 4*w^5 - 10*w^4 - 14*w^3 + 27*w^2 + 20*w - 8],\ [1997, 1997, 3*w^5 - 8*w^4 - 11*w^3 + 25*w^2 + 14*w - 10],\ [1997, 1997, -4*w^5 + 13*w^4 + 5*w^3 - 32*w^2 + w + 4],\ [1997, 1997, 2*w^5 - 7*w^4 - 3*w^3 + 18*w^2 + 4*w - 4],\ [1997, 1997, -4*w^5 + 11*w^4 + 15*w^3 - 35*w^2 - 24*w + 11]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^8 - 12*x^7 + 14*x^6 + 295*x^5 - 943*x^4 - 1065*x^3 + 5559*x^2 - 967*x - 5366 K. = NumberField(heckePol) hecke_eigenvalues_array = [1, 1, e, -573723/228952021*e^7 + 5247625/228952021*e^6 + 6547881/228952021*e^5 - 135015560/228952021*e^4 + 114348577/228952021*e^3 + 563180557/228952021*e^2 - 286781322/228952021*e - 563382596/228952021, 4687666/228952021*e^7 - 36290142/228952021*e^6 - 76919624/228952021*e^5 + 986093543/228952021*e^4 - 529790127/228952021*e^3 - 5390906146/228952021*e^2 + 4102939853/228952021*e + 5141128442/228952021, -11660334/228952021*e^7 + 75274240/228952021*e^6 + 275675227/228952021*e^5 - 2047786176/228952021*e^4 - 811610989/228952021*e^3 + 11109184830/228952021*e^2 - 2311532442/228952021*e - 11673502270/228952021, 3058449/228952021*e^7 - 23331757/228952021*e^6 - 52580106/228952021*e^5 + 634483136/228952021*e^4 - 270025326/228952021*e^3 - 3578007616/228952021*e^2 + 2538866005/228952021*e + 4155569284/228952021, -8528343/228952021*e^7 + 62613208/228952021*e^6 + 168741347/228952021*e^5 - 1718629652/228952021*e^4 + 153032415/228952021*e^3 + 9789577632/228952021*e^2 - 3740150749/228952021*e - 11181170544/228952021, 6759947/228952021*e^7 - 51590278/228952021*e^6 - 110762184/228952021*e^5 + 1374537043/228952021*e^4 - 732017682/228952021*e^3 - 7080728362/228952021*e^2 + 5592660259/228952021*e + 6765786624/228952021, 5593754/228952021*e^7 - 37155996/228952021*e^6 - 127912596/228952021*e^5 + 1017437681/228952021*e^4 + 275017721/228952021*e^3 - 5738122920/228952021*e^2 + 1685650254/228952021*e + 7353743953/228952021, -5468085/228952021*e^7 + 33295710/228952021*e^6 + 138864479/228952021*e^5 - 908674702/228952021*e^4 - 619336883/228952021*e^3 + 4856780368/228952021*e^2 - 13275743/228952021*e - 2683358290/228952021, -1294618/228952021*e^7 + 5012199/228952021*e^6 + 47798750/228952021*e^5 - 105441277/228952021*e^4 - 626701831/228952021*e^3 + 59120129/228952021*e^2 + 3205765666/228952021*e + 958874824/228952021, 2413591/228952021*e^7 - 12419252/228952021*e^6 - 69681372/228952021*e^5 + 353440551/228952021*e^4 + 424285676/228952021*e^3 - 2298504838/228952021*e^2 - 79879329/228952021*e + 5021910594/228952021, -8959535/228952021*e^7 + 69336236/228952021*e^6 + 150684838/228952021*e^5 - 1916721970/228952021*e^4 + 884819858/228952021*e^3 + 11330826803/228952021*e^2 - 6920154303/228952021*e - 14868256844/228952021, -2806425/228952021*e^7 + 22074132/228952021*e^6 + 47782273/228952021*e^5 - 597719248/228952021*e^4 + 284423654/228952021*e^3 + 3153178990/228952021*e^2 - 2983838036/228952021*e - 3018280110/228952021, -6352753/228952021*e^7 + 35046707/228952021*e^6 + 178471925/228952021*e^5 - 897801647/228952021*e^4 - 1171228567/228952021*e^3 + 3607828824/228952021*e^2 + 1290669769/228952021*e + 898955928/228952021, -4555198/228952021*e^7 + 27398527/228952021*e^6 + 102578023/228952021*e^5 - 712439262/228952021*e^4 - 144439668/228952021*e^3 + 3578811648/228952021*e^2 - 2095892046/228952021*e - 4079930194/228952021, 4432373/228952021*e^7 - 26114271/228952021*e^6 - 115426391/228952021*e^5 + 689318600/228952021*e^4 + 623924730/228952021*e^3 - 3524297040/228952021*e^2 - 1000664039/228952021*e + 5790781040/228952021, -3311508/228952021*e^7 + 21179671/228952021*e^6 + 81977662/228952021*e^5 - 581417527/228952021*e^4 - 381882874/228952021*e^3 + 3245922517/228952021*e^2 + 747248622/228952021*e - 2981763598/228952021, -1326614/228952021*e^7 + 15960653/228952021*e^6 - 9758580/228952021*e^5 - 431359550/228952021*e^4 + 974882925/228952021*e^3 + 2471733680/228952021*e^2 - 5114346313/228952021*e - 2760754234/228952021, -4542215/228952021*e^7 + 31900535/228952021*e^6 + 93517958/228952021*e^5 - 880091819/228952021*e^4 + 44642588/228952021*e^3 + 5050232991/228952021*e^2 - 2396164619/228952021*e - 4954464202/228952021, -1260044/228952021*e^7 + 2999145/228952021*e^6 + 67088374/228952021*e^5 - 100383275/228952021*e^4 - 1080791924/228952021*e^3 + 716421221/228952021*e^2 + 4356854838/228952021*e - 743993700/228952021, -11590747/228952021*e^7 + 88400264/228952021*e^6 + 204718452/228952021*e^5 - 2432361773/228952021*e^4 + 888134274/228952021*e^3 + 14158365313/228952021*e^2 - 8266823645/228952021*e - 17709725038/228952021, 417606/228952021*e^7 - 9229789/228952021*e^6 + 19548828/228952021*e^5 + 230936930/228952021*e^4 - 677104368/228952021*e^3 - 805775137/228952021*e^2 + 2931638446/228952021*e - 1159565074/228952021, 7150884/228952021*e^7 - 39401018/228952021*e^6 - 185570883/228952021*e^5 + 997280545/228952021*e^4 + 904507456/228952021*e^3 - 4046861099/228952021*e^2 - 121552006/228952021*e + 673926574/228952021, -13474058/228952021*e^7 + 95796852/228952021*e^6 + 283055249/228952021*e^5 - 2641077074/228952021*e^4 - 141518997/228952021*e^3 + 15111217040/228952021*e^2 - 4619448354/228952021*e - 17499903860/228952021, -7542683/228952021*e^7 + 52465609/228952021*e^6 + 139072061/228952021*e^5 - 1331388739/228952021*e^4 + 441711746/228952021*e^3 + 5658448976/228952021*e^2 - 4867499871/228952021*e - 3728815202/228952021, 9370095/228952021*e^7 - 62845511/228952021*e^6 - 228676950/228952021*e^5 + 1768838067/228952021*e^4 + 901827225/228952021*e^3 - 10580176152/228952021*e^2 - 216525524/228952021*e + 12588034672/228952021, -10789772/228952021*e^7 + 81469923/228952021*e^6 + 195723009/228952021*e^5 - 2247240558/228952021*e^4 + 756579063/228952021*e^3 + 12892505094/228952021*e^2 - 7835031393/228952021*e - 15021502510/228952021, -2931828/228952021*e^7 + 14422988/228952021*e^6 + 73201611/228952021*e^5 - 299979639/228952021*e^4 - 383585947/228952021*e^3 + 137778955/228952021*e^2 + 1317580707/228952021*e + 3199260234/228952021, 10885762/228952021*e^7 - 78251518/228952021*e^6 - 208693478/228952021*e^5 + 2064355801/228952021*e^4 - 358134543/228952021*e^3 - 10244128466/228952021*e^2 + 5238653710/228952021*e + 9669257320/228952021, -17387838/228952021*e^7 + 128601325/228952021*e^6 + 319694723/228952021*e^5 - 3462978172/228952021*e^4 + 1083932031/228952021*e^3 + 18528162365/228952021*e^2 - 13125684905/228952021*e - 19301865830/228952021, -12357231/228952021*e^7 + 84859357/228952021*e^6 + 267272864/228952021*e^5 - 2296854971/228952021*e^4 - 381202591/228952021*e^3 + 12536774769/228952021*e^2 - 2751263949/228952021*e - 14128901102/228952021, 2249481/228952021*e^7 - 19949012/228952021*e^6 - 42521320/228952021*e^5 + 606433966/228952021*e^4 - 155119032/228952021*e^3 - 4504291143/228952021*e^2 + 3156674768/228952021*e + 9272331744/228952021, -4521056/228952021*e^7 + 19980515/228952021*e^6 + 154035116/228952021*e^5 - 482743818/228952021*e^4 - 1479804964/228952021*e^3 + 1043351617/228952021*e^2 + 2444977403/228952021*e + 5454809818/228952021, -5112669/228952021*e^7 + 28229914/228952021*e^6 + 143370439/228952021*e^5 - 755096324/228952021*e^4 - 1009122020/228952021*e^3 + 3942681047/228952021*e^2 + 3094582003/228952021*e - 4557364318/228952021, 15048741/228952021*e^7 - 110559831/228952021*e^6 - 294697503/228952021*e^5 + 3029951527/228952021*e^4 - 353889067/228952021*e^3 - 17216999281/228952021*e^2 + 7072063724/228952021*e + 20398674184/228952021, -10983727/228952021*e^7 + 64102466/228952021*e^6 + 267499399/228952021*e^5 - 1654754849/228952021*e^4 - 973665929/228952021*e^3 + 7584838301/228952021*e^2 - 877368085/228952021*e - 3705122132/228952021, 10882410/228952021*e^7 - 77791466/228952021*e^6 - 217728733/228952021*e^5 + 2119914761/228952021*e^4 - 136823341/228952021*e^3 - 11660243778/228952021*e^2 + 3799484500/228952021*e + 11643745366/228952021, 37992737/228952021*e^7 - 276075037/228952021*e^6 - 722064622/228952021*e^5 + 7459227854/228952021*e^4 - 1674440917/228952021*e^3 - 40179718915/228952021*e^2 + 22627858555/228952021*e + 41536817516/228952021, 12039062/228952021*e^7 - 76982440/228952021*e^6 - 294120900/228952021*e^5 + 2103483546/228952021*e^4 + 1119200027/228952021*e^3 - 11429490262/228952021*e^2 + 697792326/228952021*e + 11277897890/228952021, 15884976/228952021*e^7 - 113040279/228952021*e^6 - 311104921/228952021*e^5 + 3039368674/228952021*e^4 - 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41585262714/228952021*e - 89591799816/228952021, -36262198/228952021*e^7 + 252489059/228952021*e^6 + 756228343/228952021*e^5 - 6811143963/228952021*e^4 + 193376949/228952021*e^3 + 36134770219/228952021*e^2 - 22360462704/228952021*e - 33445515336/228952021, -18457872/228952021*e^7 + 102790040/228952021*e^6 + 514951516/228952021*e^5 - 2711829349/228952021*e^4 - 3151971305/228952021*e^3 + 12728119158/228952021*e^2 + 1334266019/228952021*e - 2730684522/228952021, -33335360/228952021*e^7 + 237111659/228952021*e^6 + 691023454/228952021*e^5 - 6500583812/228952021*e^4 + 175577536/228952021*e^3 + 35572953562/228952021*e^2 - 18016715265/228952021*e - 29380201248/228952021, 5381123/228952021*e^7 - 29556808/228952021*e^6 - 151557076/228952021*e^5 + 844657917/228952021*e^4 + 875051152/228952021*e^3 - 5700859630/228952021*e^2 - 365282101/228952021*e + 10740836808/228952021, -10679477/228952021*e^7 + 116330139/228952021*e^6 - 46641689/228952021*e^5 - 3020219279/228952021*e^4 + 6607141728/228952021*e^3 + 14992774235/228952021*e^2 - 26781139221/228952021*e - 12263756444/228952021, -29059346/228952021*e^7 + 193387816/228952021*e^6 + 627133253/228952021*e^5 - 5167639977/228952021*e^4 - 648057207/228952021*e^3 + 27726944288/228952021*e^2 - 9369895353/228952021*e - 38122000256/228952021] 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 - 1])] = -1 AL_eigenvalues[ZF.ideal([17, 17, -w^2 + 2*w + 1])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]