/* 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, -2, 4, 5, -4, -2, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([43, 43, -w^5 + 3*w^4 + w^3 - 6*w^2 + 3*w + 1]) primes_array = [ [13, 13, -w^5 + 3*w^4 + 2*w^3 - 9*w^2 + w + 4],\ [13, 13, -w^2 + w + 2],\ [27, 3, 2*w^5 - 4*w^4 - 7*w^3 + 9*w^2 + 4*w - 2],\ [27, 3, -2*w^5 + 5*w^4 + 5*w^3 - 12*w^2 - w + 5],\ [29, 29, w^3 - 2*w^2 - 2*w + 3],\ [29, 29, 2*w^5 - 4*w^4 - 7*w^3 + 8*w^2 + 4*w - 2],\ [43, 43, -w^5 + 3*w^4 + w^3 - 6*w^2 + 3*w + 1],\ [43, 43, -w^4 + w^3 + 5*w^2 - 4],\ [49, 7, w^5 - 4*w^4 + 11*w^2 - 3*w - 4],\ [64, 2, -2],\ [71, 71, 2*w^5 - 6*w^4 - 4*w^3 + 17*w^2 - w - 6],\ [71, 71, 2*w^4 - 4*w^3 - 6*w^2 + 7*w + 2],\ [71, 71, -w^5 + 3*w^4 + 2*w^3 - 7*w^2 - w],\ [71, 71, -2*w^5 + 5*w^4 + 6*w^3 - 14*w^2 - 3*w + 5],\ [83, 83, -3*w^5 + 7*w^4 + 9*w^3 - 17*w^2 - 5*w + 5],\ [83, 83, 3*w^5 - 6*w^4 - 10*w^3 + 12*w^2 + 5*w - 2],\ [83, 83, -2*w^5 + 5*w^4 + 5*w^3 - 11*w^2 - 3*w + 3],\ [83, 83, 3*w^5 - 7*w^4 - 8*w^3 + 15*w^2 + 2*w - 4],\ [97, 97, -3*w^5 + 6*w^4 + 10*w^3 - 12*w^2 - 5*w + 3],\ [97, 97, w^5 - 2*w^4 - 4*w^3 + 5*w^2 + 5*w - 3],\ [97, 97, w^5 - 2*w^4 - 4*w^3 + 5*w^2 + 5*w - 1],\ [97, 97, -3*w^5 + 7*w^4 + 8*w^3 - 15*w^2 - 2*w + 3],\ [113, 113, -3*w^5 + 6*w^4 + 11*w^3 - 14*w^2 - 7*w + 3],\ [113, 113, 2*w^5 - 5*w^4 - 5*w^3 + 12*w^2 - 6],\ [113, 113, -w^5 + 2*w^4 + 3*w^3 - 4*w^2 - 1],\ [113, 113, -w^5 + 2*w^4 + 4*w^3 - 6*w^2 - 2*w + 5],\ [125, 5, 3*w^5 - 6*w^4 - 10*w^3 + 11*w^2 + 6*w - 1],\ [125, 5, -2*w^5 + 4*w^4 + 6*w^3 - 6*w^2 - 3*w - 1],\ [127, 127, -2*w^5 + 6*w^4 + 3*w^3 - 15*w^2 + 2*w + 5],\ [127, 127, 2*w^5 - 3*w^4 - 9*w^3 + 6*w^2 + 7*w - 2],\ [127, 127, -3*w^5 + 7*w^4 + 9*w^3 - 17*w^2 - 4*w + 4],\ [127, 127, -w^5 + 2*w^4 + 3*w^3 - 4*w^2 - w + 3],\ [139, 139, 3*w^5 - 7*w^4 - 9*w^3 + 16*w^2 + 6*w - 4],\ [139, 139, -w^5 + w^4 + 5*w^3 - 6*w - 1],\ [167, 167, -2*w^5 + 5*w^4 + 6*w^3 - 12*w^2 - 5*w + 4],\ [167, 167, w^5 - 4*w^4 - w^3 + 13*w^2 - 2*w - 4],\ [169, 13, -w^5 + 4*w^4 - 11*w^2 + 3*w + 5],\ [169, 13, 2*w^5 - 5*w^4 - 6*w^3 + 13*w^2 + 3*w - 3],\ [181, 181, 2*w^5 - 4*w^4 - 6*w^3 + 6*w^2 + 2*w + 1],\ [197, 197, -w^4 + 3*w^3 + 2*w^2 - 6*w - 1],\ [197, 197, -3*w^5 + 6*w^4 + 11*w^3 - 14*w^2 - 6*w + 4],\ [211, 211, 2*w^5 - 5*w^4 - 6*w^3 + 12*w^2 + 6*w - 1],\ [211, 211, -2*w^5 + 4*w^4 + 8*w^3 - 9*w^2 - 8*w + 2],\ [211, 211, -w^5 + 2*w^4 + 3*w^3 - 3*w^2 - 3*w + 3],\ [211, 211, -2*w^4 + 5*w^3 + 5*w^2 - 9*w - 3],\ [223, 223, -w^5 + 8*w^3 + w^2 - 11*w + 1],\ [223, 223, 2*w^5 - 5*w^4 - 5*w^3 + 12*w^2 + w - 1],\ [239, 239, -2*w^5 + 5*w^4 + 6*w^3 - 13*w^2 - 2*w + 3],\ [239, 239, 2*w^5 - 5*w^4 - 4*w^3 + 9*w^2 + w],\ [239, 239, -3*w^5 + 7*w^4 + 10*w^3 - 18*w^2 - 7*w + 5],\ [239, 239, w^5 - w^4 - 6*w^3 + 2*w^2 + 5*w - 2],\ [251, 251, w^5 - w^4 - 6*w^3 + 3*w^2 + 6*w - 2],\ [251, 251, -w^5 + 4*w^4 - 12*w^2 + 3*w + 6],\ [281, 281, -5*w^5 + 11*w^4 + 17*w^3 - 27*w^2 - 10*w + 9],\ [281, 281, 2*w^5 - 5*w^4 - 5*w^3 + 10*w^2 + 3*w + 1],\ [293, 293, -2*w^5 + 4*w^4 + 7*w^3 - 7*w^2 - 7*w + 2],\ [293, 293, -w^4 + 2*w^3 + 2*w^2 - 3*w + 3],\ [293, 293, -2*w^5 + 7*w^4 + w^3 - 19*w^2 + 7*w + 9],\ [293, 293, -2*w^5 + 3*w^4 + 9*w^3 - 5*w^2 - 7*w + 2],\ [307, 307, 2*w^2 - 3*w - 4],\ [307, 307, -w^5 + 4*w^4 - 13*w^2 + 6*w + 6],\ [337, 337, 4*w^5 - 9*w^4 - 12*w^3 + 20*w^2 + 6*w - 5],\ [337, 337, -2*w^5 + 3*w^4 + 10*w^3 - 6*w^2 - 12*w + 2],\ [349, 349, 4*w^5 - 9*w^4 - 11*w^3 + 19*w^2 + 2*w - 4],\ [349, 349, 5*w^5 - 11*w^4 - 15*w^3 + 24*w^2 + 5*w - 7],\ [379, 379, -2*w^5 + 5*w^4 + 4*w^3 - 10*w^2 + w + 5],\ [379, 379, -2*w^5 + 5*w^4 + 6*w^3 - 12*w^2 - 5*w + 5],\ [379, 379, -4*w^5 + 8*w^4 + 14*w^3 - 17*w^2 - 10*w + 4],\ [379, 379, -5*w^5 + 12*w^4 + 13*w^3 - 27*w^2 - 2*w + 7],\ [421, 421, 2*w^5 - 6*w^4 - 4*w^3 + 15*w^2 + w - 3],\ [421, 421, 3*w^5 - 9*w^4 - 6*w^3 + 25*w^2 - w - 9],\ [433, 433, -5*w^5 + 10*w^4 + 18*w^3 - 21*w^2 - 13*w + 4],\ [433, 433, 5*w^5 - 12*w^4 - 15*w^3 + 30*w^2 + 7*w - 9],\ [433, 433, -3*w^5 + 8*w^4 + 8*w^3 - 22*w^2 - w + 8],\ [433, 433, 2*w^5 - 3*w^4 - 10*w^3 + 8*w^2 + 9*w - 3],\ [449, 449, 3*w^5 - 9*w^4 - 6*w^3 + 24*w^2 + w - 9],\ [449, 449, 3*w^5 - 6*w^4 - 11*w^3 + 15*w^2 + 6*w - 5],\ [449, 449, 2*w^5 - 3*w^4 - 8*w^3 + 3*w^2 + 7*w],\ [449, 449, 4*w^5 - 8*w^4 - 14*w^3 + 17*w^2 + 8*w - 2],\ [449, 449, -3*w^4 + 5*w^3 + 12*w^2 - 9*w - 6],\ [449, 449, -3*w^5 + 8*w^4 + 7*w^3 - 20*w^2 - 2*w + 6],\ [461, 461, 2*w^4 - 4*w^3 - 7*w^2 + 7*w + 2],\ [461, 461, -4*w^5 + 10*w^4 + 11*w^3 - 24*w^2 - 6*w + 8],\ [463, 463, 4*w^5 - 8*w^4 - 14*w^3 + 18*w^2 + 6*w - 5],\ [463, 463, w^5 - w^4 - 5*w^3 + w^2 + 4*w - 3],\ [491, 491, -3*w^5 + 7*w^4 + 8*w^3 - 15*w^2 + 5],\ [491, 491, -2*w^5 + 6*w^4 + 3*w^3 - 14*w^2 + 3],\ [547, 547, 2*w^5 - 3*w^4 - 10*w^3 + 8*w^2 + 10*w - 5],\ [547, 547, w^5 - 4*w^4 + 12*w^2 - 2*w - 5],\ [547, 547, -3*w^5 + 5*w^4 + 13*w^3 - 11*w^2 - 11*w + 3],\ [547, 547, -2*w^5 + 6*w^4 + 3*w^3 - 15*w^2 + w + 6],\ [547, 547, -2*w^5 + 6*w^4 + 4*w^3 - 15*w^2 - w + 4],\ [547, 547, 4*w^5 - 8*w^4 - 15*w^3 + 18*w^2 + 12*w - 6],\ [587, 587, -5*w^5 + 12*w^4 + 14*w^3 - 28*w^2 - 6*w + 7],\ [587, 587, 5*w^5 - 12*w^4 - 14*w^3 + 27*w^2 + 6*w - 6],\ [601, 601, -3*w^5 + 9*w^4 + 5*w^3 - 24*w^2 + 5*w + 10],\ [601, 601, -3*w^5 + 8*w^4 + 8*w^3 - 23*w^2 - w + 9],\ [601, 601, -w^5 + 8*w^3 + 2*w^2 - 11*w - 1],\ [601, 601, 2*w^5 - 3*w^4 - 10*w^3 + 8*w^2 + 10*w - 4],\ [617, 617, w^5 - w^4 - 7*w^3 + 4*w^2 + 10*w - 3],\ [617, 617, -5*w^5 + 11*w^4 + 16*w^3 - 26*w^2 - 7*w + 8],\ [617, 617, 3*w^5 - 7*w^4 - 9*w^3 + 15*w^2 + 7*w - 2],\ [617, 617, -2*w^5 + 2*w^4 + 11*w^3 - w^2 - 13*w - 2],\ [631, 631, 2*w^4 - 4*w^3 - 7*w^2 + 8*w + 2],\ [631, 631, w^5 - w^4 - 5*w^3 - w^2 + 4*w + 5],\ [631, 631, 4*w^5 - 9*w^4 - 11*w^3 + 19*w^2 + 2*w - 6],\ [631, 631, -3*w^5 + 7*w^4 + 9*w^3 - 15*w^2 - 7*w + 3],\ [643, 643, -2*w^5 + 3*w^4 + 9*w^3 - 6*w^2 - 8*w + 4],\ [643, 643, 3*w^5 - 8*w^4 - 7*w^3 + 20*w^2 + 2*w - 5],\ [659, 659, 3*w^5 - 9*w^4 - 5*w^3 + 23*w^2 - 3*w - 10],\ [659, 659, -2*w^4 + 3*w^3 + 7*w^2 - 3*w - 5],\ [673, 673, 3*w^5 - 6*w^4 - 11*w^3 + 14*w^2 + 8*w - 2],\ [673, 673, 3*w^5 - 8*w^4 - 6*w^3 + 19*w^2 - 3*w - 8],\ [673, 673, 3*w^5 - 6*w^4 - 10*w^3 + 13*w^2 + 3*w - 2],\ [673, 673, 2*w^5 - 3*w^4 - 10*w^3 + 6*w^2 + 11*w - 2],\ [701, 701, 2*w^5 - 5*w^4 - 6*w^3 + 13*w^2 + 6*w - 4],\ [701, 701, 2*w^5 - 5*w^4 - 5*w^3 + 11*w^2 + 2*w - 6],\ [701, 701, -w^5 + w^4 + 5*w^3 - 5*w - 5],\ [701, 701, -w^5 + w^4 + 6*w^3 - 2*w^2 - 9*w + 2],\ [727, 727, -w^5 + 5*w^4 - w^3 - 16*w^2 + 4*w + 8],\ [727, 727, w^5 - w^4 - 4*w^3 - 2*w^2 + 4*w + 4],\ [727, 727, -3*w^5 + 9*w^4 + 5*w^3 - 22*w^2 + 2*w + 7],\ [727, 727, -w^5 + 2*w^4 + 3*w^3 - 5*w^2 + 5],\ [727, 727, -5*w^5 + 11*w^4 + 16*w^3 - 24*w^2 - 10*w + 6],\ [727, 727, 4*w^5 - 8*w^4 - 14*w^3 + 17*w^2 + 7*w - 4],\ [743, 743, 5*w^5 - 11*w^4 - 16*w^3 + 26*w^2 + 6*w - 8],\ [743, 743, -4*w^5 + 7*w^4 + 16*w^3 - 14*w^2 - 12*w + 1],\ [743, 743, -4*w^5 + 9*w^4 + 14*w^3 - 23*w^2 - 10*w + 7],\ [743, 743, w^5 - 3*w^4 + 6*w^2 - 6*w - 2],\ [757, 757, -w^5 + 7*w^3 + 3*w^2 - 7*w - 6],\ [757, 757, 5*w^5 - 11*w^4 - 16*w^3 + 24*w^2 + 11*w - 5],\ [769, 769, -3*w^5 + 4*w^4 + 14*w^3 - 6*w^2 - 12*w + 1],\ [769, 769, 4*w^5 - 7*w^4 - 17*w^3 + 16*w^2 + 14*w - 5],\ [797, 797, -5*w^5 + 12*w^4 + 14*w^3 - 27*w^2 - 7*w + 7],\ [797, 797, 2*w^5 - 7*w^4 - w^3 + 17*w^2 - 4*w - 7],\ [811, 811, 5*w^5 - 13*w^4 - 13*w^3 + 33*w^2 + 3*w - 10],\ [811, 811, -4*w^5 + 10*w^4 + 9*w^3 - 22*w^2 + w + 6],\ [841, 29, 3*w^5 - 6*w^4 - 9*w^3 + 10*w^2 + 4*w - 1],\ [841, 29, -w^5 + w^4 + 4*w^3 + 3*w^2 - 3*w - 7],\ [853, 853, -4*w^5 + 9*w^4 + 12*w^3 - 21*w^2 - 4*w + 5],\ [853, 853, -3*w^5 + 6*w^4 + 10*w^3 - 13*w^2 - 4*w + 2],\ [853, 853, -w^5 + w^4 + 5*w^3 + w^2 - 4*w - 6],\ [853, 853, -3*w^5 + 7*w^4 + 8*w^3 - 16*w^2 - w + 7],\ [853, 853, -4*w^5 + 10*w^4 + 10*w^3 - 24*w^2 - w + 10],\ [853, 853, -w^4 + 3*w^3 + 3*w^2 - 8*w - 2],\ [883, 883, -w^5 + 3*w^4 + 3*w^3 - 9*w^2 - 4*w + 6],\ [883, 883, -w^5 + 5*w^4 - 3*w^3 - 12*w^2 + 9*w + 3],\ [883, 883, 5*w^5 - 12*w^4 - 14*w^3 + 27*w^2 + 7*w - 6],\ [883, 883, 2*w^5 - 6*w^4 - 3*w^3 + 16*w^2 - 3*w - 9],\ [937, 937, 5*w^5 - 11*w^4 - 16*w^3 + 26*w^2 + 8*w - 7],\ [937, 937, -w^5 + 2*w^4 + 4*w^3 - 3*w^2 - 6*w],\ [937, 937, 2*w^5 - 6*w^4 - 4*w^3 + 18*w^2 - 3*w - 10],\ [937, 937, 2*w^5 - 7*w^4 - 3*w^3 + 22*w^2 - 2*w - 10],\ [953, 953, 4*w^5 - 7*w^4 - 16*w^3 + 14*w^2 + 13*w - 3],\ [953, 953, 5*w^5 - 11*w^4 - 15*w^3 + 23*w^2 + 7*w - 3],\ [953, 953, w^5 + w^4 - 9*w^3 - 6*w^2 + 12*w + 3],\ [953, 953, -3*w^5 + 7*w^4 + 10*w^3 - 17*w^2 - 9*w + 5],\ [953, 953, w^5 - 4*w^4 + w^3 + 8*w^2 - 5*w + 1],\ [953, 953, -2*w^5 + 6*w^4 + 3*w^3 - 15*w^2 + 4*w + 7],\ [967, 967, 4*w^5 - 10*w^4 - 11*w^3 + 26*w^2 + w - 9],\ [967, 967, w^5 - 3*w^4 - 3*w^3 + 10*w^2 + 3*w - 8]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^2 + 2*x - 4 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, -1/2*e - 1, 5/2*e + 4, -3/2*e + 5, -3/2*e - 4, 8, -1, 3*e + 2, -1/2*e + 2, -e - 9, 7/2*e + 1, 5*e + 2, 2*e, -4*e - 4, 3/2*e - 3, -5/2*e + 9, -6*e - 8, -6*e - 8, -5/2*e - 10, 1/2*e + 14, -4*e, 2*e + 4, 4, 9*e + 10, -4*e - 6, -1/2*e, -e + 8, -9/2*e - 9, 5/2*e - 6, 2*e + 12, 1/2*e + 11, -e - 12, -7/2*e + 2, -15/2*e - 8, 1/2*e - 4, 4*e + 2, -5/2*e + 7, 7*e + 6, e + 14, -e - 8, 9/2*e + 14, -1/2*e + 10, 15/2*e + 8, 1/2*e + 18, -2*e - 2, 3/2*e + 5, -5*e - 14, 17/2*e + 11, 13/2*e - 5, 5*e + 16, 2*e - 8, -4*e, 2*e - 18, -21/2*e - 11, 15/2*e + 1, -1/2*e + 15, 7*e - 2, -2*e + 14, 7/2*e - 8, -5/2*e + 24, -5*e + 4, 7/2*e + 25, 15/2*e + 2, -25/2*e - 14, 17/2*e + 11, 6, 6*e - 12, -9/2*e + 3, 3/2*e - 15, -2*e + 10, -16*e - 14, -6, -8*e - 26, -e + 8, e + 2, 13/2*e + 7, -17*e - 16, -8*e - 10, -27/2*e - 10, -e + 24, 9/2*e + 24, -8*e + 2, -7*e - 12, 4*e - 10, -3*e + 22, -13/2*e + 11, -7*e - 4, 8*e - 4, 8*e - 4, 8*e - 4, -1/2*e - 6, 2*e + 14, e + 28, -6*e - 10, 5*e + 12, 19/2*e - 15, 4*e - 26, 3/2*e + 9, -27/2*e - 23, e + 32, 4*e - 10, 2, -17/2*e + 11, 8*e + 14, 9/2*e + 30, 1/2*e - 13, 11*e - 6, 18*e + 18, -e - 24, 15/2*e + 16, -3/2*e + 21, 33/2*e + 25, 6*e - 4, 9*e + 20, 7/2*e - 13, -21*e - 16, 1/2*e - 31, 6*e + 2, 9/2*e - 21, -e - 6, 5/2*e + 11, -3/2*e - 21, 2, -31/2*e - 23, -21/2*e - 38, -25/2*e + 6, 5*e + 36, 6*e, -8*e - 24, 10, -12, 5/2*e - 2, -9*e - 28, 6*e + 10, 3/2*e - 26, -17*e - 28, -21/2*e + 2, 4*e + 16, 19*e + 26, -16*e - 22, e + 26, -3*e + 38, -5*e - 22, -3/2*e - 5, 8*e + 38, 16*e + 22, -5/2*e + 39, -17/2*e + 13, -7/2*e - 46, -3*e - 32, -10*e - 22, -12*e - 16, 13*e + 30, -14, -29/2*e + 2, 25/2*e + 31, -14*e - 38, 7*e - 24, -29/2*e + 2, 9*e + 6, -25/2*e + 10] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([43, 43, -w^5 + 3*w^4 + w^3 - 6*w^2 + 3*w + 1])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]