/* This code can be loaded, or copied and pasted, into Magma. It will load the data associated to the HMF, including the field, level, and Hecke and Atkin-Lehner eigenvalue data. At the *bottom* of the file, there is code to recreate the Hilbert modular form in Magma, by creating the HMF space and cutting out the corresponding Hecke irreducible subspace. From there, you can ask for more eigenvalues or modify as desired. It is commented out, as this computation may be lengthy. */ P := PolynomialRing(Rationals()); g := P![-1, -11, 10, 10, -6, -2, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [7, 7, w^4 - 2*w^3 - 3*w^2 + 6*w - 1], [7, 7, -w^4 + w^3 + 4*w^2 - w - 1], [13, 13, w^5 - 2*w^4 - 4*w^3 + 7*w^2 + 3*w - 3], [29, 29, -w^4 + w^3 + 4*w^2 - 3*w - 2], [29, 29, w^4 - 5*w^2 - 2*w + 4], [29, 29, -w^4 + w^3 + 3*w^2 - w + 2], [29, 29, -w^2 + w + 1], [41, 41, -w^5 + w^4 + 4*w^3 - w^2 - 3*w - 1], [43, 43, 2*w^3 - 2*w^2 - 7*w + 3], [43, 43, w^3 - w^2 - 2*w + 1], [64, 2, -2], [71, 71, w^5 - w^4 - 6*w^3 + 4*w^2 + 8*w - 3], [71, 71, w^5 - 2*w^4 - 2*w^3 + 5*w^2 - 3*w - 1], [83, 83, -2*w^4 + 3*w^3 + 6*w^2 - 7*w + 2], [83, 83, w^4 - w^3 - 5*w^2 + 2*w + 3], [83, 83, -w^5 + 3*w^4 + w^3 - 9*w^2 + 5*w + 3], [83, 83, w^5 - 7*w^3 + 10*w - 1], [97, 97, w^5 - 2*w^4 - 3*w^3 + 5*w^2 - w - 1], [97, 97, w^5 - 5*w^3 - w^2 + 3*w - 3], [113, 113, w^4 - w^3 - 4*w^2 + w + 4], [113, 113, w^4 - 2*w^3 - 3*w^2 + 6*w + 2], [127, 127, -w^4 + 2*w^3 + 4*w^2 - 5*w - 3], [127, 127, w^5 - 9*w^3 + w^2 + 18*w - 1], [127, 127, 2*w^4 - 4*w^3 - 5*w^2 + 10*w - 2], [127, 127, w^5 - w^4 - 6*w^3 + 3*w^2 + 9*w - 2], [127, 127, -3*w^4 + 5*w^3 + 9*w^2 - 11*w], [127, 127, -w^5 + 3*w^4 + w^3 - 8*w^2 + 4*w - 1], [139, 139, w^4 - 3*w^3 - 2*w^2 + 10*w - 3], [139, 139, -w^5 + 2*w^4 + 4*w^3 - 5*w^2 - 4*w - 3], [167, 167, -w^4 + w^3 + 5*w^2 - 4*w - 3], [167, 167, -w^5 + w^4 + 6*w^3 - 3*w^2 - 8*w + 3], [169, 13, w^4 - 3*w^3 - 2*w^2 + 8*w - 2], [169, 13, 2*w^4 - 3*w^3 - 7*w^2 + 7*w + 3], [181, 181, -w^5 + 4*w^3 + 3*w^2 - 2], [181, 181, w^5 - 3*w^4 - 3*w^3 + 11*w^2 - 4], [181, 181, w^5 - 3*w^4 - 3*w^3 + 10*w^2 + 2*w - 4], [181, 181, w^5 - 2*w^4 - 4*w^3 + 6*w^2 + 3*w - 2], [197, 197, -3*w^4 + 5*w^3 + 10*w^2 - 11*w - 3], [197, 197, -3*w^4 + 6*w^3 + 9*w^2 - 16*w - 1], [197, 197, 2*w^3 - 3*w^2 - 5*w + 4], [197, 197, -w^5 + 6*w^3 - 6*w + 1], [223, 223, 2*w^4 - 2*w^3 - 9*w^2 + 3*w + 6], [223, 223, -3*w^4 + 5*w^3 + 9*w^2 - 13*w + 1], [223, 223, w^5 - w^4 - 6*w^3 + 4*w^2 + 10*w - 4], [223, 223, w^4 - 4*w^3 - 2*w^2 + 12*w], [239, 239, 2*w^4 - w^3 - 8*w^2 + 3], [239, 239, -w^5 + 3*w^4 - 8*w^2 + 9*w + 3], [239, 239, -w^5 + 2*w^4 + 2*w^3 - 6*w^2 + 3*w + 4], [239, 239, w^4 - 6*w^2 + 6], [239, 239, 3*w^4 - 4*w^3 - 11*w^2 + 10*w + 3], [239, 239, -w^5 + 7*w^3 - 9*w - 1], [251, 251, w^5 - 2*w^4 - 2*w^3 + 5*w^2 - 3*w - 2], [251, 251, -w^5 + w^4 + 6*w^3 - 4*w^2 - 8*w + 2], [281, 281, -3*w^4 + 4*w^3 + 11*w^2 - 10*w - 4], [281, 281, 3*w^4 - 3*w^3 - 12*w^2 + 5*w + 6], [293, 293, -w^5 + 8*w^3 - 15*w + 1], [293, 293, w^5 - 4*w^4 + w^3 + 11*w^2 - 9*w + 1], [293, 293, -2*w^4 + 5*w^3 + 5*w^2 - 14*w + 3], [293, 293, -w^5 + 8*w^3 - 14*w + 2], [293, 293, -2*w^4 + 4*w^3 + 6*w^2 - 9*w + 1], [293, 293, w^5 - 4*w^4 + 12*w^2 - 5*w - 2], [307, 307, 2*w^4 - 3*w^3 - 7*w^2 + 9*w + 2], [307, 307, -w^5 + 2*w^4 + 5*w^3 - 8*w^2 - 8*w + 6], [307, 307, -w^3 + 6*w], [307, 307, w^5 - w^4 - 7*w^3 + 5*w^2 + 12*w - 3], [337, 337, -2*w^4 + 2*w^3 + 9*w^2 - 4*w - 8], [337, 337, 3*w^4 - 4*w^3 - 10*w^2 + 9*w + 3], [349, 349, 2*w^5 - 4*w^4 - 7*w^3 + 13*w^2 + 3*w - 6], [349, 349, 2*w^5 - 2*w^4 - 9*w^3 + 5*w^2 + 7*w - 2], [349, 349, w^5 - 3*w^4 + 8*w^2 - 7*w - 2], [349, 349, w^5 - 9*w^3 + 2*w^2 + 17*w - 4], [379, 379, w^5 - 3*w^4 + 6*w^2 - 8*w + 5], [379, 379, w^5 - 2*w^4 - 4*w^3 + 5*w^2 + 3*w + 1], [379, 379, w^5 - 4*w^4 + 12*w^2 - 8*w - 2], [379, 379, -w^5 + w^4 + 7*w^3 - 5*w^2 - 11*w + 5], [419, 419, w^5 - 4*w^4 + w^3 + 10*w^2 - 10*w + 3], [419, 419, 3*w^4 - 3*w^3 - 11*w^2 + 5*w + 4], [421, 421, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 8*w - 4], [421, 421, -w^5 + w^4 + 5*w^3 - 3*w^2 - 7*w + 4], [433, 433, -2*w^2 + w + 5], [433, 433, -2*w^4 + 3*w^3 + 5*w^2 - 6*w + 3], [461, 461, 2*w^3 - 2*w^2 - 5*w + 3], [461, 461, -w^5 + 5*w^3 + 2*w^2 - 4*w + 1], [461, 461, -w^5 + 3*w^4 + 2*w^3 - 10*w^2 + 3*w + 5], [461, 461, -w^5 + 3*w^4 + 3*w^3 - 11*w^2 - w + 3], [463, 463, -2*w^4 + 3*w^3 + 7*w^2 - 8*w - 4], [463, 463, -w^5 + 4*w^4 + w^3 - 12*w^2 + 3*w - 1], [491, 491, w^5 - 7*w^3 - w^2 + 10*w + 3], [491, 491, -w^5 + 4*w^4 - w^3 - 11*w^2 + 10*w + 2], [503, 503, -2*w^5 + 5*w^4 + 5*w^3 - 15*w^2 + w + 3], [503, 503, 2*w^4 - w^3 - 8*w^2 + w + 1], [503, 503, -2*w^5 + 3*w^4 + 7*w^3 - 7*w^2 - 3*w + 1], [503, 503, -2*w^5 + 2*w^4 + 10*w^3 - 5*w^2 - 11*w + 2], [547, 547, w^5 - 2*w^4 - 6*w^3 + 9*w^2 + 10*w - 5], [547, 547, -w^4 + 2*w^3 + 2*w^2 - 4*w + 5], [547, 547, -2*w^5 + 4*w^4 + 7*w^3 - 12*w^2 - 3*w + 4], [547, 547, -w^5 + w^4 + 3*w^3 - 2*w^2 + 3*w + 2], [587, 587, -w^5 - w^4 + 8*w^3 + 4*w^2 - 13*w + 1], [587, 587, w^5 - w^4 - 5*w^3 + 2*w^2 + 7*w - 2], [617, 617, w^5 - w^4 - 4*w^3 + 3*w^2 - 1], [617, 617, w^4 - 2*w^3 - 5*w^2 + 5*w + 6], [617, 617, -w^5 + w^4 + 4*w^3 - 3*w^2 + 4], [617, 617, -w^5 + 3*w^4 + 6*w^3 - 13*w^2 - 11*w + 8], [631, 631, -2*w^5 + 2*w^4 + 10*w^3 - 5*w^2 - 11*w + 1], [631, 631, -w^5 + w^4 + 4*w^3 - 2*w - 4], [631, 631, -2*w^4 + 5*w^3 + 4*w^2 - 12*w + 4], [631, 631, -2*w^5 + 5*w^4 + 5*w^3 - 15*w^2 + w + 4], [643, 643, w^5 - 2*w^4 - 5*w^3 + 7*w^2 + 6*w - 2], [643, 643, -w^5 + 2*w^4 + w^3 - 3*w^2 + 6*w - 3], [659, 659, -2*w^4 + 4*w^3 + 7*w^2 - 10*w - 2], [659, 659, w^4 - 3*w^3 - 3*w^2 + 10*w - 1], [659, 659, w^5 - 4*w^4 + 13*w^2 - 6*w - 6], [659, 659, w^5 - 2*w^4 - 5*w^3 + 7*w^2 + 7*w - 5], [673, 673, 2*w^5 - 3*w^4 - 7*w^3 + 8*w^2 + 2*w - 3], [673, 673, w^5 - 4*w^4 - w^3 + 12*w^2 - 3*w - 2], [673, 673, -2*w^5 + 3*w^4 + 9*w^3 - 10*w^2 - 8*w + 5], [673, 673, w^5 - w^4 - 4*w^3 + 3*w + 6], [701, 701, -2*w^4 + 4*w^3 + 6*w^2 - 11*w - 3], [701, 701, 2*w^4 - 3*w^3 - 7*w^2 + 6*w + 5], [727, 727, -w^5 + 2*w^4 + w^3 - 3*w^2 + 7*w - 2], [727, 727, 2*w^5 - 3*w^4 - 7*w^3 + 8*w^2 + 3*w - 4], [727, 727, w^5 - 2*w^4 - 4*w^3 + 6*w^2 + 2*w + 1], [727, 727, w^5 + 2*w^4 - 9*w^3 - 8*w^2 + 16*w + 2], [729, 3, -3], [743, 743, w^5 - 4*w^4 + 13*w^2 - 8*w - 5], [743, 743, -w^3 + w^2 + w - 2], [743, 743, w^4 - 4*w^3 - w^2 + 10*w - 2], [743, 743, 3*w^3 - 3*w^2 - 11*w + 6], [757, 757, -w^5 + 4*w^4 - 11*w^2 + 5*w - 2], [757, 757, -3*w^4 + 3*w^3 + 11*w^2 - 5*w - 2], [757, 757, w^5 - 3*w^4 - 2*w^3 + 10*w^2 - 4], [757, 757, w^5 - 8*w^3 + w^2 + 15*w - 4], [769, 769, w^5 - 5*w^3 - 3*w^2 + 4*w + 6], [769, 769, -2*w^5 + 3*w^4 + 8*w^3 - 9*w^2 - 4*w + 3], [769, 769, w^4 - 3*w^3 - w^2 + 9*w - 3], [769, 769, w^5 - 4*w^4 - w^3 + 13*w^2 - 4*w - 5], [797, 797, w^5 + w^4 - 8*w^3 - 4*w^2 + 11*w + 2], [797, 797, w^5 - 2*w^4 - 6*w^3 + 9*w^2 + 9*w - 6], [811, 811, w^5 - 3*w^4 - 4*w^3 + 11*w^2 + 4*w - 4], [811, 811, 2*w^4 - 2*w^3 - 6*w^2 + 3*w - 4], [811, 811, -2*w^5 + 3*w^4 + 9*w^3 - 8*w^2 - 9*w + 1], [811, 811, -w^5 + w^4 + 2*w^3 + w^2 + 4*w - 5], [827, 827, w^5 + w^4 - 9*w^3 - 2*w^2 + 14*w - 3], [827, 827, w^5 - 3*w^4 + w^3 + 6*w^2 - 11*w + 4], [827, 827, w^5 - 3*w^4 + 9*w^2 - 9*w - 4], [827, 827, w^5 - w^4 - 4*w^3 + 4*w + 5], [827, 827, -w^5 + 4*w^4 + 2*w^3 - 13*w^2 - w + 3], [827, 827, -w^5 + w^4 + 7*w^3 - 4*w^2 - 11*w + 3], [839, 839, -w^5 + w^4 + 7*w^3 - 4*w^2 - 13*w + 4], [839, 839, -w^5 + 3*w^4 + w^3 - 8*w^2 + 3*w - 1], [841, 29, -2*w^4 + 5*w^3 + 5*w^2 - 13*w + 1], [853, 853, -2*w^4 + 10*w^2 + 3*w - 6], [853, 853, 2*w^5 - 4*w^4 - 8*w^3 + 13*w^2 + 8*w - 6], [853, 853, -2*w^4 + w^3 + 9*w^2 - 2*w - 4], [853, 853, -2*w^5 + 3*w^4 + 8*w^3 - 8*w^2 - 7*w + 1], [881, 881, w^5 - 5*w^3 - 2*w^2 + 5*w - 1], [881, 881, -w^5 + 3*w^4 + 4*w^3 - 12*w^2 - 5*w + 5], [883, 883, 3*w^4 - 4*w^3 - 11*w^2 + 10*w + 5], [883, 883, -3*w^4 + 3*w^3 + 12*w^2 - 5*w - 7], [883, 883, w^5 - 2*w^4 - 4*w^3 + 7*w^2 + 5*w - 2], [883, 883, w^5 - w^4 - 6*w^3 + 4*w^2 + 10*w - 6], [911, 911, -w^5 + w^4 + 6*w^3 - 5*w^2 - 7*w + 4], [911, 911, 2*w^5 - 4*w^4 - 6*w^3 + 10*w^2 + 2*w + 1], [911, 911, w^5 - w^4 - 3*w^3 + 2*w^2 - 2*w - 3], [911, 911, -w^5 + 3*w^4 + 4*w^3 - 10*w^2 - 5*w + 2], [911, 911, w^5 - 2*w^4 - w^3 + 2*w^2 - 5*w + 6], [911, 911, -2*w^5 + 4*w^4 + 8*w^3 - 12*w^2 - 8*w + 3], [937, 937, w^4 - 2*w^3 - 5*w^2 + 7*w + 5], [937, 937, w^5 + w^4 - 7*w^3 - 5*w^2 + 9*w + 4], [937, 937, 3*w^4 - 4*w^3 - 9*w^2 + 7*w - 1], [937, 937, -w^5 + 4*w^4 + w^3 - 14*w^2 + 4*w + 8], [953, 953, w^5 - 3*w^4 - w^3 + 7*w^2 - 5*w + 4], [953, 953, w^5 - 2*w^4 - 3*w^3 + 4*w^2 + 2], [967, 967, w^5 - 6*w^4 + 2*w^3 + 20*w^2 - 12*w - 6], [967, 967, w^5 - 4*w^4 - 2*w^3 + 14*w^2 + w - 5], [967, 967, -w^5 + 8*w^3 - 2*w^2 - 14*w + 10], [967, 967, w^5 - 3*w^4 + 7*w^2 - 9*w], [967, 967, -w^5 + 6*w^3 + 2*w^2 - 9*w - 2], [967, 967, -w^5 + w^4 + 5*w^3 - 2*w^2 - 4*w - 3], [1009, 1009, w^5 - 3*w^4 - 2*w^3 + 9*w^2 - 2*w - 4], [1009, 1009, -2*w^5 + 4*w^4 + 6*w^3 - 11*w^2 - w + 2], [1009, 1009, 2*w^5 - 3*w^4 - 9*w^3 + 9*w^2 + 9*w - 3], [1009, 1009, -w^5 + w^4 + 4*w^3 - w^2 - 2*w - 5], [1021, 1021, w^5 - w^4 - 5*w^3 + 5*w^2 + 6*w - 10], [1021, 1021, w^5 - 8*w^3 + 3*w^2 + 14*w - 6], [1021, 1021, 2*w^5 - 4*w^4 - 7*w^3 + 12*w^2 + 4*w - 4], [1021, 1021, -2*w^5 + 3*w^4 + 8*w^3 - 8*w^2 - 6*w + 1], [1051, 1051, w^5 - w^4 - 3*w^3 + w^2 - w - 2], [1051, 1051, 3*w^4 - 6*w^3 - 8*w^2 + 16*w - 3], [1091, 1091, -w^5 + 4*w^4 - 13*w^2 + 6*w + 5], [1091, 1091, -w^5 - w^4 + 9*w^3 + 3*w^2 - 16*w + 1], [1093, 1093, w^2 + w - 5], [1093, 1093, w^4 - 3*w^3 - w^2 + 9*w - 2], [1093, 1093, 2*w^5 - 5*w^4 - 5*w^3 + 14*w^2 - w - 2], [1093, 1093, 2*w^5 - 3*w^4 - 8*w^3 + 7*w^2 + 6*w + 2], [1163, 1163, -w^3 + 3*w^2 + 2*w - 5], [1163, 1163, 2*w^4 - 2*w^3 - 6*w^2 + 3*w - 3], [1217, 1217, w^5 + w^4 - 6*w^3 - 6*w^2 + 3*w + 5], [1217, 1217, w^5 - w^4 - 5*w^3 + w^2 + 9*w], [1231, 1231, -w^5 + 4*w^4 - w^3 - 10*w^2 + 10*w - 1], [1231, 1231, -w^5 + w^4 + 5*w^3 - w^2 - 5*w - 5], [1259, 1259, w^5 - w^4 - 5*w^3 + 4*w^2 + 6*w - 7], [1259, 1259, w^5 - w^4 - 6*w^3 + 5*w^2 + 9*w - 4], [1289, 1289, w^5 + w^4 - 11*w^3 + 23*w - 7], [1289, 1289, -w^5 + w^4 + 5*w^3 - 2*w^2 - 8*w - 2], [1289, 1289, 2*w^5 - 13*w^3 - w^2 + 19*w - 1], [1289, 1289, -2*w^5 + 4*w^4 + 10*w^3 - 15*w^2 - 14*w + 9], [1301, 1301, -2*w^5 + 3*w^4 + 8*w^3 - 9*w^2 - 6*w + 4], [1301, 1301, -w^5 - w^4 + 11*w^3 + 2*w^2 - 23*w + 3], [1301, 1301, -2*w^5 + w^4 + 11*w^3 - 2*w^2 - 12*w + 1], [1301, 1301, -2*w^5 + 3*w^4 + 9*w^3 - 10*w^2 - 9*w + 6], [1303, 1303, w^5 - 4*w^4 - w^3 + 13*w^2 - 2*w - 2], [1303, 1303, w^5 - 4*w^4 - w^3 + 14*w^2 - 3*w - 4], [1331, 11, -w^4 + 4*w^3 - 2*w^2 - 10*w + 8], [1331, 11, w^5 - 5*w^4 + w^3 + 16*w^2 - 10*w - 4], [1399, 1399, -w^5 + 6*w^4 - w^3 - 21*w^2 + 8*w + 8], [1399, 1399, 2*w^4 - 4*w^3 - 6*w^2 + 13*w - 4], [1399, 1399, 2*w^5 - 4*w^4 - 9*w^3 + 15*w^2 + 9*w - 7], [1399, 1399, w^5 - 7*w^3 + w^2 + 10*w], [1427, 1427, 2*w^5 - 5*w^4 - 7*w^3 + 17*w^2 + 6*w - 6], [1427, 1427, -w^5 + 3*w^4 + 2*w^3 - 12*w^2 + 3*w + 9], [1471, 1471, w^5 - 5*w^3 - w^2 + 4*w - 3], [1471, 1471, w^5 - w^4 - 6*w^3 + 3*w^2 + 7*w - 1], [1471, 1471, -w^5 + 3*w^4 - w^3 - 6*w^2 + 12*w - 4], [1471, 1471, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 6*w - 6], [1483, 1483, -2*w^4 + 2*w^3 + 9*w^2 - 6*w - 6], [1483, 1483, -w^5 + 4*w^4 - 2*w^3 - 8*w^2 + 12*w - 7], [1483, 1483, -w^5 + 5*w^4 + w^3 - 18*w^2 + w + 3], [1483, 1483, w^5 - 4*w^4 - 4*w^3 + 17*w^2 + 7*w - 8], [1499, 1499, -2*w^5 + 6*w^4 + 6*w^3 - 21*w^2 - 2*w + 8], [1499, 1499, 2*w^5 - 5*w^4 - 7*w^3 + 17*w^2 + 5*w - 7], [1567, 1567, 2*w^5 - 7*w^4 - w^3 + 19*w^2 - 11*w], [1567, 1567, -w^5 + w^4 + 4*w^3 - 2*w^2 - w - 3], [1567, 1567, w^5 - 2*w^4 - 3*w^3 + 6*w^2 - w - 5], [1567, 1567, -w^5 + 2*w^4 + 5*w^3 - 7*w^2 - 5*w - 2], [1583, 1583, 2*w^4 - 6*w^3 - 4*w^2 + 17*w - 3], [1583, 1583, -2*w^4 + 5*w^3 + 5*w^2 - 12*w + 1], [1609, 1609, -4*w^4 + 5*w^3 + 13*w^2 - 11*w + 2], [1609, 1609, 2*w^5 - 2*w^4 - 10*w^3 + 3*w^2 + 12*w + 5], [1637, 1637, -w^5 - 2*w^4 + 10*w^3 + 6*w^2 - 18*w], [1637, 1637, -w^4 + 6*w^2 - 10], [1637, 1637, -2*w^4 + w^3 + 8*w^2 - 7], [1637, 1637, -w^5 + 4*w^4 - 3*w^3 - 11*w^2 + 17*w + 2], [1667, 1667, 2*w^4 - w^3 - 10*w^2 + 2*w + 7], [1667, 1667, 3*w^4 - 2*w^3 - 12*w^2 + 2*w + 4], [1667, 1667, w^5 - 7*w^3 + w^2 + 9*w - 2], [1667, 1667, w^5 - 2*w^4 - 2*w^3 + 6*w^2 - 4*w - 5], [1681, 41, -w^4 + 4*w^3 + w^2 - 11*w + 4], [1681, 41, -3*w^4 + 6*w^3 + 9*w^2 - 15*w + 1], [1693, 1693, w^5 - 3*w^4 - 2*w^3 + 11*w^2 - w - 7], [1693, 1693, -w^4 - 3*w^3 + 7*w^2 + 12*w - 7], [1693, 1693, -w^4 + 5*w^2 - w + 1], [1693, 1693, -w^5 - w^4 + 9*w^3 + 2*w^2 - 16*w + 4], [1709, 1709, -w^5 + 7*w^3 - 12*w + 3], [1709, 1709, -w^5 + 3*w^4 + 3*w^3 - 11*w^2 - 3*w + 5], [1723, 1723, 2*w^5 - w^4 - 10*w^3 + 11*w + 2], [1723, 1723, 2*w^5 - 6*w^4 - 6*w^3 + 21*w^2 + 4*w - 9], [1723, 1723, -w^5 + 2*w^4 + 6*w^3 - 11*w^2 - 8*w + 10], [1723, 1723, -w^5 - w^4 + 5*w^3 + 5*w^2 - 2*w + 2], [1777, 1777, w^5 - 2*w^4 - 5*w^3 + 6*w^2 + 8*w - 2], [1777, 1777, w^5 - 5*w^4 + 17*w^2 - 5*w - 4], [1777, 1777, -2*w^5 + 6*w^4 + 5*w^3 - 20*w^2 + 9], [1777, 1777, -w^5 + 6*w^4 - 3*w^3 - 17*w^2 + 11*w], [1847, 1847, 3*w^5 - 4*w^4 - 14*w^3 + 11*w^2 + 14*w - 3], [1847, 1847, 2*w^3 - 3*w^2 - 8*w + 3], [1849, 43, 2*w^4 - 5*w^3 - 5*w^2 + 13*w - 2], [1849, 43, -w^4 - w^3 + 6*w^2 + 4*w - 3], [1861, 1861, 3*w^4 - 5*w^3 - 8*w^2 + 13*w - 6], [1861, 1861, 3*w^4 - w^3 - 12*w^2 - w + 2], [1861, 1861, -w^5 + w^4 + 6*w^3 - 4*w^2 - 11*w + 6], [1861, 1861, -w^5 + 4*w^4 + 3*w^3 - 13*w^2 - 4*w + 2], [1933, 1933, -2*w^5 + 4*w^4 + 7*w^3 - 11*w^2 - 4*w + 2], [1933, 1933, -2*w^5 + 4*w^4 + 6*w^3 - 10*w^2 - w - 2], [1933, 1933, 2*w^4 - 3*w^3 - 7*w^2 + 8*w + 6], [1933, 1933, -2*w^4 + 2*w^3 + 8*w^2 - 3*w - 8], [1973, 1973, w^4 - 4*w^3 + 12*w - 4], [1973, 1973, w^3 - 2*w - 3], [1987, 1987, -3*w^5 + 6*w^4 + 10*w^3 - 19*w^2 - 3*w + 9], [1987, 1987, -w^4 + 4*w^3 + w^2 - 14*w + 1]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [-1, 3, 1, 9, -2, -1, 10, 0, 2, 13, -1, 13, 2, -6, 16, -8, -8, -13, -13, 16, -6, -13, 8, -2, -8, 8, 14, -7, -18, -5, 6, 5, -14, -7, 4, -7, -18, 0, 22, -6, -17, -8, -8, -4, -4, -17, -11, 16, -17, -6, 0, -5, 28, -6, -17, -4, -2, -5, -24, 28, 29, -32, -15, -15, -21, -24, 20, -5, -16, -16, 28, 16, -6, 4, 26, 12, -32, 3, 14, 37, 15, 18, 20, -4, -24, -19, 14, 7, -15, -40, 34, 23, 15, 32, -40, -12, 37, -33, 11, -27, 2, -27, 2, -8, -1, -12, -8, 1, 1, -12, -27, 10, 6, -1, 0, -34, -11, -46, -35, 40, -29, -4, 4, -40, 28, -48, -27, 40, 54, -1, 9, 9, -18, 31, -13, -18, -2, -2, -20, -37, 7, 46, 12, -27, 34, 31, -57, -16, -34, 21, -37, -26, -14, 7, 52, -47, -36, 1, 34, -44, 55, -40, -10, 15, 24, 2, -32, -1, -34, 32, -45, -16, 17, -29, -61, 15, 27, -50, 60, 46, 46, -20, 46, -28, -28, 3, -41, -22, 22, -2, -2, -10, 12, 64, -2, 4, -40, 14, 47, 30, -36, 4, -40, 16, 60, 36, 36, 41, 22, 0, -58, -14, 52, 24, -42, -70, 26, 40, 59, 2, 24, 10, 10, 32, -45, 20, 9, -4, 40, 44, -44, -20, -27, 28, 2, 69, -63, 14, -41, 46, -5, 61, 57, 51, -26, 36, -8, 25, -2, 30, 65, 54, -14, -66, 66, -20, -64, -58, -14, 50, 6, -62, 37, -27, 28, -65, -30, -30, -62, -40, 69, -71, -16, -10, 1, -70, 40, -40, -73]; heckeEigenvalues := AssociativeArray(); for i := 1 to #heckeEigenvaluesArray do heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i]; end for; ALEigenvalues := AssociativeArray(); ALEigenvalues[ideal] := 1; // EXAMPLE: // pp := Factorization(2*ZF)[1][1]; // heckeEigenvalues[pp]; print "To reconstruct the Hilbert newform f, type f, iso := Explode(make_newform());"; function make_newform(); M := HilbertCuspForms(F, NN); S := NewSubspace(M); // SetVerbose("ModFrmHil", 1); NFD := NewformDecomposition(S); newforms := [* Eigenform(U) : U in NFD *]; if #newforms eq 0 then; print "No Hilbert newforms at this level"; return 0; end if; print "Testing ", #newforms, " possible newforms"; newforms := [* f: f in newforms | IsIsomorphic(BaseField(f), K) *]; print #newforms, " newforms have the correct Hecke field"; if #newforms eq 0 then; print "No Hilbert newform found with the correct Hecke field"; return 0; end if; autos := Automorphisms(K); xnewforms := [* *]; for f in newforms do; if K eq RationalField() then; Append(~xnewforms, [* f, autos[1] *]); else; flag, iso := IsIsomorphic(K,BaseField(f)); for a in autos do; Append(~xnewforms, [* f, a*iso *]); end for; end if; end for; newforms := xnewforms; for P in primes do; xnewforms := [* *]; for f_iso in newforms do; f, iso := Explode(f_iso); if HeckeEigenvalue(f,P) eq iso(heckeEigenvalues[P]) then; Append(~xnewforms, f_iso); end if; end for; newforms := xnewforms; if #newforms eq 0 then; print "No Hilbert newform found which matches the Hecke eigenvalues"; return 0; else if #newforms eq 1 then; print "success: unique match"; return newforms[1]; end if; end if; end for; print #newforms, "Hilbert newforms found which match the Hecke eigenvalues"; return newforms[1]; end function;