/* 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, 5, -4, -2, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [5, 5, w + 1], [5, 5, -w + 2], [7, 7, -w^2 + 2*w + 1], [7, 7, -w^2 + 2], [9, 3, w^3 - w^2 - 4*w], [16, 2, 2], [17, 17, -w^3 + 2*w^2 + 3*w - 2], [17, 17, -w^3 + w^2 + 4*w - 2], [25, 5, w^2 - w - 3], [37, 37, 2*w - 1], [43, 43, -w^3 + 2*w^2 + 2*w - 2], [43, 43, w^3 - w^2 - 3*w + 1], [59, 59, 2*w - 5], [59, 59, -2*w - 3], [79, 79, -w^3 + 2*w^2 + 5*w - 4], [79, 79, -w^3 + w^2 + 6*w - 2], [83, 83, -w^3 + 7*w], [83, 83, -w^3 + 2*w^2 + 4*w - 2], [83, 83, -w^3 + w^2 + 5*w - 3], [83, 83, w^2 - 2*w - 6], [89, 89, w^2 - 2*w - 4], [89, 89, w^2 - 5], [101, 101, -2*w^3 + 5*w^2 + 5*w - 11], [101, 101, w^2 + w - 4], [101, 101, w^2 - 3*w - 2], [101, 101, 2*w^3 - w^2 - 9*w - 3], [109, 109, w^3 - w^2 - 6*w - 3], [109, 109, -w^3 + 2*w^2 + 5*w - 9], [121, 11, w^3 - 8*w], [121, 11, w^3 - 3*w^2 - 5*w + 7], [131, 131, w^3 - 5*w - 6], [131, 131, -w^3 + 3*w^2 + 2*w - 10], [163, 163, 2*w^2 - 3*w - 6], [163, 163, 2*w^2 - w - 7], [167, 167, w^3 - w^2 - 3*w - 4], [167, 167, -w^3 + 2*w^2 + 2*w - 7], [193, 193, 2*w^3 - 3*w^2 - 7*w + 2], [193, 193, 2*w^3 - 3*w^2 - 7*w + 6], [227, 227, 2*w^3 - 3*w^2 - 8*w + 1], [227, 227, -2*w^3 + 3*w^2 + 8*w - 8], [251, 251, 2*w^2 - w - 9], [251, 251, 2*w^2 - 3*w - 8], [257, 257, w^3 + w^2 - 8*w - 7], [257, 257, -4*w^3 + 5*w^2 + 17*w - 1], [269, 269, 2*w^3 - 2*w^2 - 10*w + 3], [269, 269, 2*w^3 - 3*w^2 - 9*w + 2], [269, 269, 2*w^3 - 3*w^2 - 9*w + 8], [269, 269, -2*w^3 + 4*w^2 + 8*w - 7], [277, 277, w^3 - w^2 - 4*w - 4], [277, 277, -w^3 + 2*w^2 + 3*w - 8], [289, 17, 2*w^2 - 2*w - 3], [293, 293, -w^2 - 2*w + 5], [293, 293, -w^3 + 3*w^2 + 3*w - 4], [293, 293, w^3 - 6*w + 1], [293, 293, -2*w^3 + 4*w^2 + 7*w - 6], [311, 311, -w^3 + 3*w^2 + w - 7], [311, 311, w^3 - 4*w - 4], [331, 331, -2*w^3 + 4*w^2 + 9*w - 10], [331, 331, 2*w^3 - 2*w^2 - 11*w + 1], [337, 337, 2*w^2 - 3*w - 4], [337, 337, -w^3 + 2*w^2 + 6*w - 3], [337, 337, w^3 - w^2 - 7*w + 4], [337, 337, 2*w^2 - w - 5], [353, 353, -3*w^3 + 4*w^2 + 13*w - 2], [353, 353, 3*w^3 - 5*w^2 - 12*w + 12], [373, 373, -2*w^3 + 6*w^2 + 3*w - 11], [373, 373, -w^3 - w^2 + 6*w + 7], [373, 373, -w^3 + 4*w^2 + w - 11], [373, 373, 2*w^3 - 9*w - 4], [379, 379, w^3 - w^2 - 7*w + 1], [379, 379, 3*w^3 - 7*w^2 - 7*w + 11], [379, 379, 3*w^3 - 2*w^2 - 12*w], [379, 379, -w^3 + 2*w^2 + 6*w - 6], [383, 383, -3*w^3 + 4*w^2 + 12*w - 8], [383, 383, 3*w^3 - 5*w^2 - 11*w + 5], [421, 421, 2*w^3 - 2*w^2 - 9*w - 4], [421, 421, -2*w^3 + 4*w^2 + 7*w - 13], [457, 457, -w^3 + 2*w^2 + 6*w - 5], [457, 457, w^3 - w^2 - 7*w + 2], [461, 461, -w^3 + 3*w^2 + 4*w - 4], [461, 461, -w^3 + 7*w - 2], [463, 463, -w^3 + 4*w^2 + 3*w - 10], [463, 463, w^3 + w^2 - 8*w - 4], [467, 467, -w^3 + 6*w - 2], [467, 467, -w^3 + 3*w^2 + 3*w - 3], [479, 479, -w^3 + w^2 + w - 4], [479, 479, w^3 - 2*w^2 - 3], [487, 487, -3*w^3 + 4*w^2 + 12*w - 3], [487, 487, -3*w^3 + 5*w^2 + 11*w - 10], [499, 499, 3*w^3 - 4*w^2 - 11*w + 8], [499, 499, 3*w^3 - 5*w^2 - 10*w + 4], [503, 503, -w^3 + 2*w^2 + w - 5], [503, 503, w^3 - w^2 - 2*w - 3], [521, 521, 2*w^3 - 4*w^2 - 6*w + 3], [521, 521, -w^3 + 10*w - 8], [521, 521, w^3 - 3*w^2 - 7*w + 1], [521, 521, -2*w^3 + 2*w^2 + 8*w - 5], [541, 541, 2*w - 7], [541, 541, 2*w + 5], [547, 547, w^3 - 4*w^2 + 7], [547, 547, w^3 + w^2 - 5*w - 4], [563, 563, -2*w^3 + 4*w^2 + 7*w - 5], [563, 563, -2*w^3 + 2*w^2 + 9*w - 4], [587, 587, -w^3 - w^2 + 4*w + 6], [587, 587, w^3 - 4*w^2 + w + 8], [593, 593, w^2 + w - 7], [593, 593, 2*w^3 - 5*w^2 - 4*w + 5], [593, 593, -2*w^3 + w^2 + 8*w - 2], [593, 593, w^2 - 3*w - 5], [613, 613, 2*w^3 - 3*w^2 - 11*w + 2], [613, 613, w^3 + w^2 - 6*w - 3], [613, 613, 3*w^3 - 4*w^2 - 12*w + 4], [613, 613, 2*w^3 - 3*w^2 - 11*w + 10], [631, 631, -w^3 - w^2 + 7*w + 6], [631, 631, -w^3 + 4*w^2 + 2*w - 11], [647, 647, w^2 - 4*w - 3], [647, 647, w^2 + 2*w - 6], [677, 677, w^2 + w - 8], [677, 677, -w^3 + 7*w - 3], [677, 677, -w^3 + 3*w^2 + 4*w - 3], [677, 677, w^2 - 3*w - 6], [709, 709, w^3 - 7*w - 8], [709, 709, w^3 - 3*w^2 - 4*w + 14], [739, 739, w^3 - w^2 - 4*w - 5], [739, 739, -3*w^3 + 3*w^2 + 16*w - 3], [739, 739, -3*w^3 + 6*w^2 + 13*w - 13], [739, 739, -w^3 + 2*w^2 + 3*w - 9], [751, 751, -w - 5], [751, 751, w^3 + w^2 - 7*w - 4], [751, 751, -w^3 + 4*w^2 + 2*w - 9], [751, 751, w - 6], [757, 757, -w^3 + 4*w^2 + 2*w - 10], [757, 757, w^3 + w^2 - 7*w - 5], [761, 761, -3*w^3 + 2*w^2 + 17*w - 1], [761, 761, -3*w^3 + 6*w^2 + 9*w - 17], [761, 761, 2*w^3 - 5*w^2 - 9*w + 9], [761, 761, 2*w^3 - 2*w^2 - 7*w - 6], [773, 773, -3*w^3 + 8*w^2 + 6*w - 12], [773, 773, w^3 - 4*w - 6], [773, 773, -w^3 + 3*w^2 + w - 9], [773, 773, -3*w^3 + w^2 + 13*w + 1], [797, 797, 3*w^3 - 2*w^2 - 13*w + 1], [797, 797, 3*w^3 - 7*w^2 - 8*w + 11], [823, 823, 2*w^3 - 3*w^2 - 6*w - 4], [823, 823, -2*w^3 + 3*w^2 + 7*w + 3], [823, 823, w^3 - 2*w^2 - 7*w - 3], [823, 823, -3*w^3 + 5*w^2 + 9*w - 2], [857, 857, 2*w^3 - w^2 - 10*w + 1], [857, 857, -2*w^3 + 5*w^2 + 6*w - 8], [883, 883, 3*w^3 - 3*w^2 - 12*w + 1], [883, 883, 3*w^3 - 6*w^2 - 9*w + 11], [887, 887, -2*w^3 + 5*w^2 + 7*w - 9], [887, 887, -4*w^3 + 6*w^2 + 17*w - 7], [887, 887, -2*w^3 + 5*w^2 + 9*w - 16], [887, 887, 2*w^3 - w^2 - 11*w + 1], [907, 907, -3*w^3 + 3*w^2 + 16*w - 2], [907, 907, -3*w^3 + 6*w^2 + 13*w - 14], [919, 919, 2*w^3 - 2*w^2 - 7*w + 9], [919, 919, -2*w^3 + 2*w^2 + 12*w - 5], [929, 929, -2*w^3 + 4*w^2 + 8*w - 5], [929, 929, 3*w^3 - 4*w^2 - 11*w + 12], [929, 929, 3*w^3 - 5*w^2 - 10*w], [929, 929, -2*w^3 + 2*w^2 + 10*w - 5], [967, 967, 3*w^3 - 5*w^2 - 9*w + 11], [967, 967, -3*w^3 + 7*w^2 + 6*w - 10], [971, 971, 2*w^3 - 2*w^2 - 7*w - 5], [971, 971, -2*w^3 + w^2 + 8*w - 3], [971, 971, 2*w^3 - 5*w^2 - 4*w + 4], [971, 971, -2*w^3 + 4*w^2 + 5*w - 12], [983, 983, 4*w - 7], [983, 983, 2*w^2 - 4*w - 7], [983, 983, 2*w^2 - 9], [983, 983, 4*w + 3], [991, 991, 2*w^3 - 6*w^2 - 5*w + 14], [991, 991, -2*w^3 + 11*w + 5]]; primes := [ideal : I in primesArray]; heckePol := x^8 - 36*x^6 + 432*x^4 - 2020*x^2 + 3008; K := NumberField(heckePol); heckeEigenvaluesArray := [e, e, 1/37*e^6 - 53/74*e^4 + 171/37*e^2 - 192/37, 1/37*e^6 - 53/74*e^4 + 171/37*e^2 - 192/37, -1, 7/74*e^6 - 102/37*e^4 + 839/37*e^2 - 1819/37, -3/148*e^7 + 49/74*e^5 - 230/37*e^3 + 551/37*e, -3/148*e^7 + 49/74*e^5 - 230/37*e^3 + 551/37*e, 2/37*e^6 - 53/37*e^4 + 379/37*e^2 - 606/37, -11/74*e^6 + 155/37*e^4 - 1218/37*e^2 + 2462/37, -3/37*e^6 + 159/74*e^4 - 587/37*e^2 + 1316/37, -3/37*e^6 + 159/74*e^4 - 587/37*e^2 + 1316/37, 1/37*e^7 - 53/74*e^5 + 171/37*e^3 - 118/37*e, 1/37*e^7 - 53/74*e^5 + 171/37*e^3 - 118/37*e, 1/37*e^6 - 45/37*e^4 + 504/37*e^2 - 1080/37, 1/37*e^6 - 45/37*e^4 + 504/37*e^2 - 1080/37, -1/37*e^7 + 53/74*e^5 - 171/37*e^3 + 266/37*e, 3/74*e^7 - 49/37*e^5 + 460/37*e^3 - 1102/37*e, 3/74*e^7 - 49/37*e^5 + 460/37*e^3 - 1102/37*e, -1/37*e^7 + 53/74*e^5 - 171/37*e^3 + 266/37*e, 3/148*e^7 - 49/74*e^5 + 230/37*e^3 - 625/37*e, 3/148*e^7 - 49/74*e^5 + 230/37*e^3 - 625/37*e, -3*e, 3/148*e^7 - 49/74*e^5 + 230/37*e^3 - 551/37*e, 3/148*e^7 - 49/74*e^5 + 230/37*e^3 - 551/37*e, -3*e, -7/74*e^6 + 102/37*e^4 - 876/37*e^2 + 1930/37, -7/74*e^6 + 102/37*e^4 - 876/37*e^2 + 1930/37, 7/74*e^6 - 102/37*e^4 + 876/37*e^2 - 2226/37, 7/74*e^6 - 102/37*e^4 + 876/37*e^2 - 2226/37, -2/37*e^7 + 53/37*e^5 - 342/37*e^3 + 310/37*e, -2/37*e^7 + 53/37*e^5 - 342/37*e^3 + 310/37*e, 1/37*e^6 - 8/37*e^4 - 236/37*e^2 + 1436/37, 1/37*e^6 - 8/37*e^4 - 236/37*e^2 + 1436/37, -3/74*e^7 + 49/37*e^5 - 460/37*e^3 + 1176/37*e, -3/74*e^7 + 49/37*e^5 - 460/37*e^3 + 1176/37*e, 2/37*e^6 - 53/37*e^4 + 342/37*e^2 - 458/37, 2/37*e^6 - 53/37*e^4 + 342/37*e^2 - 458/37, 3/74*e^7 - 49/37*e^5 + 497/37*e^3 - 1546/37*e, 3/74*e^7 - 49/37*e^5 + 497/37*e^3 - 1546/37*e, 6/37*e^7 - 355/74*e^5 + 1507/37*e^3 - 3594/37*e, 6/37*e^7 - 355/74*e^5 + 1507/37*e^3 - 3594/37*e, -7/74*e^7 + 102/37*e^5 - 839/37*e^3 + 1819/37*e, -7/74*e^7 + 102/37*e^5 - 839/37*e^3 + 1819/37*e, e^3 - 11*e, 5/148*e^7 - 57/74*e^5 + 112/37*e^3 + 389/37*e, 5/148*e^7 - 57/74*e^5 + 112/37*e^3 + 389/37*e, e^3 - 11*e, 8/37*e^6 - 249/37*e^4 + 2145/37*e^2 - 4718/37, 8/37*e^6 - 249/37*e^4 + 2145/37*e^2 - 4718/37, -1/37*e^6 + 8/37*e^4 + 236/37*e^2 - 918/37, -7/74*e^7 + 102/37*e^5 - 839/37*e^3 + 2041/37*e, -9/148*e^7 + 147/74*e^5 - 690/37*e^3 + 1653/37*e, -9/148*e^7 + 147/74*e^5 - 690/37*e^3 + 1653/37*e, -7/74*e^7 + 102/37*e^5 - 839/37*e^3 + 2041/37*e, -1/74*e^7 + 4/37*e^5 + 118/37*e^3 - 940/37*e, -1/74*e^7 + 4/37*e^5 + 118/37*e^3 - 940/37*e, 7/74*e^6 - 241/74*e^4 + 1209/37*e^2 - 3188/37, 7/74*e^6 - 241/74*e^4 + 1209/37*e^2 - 3188/37, -11/37*e^6 + 310/37*e^4 - 2362/37*e^2 + 4554/37, -9/74*e^6 + 147/37*e^4 - 1380/37*e^2 + 3306/37, -9/74*e^6 + 147/37*e^4 - 1380/37*e^2 + 3306/37, -11/37*e^6 + 310/37*e^4 - 2362/37*e^2 + 4554/37, -5/148*e^7 + 57/74*e^5 - 112/37*e^3 - 241/37*e, -5/148*e^7 + 57/74*e^5 - 112/37*e^3 - 241/37*e, -2/37*e^6 + 53/37*e^4 - 416/37*e^2 + 1198/37, -16/37*e^6 + 461/37*e^4 - 3735/37*e^2 + 8178/37, -16/37*e^6 + 461/37*e^4 - 3735/37*e^2 + 8178/37, -2/37*e^6 + 53/37*e^4 - 416/37*e^2 + 1198/37, -3/74*e^6 + 61/74*e^4 + 21/37*e^2 - 1340/37, 15/74*e^6 - 379/74*e^4 + 1153/37*e^2 - 996/37, 15/74*e^6 - 379/74*e^4 + 1153/37*e^2 - 996/37, -3/74*e^6 + 61/74*e^4 + 21/37*e^2 - 1340/37, -7/74*e^7 + 102/37*e^5 - 839/37*e^3 + 1856/37*e, -7/74*e^7 + 102/37*e^5 - 839/37*e^3 + 1856/37*e, 7/37*e^6 - 204/37*e^4 + 1752/37*e^2 - 4526/37, 7/37*e^6 - 204/37*e^4 + 1752/37*e^2 - 4526/37, 2/37*e^6 - 90/37*e^4 + 1156/37*e^2 - 3122/37, 2/37*e^6 - 90/37*e^4 + 1156/37*e^2 - 3122/37, 4/37*e^7 - 106/37*e^5 + 684/37*e^3 - 583/37*e, 4/37*e^7 - 106/37*e^5 + 684/37*e^3 - 583/37*e, -11/37*e^6 + 310/37*e^4 - 2288/37*e^2 + 3592/37, -11/37*e^6 + 310/37*e^4 - 2288/37*e^2 + 3592/37, 7/74*e^7 - 102/37*e^5 + 876/37*e^3 - 2374/37*e, 7/74*e^7 - 102/37*e^5 + 876/37*e^3 - 2374/37*e, 1/74*e^7 - 4/37*e^5 - 118/37*e^3 + 644/37*e, 1/74*e^7 - 4/37*e^5 - 118/37*e^3 + 644/37*e, -9/74*e^6 + 257/74*e^4 - 1121/37*e^2 + 3528/37, -9/74*e^6 + 257/74*e^4 - 1121/37*e^2 + 3528/37, -11/74*e^6 + 273/74*e^4 - 1033/37*e^2 + 3572/37, -11/74*e^6 + 273/74*e^4 - 1033/37*e^2 + 3572/37, 11/74*e^7 - 347/74*e^5 + 1625/37*e^3 - 4164/37*e, 11/74*e^7 - 347/74*e^5 + 1625/37*e^3 - 4164/37*e, -11/148*e^7 + 155/74*e^5 - 572/37*e^3 + 935/37*e, -2/37*e^7 + 53/37*e^5 - 305/37*e^3 - 97/37*e, -2/37*e^7 + 53/37*e^5 - 305/37*e^3 - 97/37*e, -11/148*e^7 + 155/74*e^5 - 572/37*e^3 + 935/37*e, 1/37*e^6 - 45/37*e^4 + 615/37*e^2 - 2042/37, 1/37*e^6 - 45/37*e^4 + 615/37*e^2 - 2042/37, 5/37*e^6 - 114/37*e^4 + 670/37*e^2 - 1700/37, 5/37*e^6 - 114/37*e^4 + 670/37*e^2 - 1700/37, -5/74*e^7 + 57/37*e^5 - 224/37*e^3 - 630/37*e, -5/74*e^7 + 57/37*e^5 - 224/37*e^3 - 630/37*e, 11/74*e^7 - 155/37*e^5 + 1218/37*e^3 - 2610/37*e, 11/74*e^7 - 155/37*e^5 + 1218/37*e^3 - 2610/37*e, 9/148*e^7 - 147/74*e^5 + 764/37*e^3 - 2541/37*e, -e^3 + 15*e, -e^3 + 15*e, 9/148*e^7 - 147/74*e^5 + 764/37*e^3 - 2541/37*e, -1/37*e^6 + 45/37*e^4 - 652/37*e^2 + 1746/37, 14/37*e^6 - 371/37*e^4 + 2579/37*e^2 - 4094/37, 14/37*e^6 - 371/37*e^4 + 2579/37*e^2 - 4094/37, -1/37*e^6 + 45/37*e^4 - 652/37*e^2 + 1746/37, -7/37*e^6 + 167/37*e^4 - 1012/37*e^2 + 1936/37, -7/37*e^6 + 167/37*e^4 - 1012/37*e^2 + 1936/37, -10/37*e^7 + 302/37*e^5 - 2598/37*e^3 + 6212/37*e, -10/37*e^7 + 302/37*e^5 - 2598/37*e^3 + 6212/37*e, -3/37*e^7 + 98/37*e^5 - 920/37*e^3 + 2241/37*e, -5/148*e^7 + 57/74*e^5 - 112/37*e^3 - 463/37*e, -5/148*e^7 + 57/74*e^5 - 112/37*e^3 - 463/37*e, -3/37*e^7 + 98/37*e^5 - 920/37*e^3 + 2241/37*e, 7/74*e^6 - 102/37*e^4 + 876/37*e^2 - 2670/37, 7/74*e^6 - 102/37*e^4 + 876/37*e^2 - 2670/37, -6/37*e^6 + 196/37*e^4 - 1840/37*e^2 + 4260/37, -18/37*e^6 + 514/37*e^4 - 4188/37*e^2 + 8932/37, -18/37*e^6 + 514/37*e^4 - 4188/37*e^2 + 8932/37, -6/37*e^6 + 196/37*e^4 - 1840/37*e^2 + 4260/37, 1/37*e^6 - 127/74*e^4 + 763/37*e^2 - 784/37, -5/74*e^6 + 151/74*e^4 - 705/37*e^2 + 2256/37, -5/74*e^6 + 151/74*e^4 - 705/37*e^2 + 2256/37, 1/37*e^6 - 127/74*e^4 + 763/37*e^2 - 784/37, -13/37*e^6 + 363/37*e^4 - 2852/37*e^2 + 5678/37, -13/37*e^6 + 363/37*e^4 - 2852/37*e^2 + 5678/37, -17/148*e^7 + 253/74*e^5 - 1106/37*e^3 + 2999/37*e, -5/74*e^7 + 57/37*e^5 - 261/37*e^3 - 297/37*e, -5/74*e^7 + 57/37*e^5 - 261/37*e^3 - 297/37*e, -17/148*e^7 + 253/74*e^5 - 1106/37*e^3 + 2999/37*e, 23/148*e^7 - 351/74*e^5 + 1566/37*e^3 - 4027/37*e, 1/37*e^7 - 45/37*e^5 + 578/37*e^3 - 1931/37*e, 1/37*e^7 - 45/37*e^5 + 578/37*e^3 - 1931/37*e, 23/148*e^7 - 351/74*e^5 + 1566/37*e^3 - 4027/37*e, 1/148*e^7 - 41/74*e^5 + 348/37*e^3 - 1343/37*e, 1/148*e^7 - 41/74*e^5 + 348/37*e^3 - 1343/37*e, 5/37*e^6 - 151/37*e^4 + 1114/37*e^2 - 960/37, 4/37*e^6 - 143/37*e^4 + 1572/37*e^2 - 4320/37, 4/37*e^6 - 143/37*e^4 + 1572/37*e^2 - 4320/37, 5/37*e^6 - 151/37*e^4 + 1114/37*e^2 - 960/37, -17/148*e^7 + 253/74*e^5 - 1106/37*e^3 + 2851/37*e, -17/148*e^7 + 253/74*e^5 - 1106/37*e^3 + 2851/37*e, 10/37*e^6 - 567/74*e^4 + 1969/37*e^2 - 2660/37, 10/37*e^6 - 567/74*e^4 + 1969/37*e^2 - 2660/37, 5/37*e^7 - 151/37*e^5 + 1299/37*e^3 - 3032/37*e, 3/74*e^7 - 61/74*e^5 + 53/37*e^3 + 452/37*e, 3/74*e^7 - 61/74*e^5 + 53/37*e^3 + 452/37*e, 5/37*e^7 - 151/37*e^5 + 1299/37*e^3 - 3032/37*e, 13/37*e^6 - 763/74*e^4 + 3111/37*e^2 - 7084/37, 13/37*e^6 - 763/74*e^4 + 3111/37*e^2 - 7084/37, -33/74*e^6 + 1041/74*e^4 - 4653/37*e^2 + 10864/37, -33/74*e^6 + 1041/74*e^4 - 4653/37*e^2 + 10864/37, 4/37*e^7 - 106/37*e^5 + 684/37*e^3 - 657/37*e, -17/148*e^7 + 253/74*e^5 - 1106/37*e^3 + 3295/37*e, -17/148*e^7 + 253/74*e^5 - 1106/37*e^3 + 3295/37*e, 4/37*e^7 - 106/37*e^5 + 684/37*e^3 - 657/37*e, -18/37*e^6 + 551/37*e^4 - 5002/37*e^2 + 12040/37, -18/37*e^6 + 551/37*e^4 - 5002/37*e^2 + 12040/37, -5/37*e^7 + 151/37*e^5 - 1299/37*e^3 + 2662/37*e, 1/74*e^7 - 4/37*e^5 - 118/37*e^3 + 718/37*e, 1/74*e^7 - 4/37*e^5 - 118/37*e^3 + 718/37*e, -5/37*e^7 + 151/37*e^5 - 1299/37*e^3 + 2662/37*e, -17/74*e^7 + 253/37*e^5 - 2138/37*e^3 + 5184/37*e, 5/74*e^7 - 57/37*e^5 + 261/37*e^3 - 36/37*e, 5/74*e^7 - 57/37*e^5 + 261/37*e^3 - 36/37*e, -17/74*e^7 + 253/37*e^5 - 2138/37*e^3 + 5184/37*e, -11/74*e^6 + 347/74*e^4 - 1477/37*e^2 + 3128/37, -11/74*e^6 + 347/74*e^4 - 1477/37*e^2 + 3128/37]; 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;