/* 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, -9, 2, 11, -5, -2, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [13, 13, 3*w^5 - 2*w^4 - 17*w^3 + 10*w^2 + 16*w - 3], [13, 13, w^5 - w^4 - 5*w^3 + 5*w^2 + 3*w - 3], [13, 13, 2*w^5 - w^4 - 12*w^3 + 4*w^2 + 13*w], [13, 13, -w^2 + 3], [41, 41, -2*w^5 + w^4 + 11*w^3 - 4*w^2 - 10*w - 1], [41, 41, -4*w^5 + 3*w^4 + 23*w^3 - 14*w^2 - 22*w + 4], [41, 41, w^5 - 6*w^3 + w^2 + 7*w - 2], [41, 41, -3*w^5 + w^4 + 18*w^3 - 4*w^2 - 20*w - 1], [43, 43, 3*w^5 - 2*w^4 - 17*w^3 + 11*w^2 + 16*w - 6], [43, 43, -2*w^5 + w^4 + 12*w^3 - 6*w^2 - 14*w + 5], [49, 7, 2*w^5 - w^4 - 12*w^3 + 5*w^2 + 13*w - 4], [64, 2, -2], [71, 71, -2*w^5 + w^4 + 12*w^3 - 4*w^2 - 12*w - 1], [71, 71, -3*w^5 + 2*w^4 + 17*w^3 - 9*w^2 - 15*w], [83, 83, 4*w^5 - 2*w^4 - 24*w^3 + 9*w^2 + 26*w - 1], [83, 83, -4*w^5 + 2*w^4 + 23*w^3 - 9*w^2 - 23*w], [97, 97, -w^5 + 7*w^3 + w^2 - 11*w - 3], [97, 97, -2*w^5 + w^4 + 11*w^3 - 6*w^2 - 9*w + 3], [113, 113, -4*w^5 + 2*w^4 + 24*w^3 - 9*w^2 - 27*w + 2], [113, 113, 3*w^5 - 2*w^4 - 18*w^3 + 10*w^2 + 20*w - 6], [113, 113, -2*w^5 + w^4 + 12*w^3 - 5*w^2 - 12*w + 3], [113, 113, 3*w^5 - w^4 - 18*w^3 + 4*w^2 + 20*w], [127, 127, -w^5 + w^4 + 6*w^3 - 5*w^2 - 7*w], [127, 127, -2*w^5 + w^4 + 11*w^3 - 4*w^2 - 10*w - 2], [127, 127, -4*w^5 + 3*w^4 + 23*w^3 - 14*w^2 - 22*w + 5], [127, 127, -8*w^5 + 5*w^4 + 46*w^3 - 23*w^2 - 45*w + 4], [139, 139, w^3 + w^2 - 4*w - 1], [139, 139, 5*w^5 - 2*w^4 - 29*w^3 + 9*w^2 + 29*w + 1], [139, 139, -w^5 + w^4 + 5*w^3 - 6*w^2 - 2*w + 5], [139, 139, w^5 - 6*w^3 + 5*w + 2], [167, 167, w^4 - 4*w^2 + 1], [167, 167, -3*w^5 + w^4 + 18*w^3 - 4*w^2 - 21*w - 1], [169, 13, -3*w^5 + w^4 + 18*w^3 - 5*w^2 - 20*w], [181, 181, -w^4 + w^3 + 5*w^2 - 3*w - 3], [181, 181, -5*w^5 + 3*w^4 + 29*w^3 - 15*w^2 - 30*w + 6], [181, 181, -w^5 + w^4 + 5*w^3 - 6*w^2 - 2*w + 4], [181, 181, w^3 + w^2 - 4*w - 2], [197, 197, w^5 - w^4 - 6*w^3 + 4*w^2 + 6*w + 2], [197, 197, -4*w^5 + 2*w^4 + 24*w^3 - 9*w^2 - 26*w + 4], [197, 197, -2*w^5 + 2*w^4 + 12*w^3 - 11*w^2 - 12*w + 5], [197, 197, 2*w^5 - w^4 - 12*w^3 + 6*w^2 + 13*w - 7], [197, 197, w^5 - 6*w^3 - w^2 + 7*w + 1], [197, 197, -4*w^5 + 3*w^4 + 24*w^3 - 14*w^2 - 25*w + 2], [211, 211, 2*w^5 - w^4 - 13*w^3 + 5*w^2 + 18*w - 1], [211, 211, -2*w^5 + w^4 + 12*w^3 - 4*w^2 - 12*w - 4], [223, 223, 4*w^5 - w^4 - 23*w^3 + 5*w^2 + 24*w - 1], [223, 223, -3*w^5 + w^4 + 17*w^3 - 4*w^2 - 15*w - 4], [223, 223, 3*w^5 - 3*w^4 - 16*w^3 + 15*w^2 + 11*w - 5], [223, 223, 3*w^5 - 3*w^4 - 16*w^3 + 14*w^2 + 13*w - 5], [239, 239, -w^5 + w^4 + 6*w^3 - 6*w^2 - 7*w + 4], [239, 239, 2*w^5 - w^4 - 12*w^3 + 4*w^2 + 12*w + 2], [251, 251, 3*w^5 - 3*w^4 - 16*w^3 + 14*w^2 + 13*w - 4], [251, 251, 5*w^5 - 3*w^4 - 29*w^3 + 14*w^2 + 28*w - 1], [251, 251, 6*w^5 - 4*w^4 - 35*w^3 + 19*w^2 + 37*w - 7], [251, 251, -8*w^5 + 5*w^4 + 46*w^3 - 24*w^2 - 46*w + 8], [281, 281, -6*w^5 + 4*w^4 + 35*w^3 - 18*w^2 - 35*w + 3], [281, 281, 3*w^5 - 3*w^4 - 17*w^3 + 15*w^2 + 16*w - 10], [281, 281, -3*w^5 + w^4 + 17*w^3 - 5*w^2 - 18*w + 3], [281, 281, 5*w^5 - 3*w^4 - 30*w^3 + 14*w^2 + 33*w - 5], [293, 293, -w^5 + w^4 + 6*w^3 - 5*w^2 - 8*w + 3], [307, 307, 5*w^5 - 3*w^4 - 30*w^3 + 13*w^2 + 32*w], [307, 307, -4*w^5 + 3*w^4 + 22*w^3 - 15*w^2 - 20*w + 5], [307, 307, 3*w^5 - 2*w^4 - 17*w^3 + 8*w^2 + 15*w], [307, 307, -2*w^5 + w^4 + 12*w^3 - 5*w^2 - 11*w - 1], [307, 307, 5*w^5 - 4*w^4 - 28*w^3 + 20*w^2 + 26*w - 10], [307, 307, -4*w^5 + 2*w^4 + 23*w^3 - 8*w^2 - 22*w - 3], [337, 337, -w^5 + w^4 + 5*w^3 - 4*w^2 - 3*w + 3], [337, 337, 5*w^5 - 3*w^4 - 29*w^3 + 14*w^2 + 29*w], [337, 337, 5*w^5 - 3*w^4 - 28*w^3 + 15*w^2 + 27*w - 5], [337, 337, -4*w^5 + 4*w^4 + 22*w^3 - 20*w^2 - 19*w + 10], [349, 349, -8*w^5 + 5*w^4 + 46*w^3 - 25*w^2 - 45*w + 9], [349, 349, -w^4 + 2*w^3 + 5*w^2 - 7*w - 3], [419, 419, 5*w^5 - 3*w^4 - 29*w^3 + 13*w^2 + 28*w - 1], [419, 419, -6*w^5 + 4*w^4 + 35*w^3 - 20*w^2 - 35*w + 10], [421, 421, -6*w^5 + 3*w^4 + 34*w^3 - 15*w^2 - 32*w + 3], [421, 421, -5*w^5 + 3*w^4 + 29*w^3 - 14*w^2 - 28*w], [421, 421, w^5 - 5*w^3 - w^2 + 2*w + 3], [421, 421, -6*w^5 + 3*w^4 + 35*w^3 - 14*w^2 - 37*w + 2], [433, 433, -4*w^5 + 4*w^4 + 22*w^3 - 19*w^2 - 19*w + 6], [433, 433, 7*w^5 - 4*w^4 - 40*w^3 + 19*w^2 + 40*w - 6], [449, 449, 2*w^4 - w^3 - 10*w^2 + 4*w + 6], [449, 449, -4*w^5 + 4*w^4 + 23*w^3 - 20*w^2 - 21*w + 10], [461, 461, -6*w^5 + 3*w^4 + 36*w^3 - 15*w^2 - 40*w + 4], [461, 461, -7*w^5 + 5*w^4 + 40*w^3 - 23*w^2 - 39*w + 6], [461, 461, 4*w^5 - 2*w^4 - 22*w^3 + 9*w^2 + 19*w + 1], [461, 461, 5*w^5 - 2*w^4 - 30*w^3 + 10*w^2 + 32*w - 1], [463, 463, 4*w^5 - 2*w^4 - 23*w^3 + 10*w^2 + 23*w], [463, 463, -6*w^5 + 4*w^4 + 35*w^3 - 20*w^2 - 34*w + 7], [491, 491, 7*w^5 - 5*w^4 - 41*w^3 + 24*w^2 + 41*w - 8], [491, 491, -2*w^5 + w^4 + 12*w^3 - 4*w^2 - 15*w - 2], [491, 491, -w^5 + 6*w^3 + w^2 - 7*w], [491, 491, w^5 - w^4 - 6*w^3 + 4*w^2 + 6*w + 3], [491, 491, 5*w^5 - 3*w^4 - 29*w^3 + 16*w^2 + 30*w - 9], [491, 491, -3*w^5 + 2*w^4 + 18*w^3 - 8*w^2 - 20*w - 2], [503, 503, 3*w^5 - 2*w^4 - 18*w^3 + 10*w^2 + 20*w - 7], [503, 503, 6*w^5 - 3*w^4 - 35*w^3 + 14*w^2 + 37*w - 3], [547, 547, 2*w^5 - 12*w^3 - w^2 + 15*w + 3], [547, 547, -w^5 + 2*w^4 + 6*w^3 - 9*w^2 - 6*w + 3], [547, 547, 7*w^5 - 4*w^4 - 41*w^3 + 19*w^2 + 41*w - 3], [547, 547, 2*w^5 - w^4 - 13*w^3 + 6*w^2 + 17*w - 3], [547, 547, -4*w^5 + 3*w^4 + 23*w^3 - 13*w^2 - 21*w], [547, 547, -7*w^5 + 4*w^4 + 41*w^3 - 20*w^2 - 43*w + 7], [587, 587, 2*w^4 - w^3 - 10*w^2 + 4*w + 7], [587, 587, 5*w^5 - 2*w^4 - 29*w^3 + 10*w^2 + 31*w - 2], [601, 601, w^3 + 2*w^2 - 4*w - 4], [601, 601, w^5 - 7*w^3 - 2*w^2 + 11*w + 5], [617, 617, -7*w^5 + 4*w^4 + 41*w^3 - 20*w^2 - 43*w + 9], [617, 617, -2*w^5 + 13*w^3 - 16*w], [643, 643, -2*w^5 + 2*w^4 + 12*w^3 - 8*w^2 - 13*w], [643, 643, -3*w^5 + w^4 + 18*w^3 - 3*w^2 - 21*w - 3], [659, 659, 5*w^5 - 2*w^4 - 30*w^3 + 8*w^2 + 33*w + 2], [659, 659, w^5 - 6*w^3 + 2*w^2 + 7*w - 4], [673, 673, -2*w^5 + 2*w^4 + 10*w^3 - 11*w^2 - 5*w + 6], [673, 673, 6*w^5 - 4*w^4 - 34*w^3 + 20*w^2 + 33*w - 9], [673, 673, -3*w^5 + 2*w^4 + 17*w^3 - 9*w^2 - 16*w + 5], [673, 673, 7*w^5 - 4*w^4 - 40*w^3 + 20*w^2 + 38*w - 6], [673, 673, -3*w^5 + 2*w^4 + 16*w^3 - 11*w^2 - 12*w + 6], [673, 673, w^5 + w^4 - 6*w^3 - 6*w^2 + 8*w + 5], [701, 701, 5*w^5 - 4*w^4 - 28*w^3 + 20*w^2 + 26*w - 9], [701, 701, 4*w^5 - 3*w^4 - 22*w^3 + 15*w^2 + 20*w - 6], [727, 727, -2*w^5 + 2*w^4 + 11*w^3 - 8*w^2 - 9*w - 2], [727, 727, -3*w^5 + 3*w^4 + 18*w^3 - 15*w^2 - 20*w + 9], [729, 3, -3], [743, 743, -3*w^5 + w^4 + 18*w^3 - 4*w^2 - 22*w], [743, 743, -5*w^5 + 2*w^4 + 29*w^3 - 9*w^2 - 31*w - 1], [743, 743, -5*w^5 + 3*w^4 + 29*w^3 - 14*w^2 - 31*w + 6], [743, 743, -3*w^5 + 3*w^4 + 17*w^3 - 14*w^2 - 17*w + 3], [757, 757, -6*w^5 + 3*w^4 + 35*w^3 - 13*w^2 - 37*w + 2], [757, 757, 3*w^5 - 3*w^4 - 17*w^3 + 13*w^2 + 16*w - 1], [757, 757, -8*w^5 + 5*w^4 + 47*w^3 - 25*w^2 - 48*w + 10], [757, 757, 4*w^5 - 2*w^4 - 25*w^3 + 10*w^2 + 29*w - 4], [797, 797, -6*w^5 + 4*w^4 + 35*w^3 - 19*w^2 - 34*w + 3], [797, 797, -w^5 + w^4 + 7*w^3 - 6*w^2 - 10*w + 3], [811, 811, 5*w^5 - 4*w^4 - 28*w^3 + 18*w^2 + 25*w - 4], [811, 811, -4*w^5 + 3*w^4 + 22*w^3 - 15*w^2 - 17*w + 7], [827, 827, w^5 - w^4 - 5*w^3 + 5*w^2 + w - 4], [827, 827, -6*w^5 + 3*w^4 + 36*w^3 - 14*w^2 - 39*w + 3], [827, 827, -6*w^5 + 3*w^4 + 36*w^3 - 14*w^2 - 39*w + 2], [827, 827, -w^5 - w^4 + 7*w^3 + 6*w^2 - 11*w - 4], [839, 839, -2*w^5 + 2*w^4 + 10*w^3 - 10*w^2 - 6*w + 7], [839, 839, -6*w^5 + 4*w^4 + 34*w^3 - 20*w^2 - 32*w + 5], [841, 29, -4*w^5 + 3*w^4 + 24*w^3 - 15*w^2 - 25*w + 7], [841, 29, 5*w^5 - 3*w^4 - 30*w^3 + 15*w^2 + 32*w - 6], [841, 29, 5*w^5 - 2*w^4 - 30*w^3 + 10*w^2 + 33*w - 2], [853, 853, -5*w^5 + 3*w^4 + 29*w^3 - 16*w^2 - 29*w + 9], [853, 853, -5*w^5 + 3*w^4 + 29*w^3 - 16*w^2 - 30*w + 7], [881, 881, -3*w^5 + 3*w^4 + 16*w^3 - 14*w^2 - 13*w + 3], [881, 881, 4*w^5 - 2*w^4 - 24*w^3 + 9*w^2 + 25*w + 2], [881, 881, -8*w^5 + 5*w^4 + 46*w^3 - 24*w^2 - 46*w + 9], [881, 881, 5*w^5 - 2*w^4 - 30*w^3 + 8*w^2 + 34*w + 2], [883, 883, 3*w^5 - w^4 - 17*w^3 + 6*w^2 + 16*w - 5], [883, 883, 7*w^5 - 5*w^4 - 40*w^3 + 24*w^2 + 37*w - 8], [911, 911, w^5 - 8*w^3 + 14*w], [911, 911, -3*w^5 + 2*w^4 + 17*w^3 - 9*w^2 - 17*w - 1], [911, 911, -8*w^5 + 5*w^4 + 46*w^3 - 25*w^2 - 44*w + 9], [911, 911, 2*w^5 - 2*w^4 - 12*w^3 + 9*w^2 + 11*w], [937, 937, 5*w^5 - 2*w^4 - 30*w^3 + 9*w^2 + 32*w - 1], [937, 937, 3*w^5 - 3*w^4 - 17*w^3 + 15*w^2 + 15*w - 4], [967, 967, -w^5 + 7*w^3 - 9*w + 2], [967, 967, 6*w^5 - 4*w^4 - 35*w^3 + 20*w^2 + 36*w - 11]]; primes := [ideal : I in primesArray]; heckePol := x^10 - 2*x^9 - 61*x^8 + 150*x^7 + 982*x^6 - 3096*x^5 - 2013*x^4 + 10564*x^3 - 580*x^2 - 9416*x + 1456; K := NumberField(heckePol); heckeEigenvaluesArray := [e, e, -249450329/1561462156604*e^9 - 1574522153/780731078302*e^8 + 18682232881/1561462156604*e^7 + 90312154355/780731078302*e^6 - 39277322381/111533011186*e^5 - 647717390956/390365539151*e^4 + 7182695299093/1561462156604*e^3 + 626615278650/390365539151*e^2 - 2613153633425/390365539151*e + 180949662384/55766505593, -249450329/1561462156604*e^9 - 1574522153/780731078302*e^8 + 18682232881/1561462156604*e^7 + 90312154355/780731078302*e^6 - 39277322381/111533011186*e^5 - 647717390956/390365539151*e^4 + 7182695299093/1561462156604*e^3 + 626615278650/390365539151*e^2 - 2613153633425/390365539151*e + 180949662384/55766505593, -3210047721/1561462156604*e^9 + 632027281/780731078302*e^8 + 201550461341/1561462156604*e^7 - 84157610479/780731078302*e^6 - 258818040179/111533011186*e^5 + 1188782656610/390365539151*e^4 + 16485576456465/1561462156604*e^3 - 4349721961251/390365539151*e^2 - 4580339003552/390365539151*e + 257564496808/55766505593, -3210047721/1561462156604*e^9 + 632027281/780731078302*e^8 + 201550461341/1561462156604*e^7 - 84157610479/780731078302*e^6 - 258818040179/111533011186*e^5 + 1188782656610/390365539151*e^4 + 16485576456465/1561462156604*e^3 - 4349721961251/390365539151*e^2 - 4580339003552/390365539151*e + 257564496808/55766505593, -12164277169/7807310783020*e^9 - 2972779856/1951827695755*e^8 + 740084960717/7807310783020*e^7 + 69994951409/1951827695755*e^6 - 186554964711/111533011186*e^5 + 1450660439831/1951827695755*e^4 + 67147958117729/7807310783020*e^3 - 29554207783757/3903655391510*e^2 - 21761170194903/1951827695755*e + 2903223230266/278832527965, -12164277169/7807310783020*e^9 - 2972779856/1951827695755*e^8 + 740084960717/7807310783020*e^7 + 69994951409/1951827695755*e^6 - 186554964711/111533011186*e^5 + 1450660439831/1951827695755*e^4 + 67147958117729/7807310783020*e^3 - 29554207783757/3903655391510*e^2 - 21761170194903/1951827695755*e + 2903223230266/278832527965, -49914469183/15614621566040*e^9 - 6776563797/3903655391510*e^8 + 3006956162339/15614621566040*e^7 + 5364819034/1951827695755*e^6 - 728125133955/223066022372*e^5 + 8149880708507/3903655391510*e^4 + 219198911878843/15614621566040*e^3 - 53081483918989/7807310783020*e^2 - 37504947529333/1951827695755*e + 1436269741226/278832527965, -49914469183/15614621566040*e^9 - 6776563797/3903655391510*e^8 + 3006956162339/15614621566040*e^7 + 5364819034/1951827695755*e^6 - 728125133955/223066022372*e^5 + 8149880708507/3903655391510*e^4 + 219198911878843/15614621566040*e^3 - 53081483918989/7807310783020*e^2 - 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2697094092123/15614621566040*e^7 + 1238889610779/3903655391510*e^6 + 436346701219/223066022372*e^5 - 20845406012369/3903655391510*e^4 + 188121564952189/15614621566040*e^3 - 120936106034017/7807310783020*e^2 - 83462679258494/1951827695755*e + 6127729737758/278832527965, -18413251977/7807310783020*e^9 - 12398046721/3903655391510*e^8 + 1223187613181/7807310783020*e^7 + 593469994759/3903655391510*e^6 - 362988960457/111533011186*e^5 - 3425616205672/1951827695755*e^4 + 180642042372137/7807310783020*e^3 + 16891117281992/1951827695755*e^2 - 85074068844824/1951827695755*e - 5626368264212/278832527965, -18413251977/7807310783020*e^9 - 12398046721/3903655391510*e^8 + 1223187613181/7807310783020*e^7 + 593469994759/3903655391510*e^6 - 362988960457/111533011186*e^5 - 3425616205672/1951827695755*e^4 + 180642042372137/7807310783020*e^3 + 16891117281992/1951827695755*e^2 - 85074068844824/1951827695755*e - 5626368264212/278832527965, 259739142253/15614621566040*e^9 + 102101658489/7807310783020*e^8 - 15581663203469/15614621566040*e^7 - 2319175339811/7807310783020*e^6 + 3733221484443/223066022372*e^5 - 9431135720236/1951827695755*e^4 - 1082007423176913/15614621566040*e^3 - 4359978819269/1951827695755*e^2 + 180517771266463/1951827695755*e + 1070230654674/278832527965, 259739142253/15614621566040*e^9 + 102101658489/7807310783020*e^8 - 15581663203469/15614621566040*e^7 - 2319175339811/7807310783020*e^6 + 3733221484443/223066022372*e^5 - 9431135720236/1951827695755*e^4 - 1082007423176913/15614621566040*e^3 - 4359978819269/1951827695755*e^2 + 180517771266463/1951827695755*e + 1070230654674/278832527965]; 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;