/* 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![3, 2, -5, -2, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [3, 3, w], [4, 2, -w^3 + 3*w^2 + w - 2], [11, 11, -w^3 + 3*w^2 + 2*w - 5], [13, 13, w^3 - 2*w^2 - 4*w + 2], [13, 13, -w + 2], [17, 17, w^3 - 3*w^2 - 2*w + 2], [19, 19, -w^2 + w + 4], [19, 19, w^3 - 2*w^2 - 3*w + 2], [23, 23, w^2 - 2*w - 1], [27, 3, w^3 - 2*w^2 - 5*w - 1], [29, 29, -w^3 + 2*w^2 + 5*w - 1], [37, 37, -w^3 + 2*w^2 + 5*w - 4], [41, 41, 2*w^3 - 5*w^2 - 6*w + 4], [53, 53, w^3 - 2*w^2 - 3*w - 2], [59, 59, w - 4], [67, 67, 2*w^2 - 3*w - 8], [73, 73, 2*w^3 - 5*w^2 - 6*w + 5], [89, 89, -2*w^3 + 6*w^2 + 5*w - 7], [97, 97, -2*w^3 + 6*w^2 + 3*w - 7], [97, 97, -w^3 + 3*w^2 + 3*w - 1], [103, 103, w^2 - 4*w - 4], [113, 113, 2*w^3 - 5*w^2 - 4*w + 1], [113, 113, -w^3 + 3*w^2 + w - 5], [139, 139, 2*w^2 - 3*w - 4], [139, 139, w^3 - w^2 - 6*w - 1], [149, 149, 2*w^3 - 5*w^2 - 5*w + 2], [149, 149, w^3 - 4*w^2 + 10], [151, 151, -3*w - 1], [157, 157, w^2 - w - 8], [157, 157, w^3 - 7*w - 8], [163, 163, -3*w^3 + 8*w^2 + 8*w - 10], [163, 163, w^3 - 3*w^2 - 4*w + 7], [163, 163, -w^3 + 4*w^2 + w - 8], [163, 163, -2*w^3 + 5*w^2 + 7*w - 4], [167, 167, -2*w^3 + 5*w^2 + 8*w - 4], [169, 13, -2*w^3 + 4*w^2 + 7*w - 5], [173, 173, 2*w^3 - 4*w^2 - 9*w + 4], [173, 173, -w^3 + 5*w^2 - 2*w - 5], [181, 181, 3*w^3 - 7*w^2 - 12*w + 7], [191, 191, w^3 - 3*w^2 - 3*w - 1], [199, 199, 2*w^3 - 3*w^2 - 10*w - 1], [199, 199, 2*w^3 - 6*w^2 - 6*w + 13], [223, 223, -2*w^3 + 6*w^2 + 4*w - 11], [223, 223, 3*w^3 - 7*w^2 - 12*w + 8], [223, 223, -2*w^3 + 5*w^2 + 6*w - 1], [229, 229, -w^3 + 3*w^2 + 2*w + 1], [229, 229, 3*w^3 - 4*w^2 - 15*w - 5], [233, 233, -3*w^3 + 7*w^2 + 11*w - 5], [233, 233, 2*w^3 - 5*w^2 - 4*w + 5], [241, 241, w^3 - 3*w^2 - 5*w + 7], [257, 257, w^3 - 3*w^2 - 4*w + 8], [263, 263, w^2 - 4*w - 5], [271, 271, -2*w^3 + 6*w^2 + 5*w - 13], [277, 277, -2*w^3 + 6*w^2 + 3*w - 8], [277, 277, w^3 - 2*w^2 - 2*w - 2], [281, 281, 2*w^3 - 5*w^2 - 4*w + 2], [283, 283, -3*w^2 + 7*w + 10], [283, 283, -w^3 + 4*w^2 - w - 7], [293, 293, 2*w^3 - 5*w^2 - 3*w + 4], [293, 293, w^3 - 8*w - 4], [307, 307, -w^3 + 2*w^2 + 6*w - 2], [311, 311, w^2 - 5], [317, 317, -w^3 + 5*w^2 - 2*w - 11], [331, 331, -w^3 + 3*w^2 - 5], [349, 349, -2*w^2 + 5*w + 2], [353, 353, -2*w^3 + 5*w^2 + 4*w - 4], [353, 353, 3*w^3 - 7*w^2 - 10*w + 2], [353, 353, -2*w^3 + 7*w^2 + 2*w - 10], [353, 353, w^3 + w^2 - 9*w - 13], [361, 19, -w^3 + w^2 + 5*w - 1], [373, 373, -w^3 + 3*w^2 + 4*w - 11], [373, 373, 2*w^3 - 5*w^2 - 5*w - 2], [383, 383, w^3 - 5*w^2 + w + 13], [389, 389, 3*w^3 - 10*w^2 - 4*w + 16], [397, 397, -w^3 + 2*w^2 + 3*w + 5], [397, 397, -3*w^3 + 7*w^2 + 10*w - 7], [419, 419, -w^3 + 2*w^2 + 4*w + 4], [419, 419, 3*w^3 - 7*w^2 - 12*w + 5], [421, 421, -w^2 + 5*w - 2], [421, 421, 3*w^3 - 7*w^2 - 10*w + 5], [431, 431, w^2 - 10], [439, 439, w^3 - w^2 - 7*w - 7], [461, 461, -2*w^3 + 7*w^2 + 5*w - 10], [467, 467, -3*w^3 + 6*w^2 + 14*w - 4], [479, 479, 3*w^2 - 6*w - 5], [479, 479, -4*w^3 + 11*w^2 + 10*w - 10], [491, 491, -w^3 + w^2 + 6*w - 1], [499, 499, -4*w^3 + 9*w^2 + 15*w - 10], [499, 499, 2*w^3 - 5*w^2 - 7*w + 2], [503, 503, -2*w^3 + 7*w^2 - 8], [521, 521, 2*w^2 - w - 7], [521, 521, -3*w^3 + 7*w^2 + 11*w - 11], [569, 569, 2*w^2 - 5*w - 1], [569, 569, -4*w - 1], [571, 571, -w^3 + 3*w^2 - 7], [577, 577, -w^3 + w^2 + 8*w - 4], [587, 587, -3*w^3 + 8*w^2 + 7*w - 8], [593, 593, 3*w^2 - 4*w - 11], [593, 593, -3*w^3 + 9*w^2 + 6*w - 14], [601, 601, -3*w^3 + 9*w^2 + 4*w - 10], [601, 601, 5*w^3 - 9*w^2 - 24*w - 2], [607, 607, 2*w^2 - 7*w - 5], [613, 613, w^3 - 6*w - 2], [613, 613, -w^2 + 3*w + 8], [617, 617, -w^2 + 4*w - 5], [619, 619, 3*w^3 - 6*w^2 - 13*w + 5], [625, 5, -5], [631, 631, -2*w^3 + 4*w^2 + 8*w - 7], [647, 647, -2*w^3 + 4*w^2 + 5*w - 1], [647, 647, w^3 - 3*w^2 - 2*w - 2], [653, 653, w^3 - 4*w^2 - w + 13], [653, 653, w^3 - 2*w^2 - 7*w + 5], [659, 659, -4*w^3 + 10*w^2 + 15*w - 14], [659, 659, 5*w^3 - 12*w^2 - 17*w + 14], [661, 661, w^2 - 5*w - 2], [661, 661, -w^3 + 5*w^2 - 2*w - 7], [673, 673, -w^3 + w^2 + 6*w - 2], [683, 683, 2*w^3 - 8*w^2 + w + 8], [683, 683, 5*w^3 - 12*w^2 - 19*w + 14], [701, 701, -2*w^3 + 6*w^2 + 5*w - 4], [709, 709, -w^2 + 3*w - 4], [709, 709, -2*w^3 + 5*w^2 + 9*w - 8], [719, 719, w^2 - 7], [719, 719, -2*w^3 + 3*w^2 + 8*w + 5], [733, 733, -2*w^3 + 7*w^2 + 3*w - 10], [739, 739, 4*w^3 - 11*w^2 - 13*w + 14], [739, 739, -5*w^3 + 10*w^2 + 21*w - 4], [739, 739, 3*w^3 - 8*w^2 - 6*w + 8], [739, 739, -w^3 + w^2 + 8*w - 5], [743, 743, -w - 5], [743, 743, w^3 - w^2 - 8*w - 8], [751, 751, -2*w^3 + 5*w^2 + 8*w - 2], [751, 751, -2*w^3 + 6*w^2 - 1], [757, 757, -2*w^3 + 6*w^2 + 3*w - 10], [757, 757, 5*w^2 - 9*w - 22], [761, 761, -2*w^3 + 4*w^2 + 9*w - 7], [761, 761, 2*w^3 - 3*w^2 - 12*w - 1], [769, 769, 5*w^3 - 13*w^2 - 14*w + 10], [773, 773, -3*w^3 + 5*w^2 + 16*w + 1], [773, 773, w^3 - 11*w + 2], [787, 787, -w^3 + 4*w^2 - w - 11], [797, 797, -5*w^3 + 12*w^2 + 17*w - 11], [809, 809, -2*w^3 + 7*w^2 + 4*w - 11], [809, 809, 4*w^3 - 9*w^2 - 14*w + 4], [821, 821, -w^3 + 5*w^2 - 2*w - 17], [827, 827, -3*w^3 + 8*w^2 + 9*w - 7], [829, 829, 2*w^3 - 6*w^2 - 5*w + 2], [829, 829, -w^3 + 4*w^2 - w - 10], [839, 839, w^2 - 8], [853, 853, -5*w^3 + 12*w^2 + 18*w - 16], [853, 853, w^3 - 6*w - 8], [863, 863, 5*w^3 - 11*w^2 - 19*w + 7], [881, 881, -3*w^3 + 9*w^2 + 9*w - 11], [907, 907, -w^3 + 2*w^2 + 7*w - 4], [907, 907, -2*w^3 + 3*w^2 + 7*w + 4], [929, 929, -w^3 + 5*w^2 - 3*w - 11], [937, 937, 2*w^3 - 5*w^2 - 6*w + 11], [937, 937, -2*w^3 + 6*w^2 + 3*w - 11], [941, 941, -3*w^3 + 8*w^2 + 11*w - 8], [953, 953, -4*w^3 + 10*w^2 + 10*w - 13], [953, 953, w^3 - 5*w^2 - w + 13], [967, 967, 3*w^3 - 7*w^2 - 9*w + 7], [967, 967, -2*w^3 + 5*w^2 + 8*w + 1], [967, 967, 4*w^3 - 13*w^2 - 6*w + 20], [967, 967, -2*w^2 + 3*w + 13], [971, 971, -6*w^3 + 13*w^2 + 25*w - 8], [977, 977, w^3 - w^2 - 4*w - 10], [977, 977, 2*w^3 - 6*w^2 - 7*w + 13], [983, 983, -4*w^3 + 10*w^2 + 11*w - 13], [983, 983, -w^3 + 5*w^2 - w - 7], [983, 983, 3*w^3 - 9*w^2 - 7*w + 19], [983, 983, -2*w^3 + 3*w^2 + 10*w - 1], [997, 997, 5*w^3 - 12*w^2 - 17*w + 13], [997, 997, -2*w^3 + 3*w^2 + 13*w - 4], [997, 997, w^3 - 7*w - 2], [997, 997, w^3 - 9*w - 2]]; primes := [ideal : I in primesArray]; heckePol := x^12 - 6*x^11 - 4*x^10 + 78*x^9 - 62*x^8 - 336*x^7 + 436*x^6 + 512*x^5 - 832*x^4 - 152*x^3 + 316*x^2 + 72*x + 4; K := NumberField(heckePol); heckeEigenvaluesArray := [e, -269/222*e^11 + 1327/222*e^10 + 399/37*e^9 - 2994/37*e^8 - 760/111*e^7 + 41177/111*e^6 - 15563/111*e^5 - 70438/111*e^4 + 11127/37*e^3 + 29573/111*e^2 - 2366/111*e - 236/111, 31/74*e^11 - 225/74*e^10 + 30/37*e^9 + 1404/37*e^8 - 2232/37*e^7 - 5582/37*e^6 + 12729/37*e^5 + 6740/37*e^4 - 23439/37*e^3 + 1005/37*e^2 + 9673/37*e + 1146/37, 193/222*e^11 - 773/222*e^10 - 853/74*e^9 + 1855/37*e^8 + 6269/111*e^7 - 27880/111*e^6 - 15344/111*e^5 + 55418/111*e^4 + 7519/37*e^3 - 33826/111*e^2 - 19841/111*e - 2120/111, -581/222*e^11 + 3169/222*e^10 + 623/37*e^9 - 6963/37*e^8 + 8162/111*e^7 + 91757/111*e^6 - 79484/111*e^5 - 143830/111*e^4 + 51404/37*e^3 + 42539/111*e^2 - 45194/111*e - 5246/111, -1, -281/222*e^11 + 1099/222*e^10 + 641/37*e^9 - 2650/37*e^8 - 9997/111*e^7 + 40151/111*e^6 + 26545/111*e^5 - 81193/111*e^4 - 13342/37*e^3 + 52451/111*e^2 + 31225/111*e + 2056/111, -385/111*e^11 + 2120/111*e^10 + 790/37*e^9 - 9278/37*e^8 + 12224/111*e^7 + 121496/111*e^6 - 111698/111*e^5 - 188152/111*e^4 + 71898/37*e^3 + 52718/111*e^2 - 66113/111*e - 7664/111, -257/74*e^11 + 1333/74*e^10 + 989/37*e^9 - 8904/37*e^8 + 1410/37*e^7 + 39946/37*e^6 - 25000/37*e^5 - 65381/37*e^4 + 50437/37*e^3 + 23419/37*e^2 - 11722/37*e - 1418/37, -8/37*e^11 - 41/37*e^10 + 450/37*e^9 + 266/37*e^8 - 5249/37*e^7 + 889/37*e^6 + 23436/37*e^5 - 8568/37*e^4 - 41266/37*e^3 + 13410/37*e^2 + 20220/37*e + 2168/37, 40/37*e^11 - 737/74*e^10 + 451/37*e^9 + 4368/37*e^8 - 10829/37*e^7 - 15582/37*e^6 + 56276/37*e^5 + 12056/37*e^4 - 101584/37*e^3 + 15164/37*e^2 + 44162/37*e + 4626/37, -61/37*e^11 + 605/74*e^10 + 536/37*e^9 - 4086/37*e^8 - 281/37*e^7 + 18725/37*e^6 - 7207/37*e^5 - 32253/37*e^4 + 15174/37*e^3 + 14256/37*e^2 - 1130/37*e - 378/37, -169/74*e^11 + 522/37*e^10 + 253/37*e^9 - 6667/37*e^8 + 6322/37*e^7 + 27712/37*e^6 - 41307/37*e^5 - 38052/37*e^4 + 76305/37*e^3 + 3018/37*e^2 - 27102/37*e - 2760/37, 89/74*e^11 - 677/74*e^10 + 331/74*e^9 + 4149/37*e^8 - 7444/37*e^7 - 15904/37*e^6 + 41123/37*e^5 + 16931/37*e^4 - 75028/37*e^3 + 7089/37*e^2 + 31521/37*e + 3240/37, 131/74*e^11 - 693/74*e^10 - 470/37*e^9 + 4607/37*e^8 - 1218/37*e^7 - 20490/37*e^6 + 15290/37*e^5 + 32910/37*e^4 - 30798/37*e^3 - 11083/37*e^2 + 8876/37*e + 1508/37, -359/222*e^11 + 2059/222*e^10 + 290/37*e^9 - 4410/37*e^8 + 8828/111*e^7 + 55571/111*e^6 - 66386/111*e^5 - 78340/111*e^4 + 42006/37*e^3 + 9905/111*e^2 - 43862/111*e - 3692/111, 733/222*e^11 - 2416/111*e^10 - 124/37*e^9 + 10129/37*e^8 - 37384/111*e^7 - 122680/111*e^6 + 226213/111*e^5 + 155000/111*e^4 - 138905/37*e^3 + 11033/111*e^2 + 160648/111*e + 17062/111, 7/2*e^11 - 19*e^10 - 23*e^9 + 251*e^8 - 91*e^7 - 1109*e^6 + 919*e^5 + 1772*e^4 - 1781*e^3 - 591*e^2 + 512*e + 74, 145/222*e^11 - 121/111*e^10 - 587/37*e^9 + 826/37*e^8 + 15719/111*e^7 - 17332/111*e^6 - 63140/111*e^5 + 49028/111*e^4 + 36072/37*e^3 - 46876/111*e^2 - 56750/111*e - 6938/111, 115/37*e^11 - 516/37*e^10 - 1261/37*e^9 + 7156/37*e^8 + 3901/37*e^7 - 34022/37*e^6 - 1765/37*e^5 + 62226/37*e^4 - 2529/37*e^3 - 31976/37*e^2 - 9148/37*e - 714/37, 1303/222*e^11 - 3550/111*e^10 - 1407/37*e^9 + 15619/37*e^8 - 17929/111*e^7 - 206257/111*e^6 + 176686/111*e^5 + 324815/111*e^4 - 114581/37*e^3 - 98428/111*e^2 + 101704/111*e + 12310/111, 254/37*e^11 - 1205/37*e^10 - 2503/37*e^9 + 16548/37*e^8 + 4753/37*e^7 - 77704/37*e^6 + 14000/37*e^5 + 139500/37*e^4 - 36364/37*e^3 - 68440/37*e^2 - 6954/37*e + 430/37, 1/37*e^11 - 240/37*e^10 + 1100/37*e^9 + 2381/37*e^8 - 15129/37*e^7 - 4750/37*e^6 + 70497/37*e^5 - 11953/37*e^4 - 124351/37*e^3 + 32632/37*e^2 + 58060/37*e + 6056/37, -553/74*e^11 + 3257/74*e^10 + 1211/37*e^9 - 21077/37*e^8 + 15544/37*e^7 + 89711/37*e^6 - 111876/37*e^5 - 130834/37*e^4 + 210573/37*e^3 + 23863/37*e^2 - 72661/37*e - 8078/37, 194/111*e^11 - 1864/111*e^10 + 858/37*e^9 + 7291/37*e^8 - 57832/111*e^7 - 76087/111*e^6 + 297847/111*e^5 + 50339/111*e^4 - 179136/37*e^3 + 89300/111*e^2 + 236530/111*e + 24904/111, 217/37*e^11 - 2447/74*e^10 - 2379/74*e^9 + 15956/37*e^8 - 8974/37*e^7 - 68750/37*e^6 + 73163/37*e^5 + 103018/37*e^4 - 140630/37*e^3 - 23374/37*e^2 + 46659/37*e + 5758/37, -475/74*e^11 + 2593/74*e^10 + 3085/74*e^9 - 17166/37*e^8 + 6524/37*e^7 + 75993/37*e^6 - 64492/37*e^5 - 121329/37*e^4 + 125448/37*e^3 + 39547/37*e^2 - 36757/37*e - 4920/37, 266/37*e^11 - 1495/37*e^10 - 1513/37*e^9 + 19590/37*e^8 - 10147/37*e^7 - 85272/37*e^6 + 84925/37*e^5 + 131336/37*e^4 - 161685/37*e^3 - 35534/37*e^2 + 50110/37*e + 5836/37, -793/111*e^11 + 8383/222*e^10 + 3819/74*e^9 - 18557/37*e^8 + 13589/111*e^7 + 247754/111*e^6 - 175646/111*e^5 - 400234/111*e^4 + 115297/37*e^3 + 137561/111*e^2 - 87395/111*e - 11426/111, -101/222*e^11 + 1189/222*e^10 - 399/37*e^9 - 2260/37*e^8 + 22442/111*e^7 + 21797/111*e^6 - 113012/111*e^5 - 6226/111*e^4 + 68238/37*e^3 - 40303/111*e^2 - 92576/111*e - 10124/111, 5/111*e^11 - 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104416/37*e^2 - 42506/37*e - 2676/37, 211/222*e^11 + 2011/222*e^10 - 5501/74*e^9 - 2583/37*e^8 + 100988/111*e^7 - 7138/111*e^6 - 458459/111*e^5 + 154856/111*e^4 + 271199/37*e^3 - 264502/111*e^2 - 398399/111*e - 40856/111, 503/37*e^11 - 4011/74*e^10 - 13427/74*e^9 + 28850/37*e^8 + 33499/37*e^7 - 144534/37*e^6 - 84725/37*e^5 + 288038/37*e^4 + 126524/37*e^3 - 178384/37*e^2 - 106443/37*e - 10698/37, -2899/111*e^11 + 31363/222*e^10 + 12827/74*e^9 - 69012/37*e^8 + 73097/111*e^7 + 912380/111*e^6 - 753476/111*e^5 - 1443418/111*e^4 + 490224/37*e^3 + 452018/111*e^2 - 427145/111*e - 52310/111, 632/37*e^11 - 6731/74*e^10 - 4470/37*e^9 + 44735/37*e^8 - 12272/37*e^7 - 199398/37*e^6 + 147407/37*e^5 + 323263/37*e^4 - 289914/37*e^3 - 112634/37*e^2 + 76500/37*e + 10620/37]; 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;