/* 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![2, 1, -7, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, w], [2, 2, -1/2*w^3 + w^2 + 3/2*w], [11, 11, 1/2*w^3 - w^2 - 5/2*w + 4], [11, 11, -w^3 + w^2 + 6*w + 1], [13, 13, -w^3 + w^2 + 6*w - 3], [19, 19, -1/2*w^3 + w^2 + 5/2*w], [23, 23, -1/2*w^3 + 2*w^2 - 1/2*w - 2], [31, 31, -w^3 + w^2 + 6*w - 1], [31, 31, 1/2*w^3 - 7/2*w], [43, 43, 1/2*w^3 - w^2 - 9/2*w + 2], [67, 67, w^2 - w - 5], [67, 67, -1/2*w^3 + 11/2*w + 2], [73, 73, w^2 + w + 1], [79, 79, 1/2*w^3 + w^2 - 5/2*w - 2], [81, 3, -3], [83, 83, 2*w^3 - 2*w^2 - 12*w + 5], [83, 83, -1/2*w^3 - w^2 + 1/2*w + 2], [89, 89, -3/2*w^3 + 2*w^2 + 17/2*w - 2], [89, 89, 3/2*w^3 - 2*w^2 - 21/2*w + 2], [97, 97, 1/2*w^3 - w^2 - 1/2*w - 2], [97, 97, -w^3 + w^2 + 8*w + 1], [103, 103, 2*w^3 - 2*w^2 - 14*w + 1], [103, 103, w^2 - w - 3], [107, 107, -3/2*w^3 + 2*w^2 + 21/2*w - 4], [109, 109, 5/2*w^3 - 3*w^2 - 33/2*w + 4], [109, 109, -1/2*w^3 + w^2 + 9/2*w - 4], [113, 113, -w^3 + w^2 + 8*w - 3], [113, 113, 1/2*w^3 - w^2 - 1/2*w + 2], [121, 11, 5/2*w^3 - 2*w^2 - 31/2*w + 2], [127, 127, -3/2*w^3 + w^2 + 23/2*w], [127, 127, -1/2*w^3 + 11/2*w], [131, 131, w^2 - 3*w - 3], [137, 137, 1/2*w^3 - w^2 - 5/2*w + 6], [139, 139, -3/2*w^3 + 2*w^2 + 13/2*w + 2], [151, 151, -1/2*w^3 + 2*w^2 + 3/2*w - 10], [157, 157, 7/2*w^3 - 5*w^2 - 47/2*w + 10], [157, 157, 7/2*w^3 - 4*w^2 - 45/2*w + 4], [167, 167, 1/2*w^3 + w^2 - 5/2*w - 4], [167, 167, 2*w + 5], [173, 173, -w^3 + w^2 + 4*w - 7], [181, 181, w^3 - 2*w^2 - 9*w + 1], [193, 193, -1/2*w^3 + 2*w^2 + 3/2*w - 8], [197, 197, -1/2*w^3 - w^2 + 13/2*w + 2], [197, 197, -w^3 + w^2 + 4*w - 1], [197, 197, w^3 - 2*w^2 - 7*w + 7], [197, 197, -2*w^3 + 3*w^2 + 13*w - 11], [199, 199, -2*w^3 + 3*w^2 + 11*w - 5], [239, 239, -w^3 + 2*w^2 + 5*w - 1], [239, 239, 7/2*w^3 - 4*w^2 - 45/2*w + 6], [241, 241, 1/2*w^3 - 2*w^2 - 7/2*w + 8], [257, 257, 3/2*w^3 - 2*w^2 - 17/2*w - 4], [257, 257, 1/2*w^3 - 7/2*w - 6], [257, 257, 7/2*w^3 - 5*w^2 - 43/2*w + 10], [257, 257, 3*w^3 - 4*w^2 - 19*w + 7], [269, 269, 1/2*w^3 + w^2 + 3/2*w + 2], [269, 269, -2*w^3 + 3*w^2 + 11*w - 11], [277, 277, 5/2*w^3 - 4*w^2 - 35/2*w + 8], [281, 281, -3/2*w^3 + w^2 + 23/2*w - 2], [283, 283, -w^3 - w^2 + 8*w + 11], [283, 283, -3*w^2 + 3*w + 19], [311, 311, 3/2*w^3 - 2*w^2 - 17/2*w - 2], [331, 331, -4*w^3 + 7*w^2 + 25*w - 21], [337, 337, -5/2*w^3 + 2*w^2 + 35/2*w + 4], [337, 337, -2*w^3 + 3*w^2 + 11*w - 9], [347, 347, 2*w^3 - 4*w^2 - 12*w + 13], [347, 347, 5/2*w^3 - 5*w^2 - 29/2*w + 20], [347, 347, -1/2*w^3 - 1/2*w - 6], [347, 347, -1/2*w^3 + 3*w^2 + 1/2*w - 16], [349, 349, 1/2*w^3 + w^2 - 9/2*w - 4], [353, 353, -1/2*w^3 + 2*w^2 + 7/2*w], [359, 359, 3/2*w^3 - w^2 - 19/2*w + 2], [373, 373, -2*w^3 + 2*w^2 + 12*w + 1], [373, 373, -1/2*w^3 + w^2 + 9/2*w - 6], [379, 379, w^2 - w - 9], [379, 379, -2*w^3 + 3*w^2 + 13*w - 5], [383, 383, w^2 + w - 5], [389, 389, 9/2*w^3 - 6*w^2 - 59/2*w + 14], [397, 397, 2*w^2 - 2*w - 15], [401, 401, -1/2*w^3 + 2*w^2 + 3/2*w - 4], [419, 419, -3/2*w^3 + w^2 + 15/2*w - 2], [419, 419, w^3 + w^2 - 8*w - 15], [431, 431, -5/2*w^3 + 4*w^2 + 27/2*w - 16], [431, 431, 2*w^3 - 4*w^2 - 10*w + 11], [443, 443, 1/2*w^3 - 3*w^2 - 5/2*w + 14], [449, 449, 1/2*w^3 + w^2 - 9/2*w - 6], [457, 457, -w^3 + 2*w^2 + 7*w - 9], [461, 461, 4*w^3 - 5*w^2 - 27*w + 7], [487, 487, 2*w^2 - 9], [499, 499, 3/2*w^3 - w^2 - 15/2*w - 2], [499, 499, 3/2*w^3 - 4*w^2 - 17/2*w + 16], [503, 503, -5/2*w^3 + 3*w^2 + 37/2*w], [509, 509, -3*w^3 + 3*w^2 + 18*w + 7], [521, 521, -1/2*w^3 - 2*w^2 + 11/2*w + 18], [521, 521, -5/2*w^3 + 4*w^2 + 31/2*w - 8], [523, 523, -1/2*w^3 + 2*w^2 + 11/2*w], [523, 523, -1/2*w^3 - w^2 + 9/2*w], [547, 547, -7/2*w^3 + 6*w^2 + 45/2*w - 18], [547, 547, -w^3 - w^2 + 12*w + 5], [547, 547, 3/2*w^3 - w^2 - 19/2*w + 4], [547, 547, w^3 - w^2 - 6*w - 5], [557, 557, 1/2*w^3 + w^2 - 9/2*w - 12], [557, 557, 3/2*w^3 - 2*w^2 - 21/2*w], [557, 557, 2*w^3 - 2*w^2 - 12*w + 1], [557, 557, w^3 - 3*w^2 - 6*w + 11], [569, 569, w^3 - 2*w^2 - 7*w + 1], [577, 577, w^3 + w^2 - 6*w - 5], [599, 599, -3/2*w^3 + 3*w^2 + 15/2*w - 10], [607, 607, -5/2*w^3 + 3*w^2 + 25/2*w + 4], [613, 613, w^3 - 9*w + 1], [619, 619, 3*w^2 - w - 17], [619, 619, w^3 - 3*w^2 - 6*w + 13], [625, 5, -5], [631, 631, 11/2*w^3 - 7*w^2 - 71/2*w + 16], [631, 631, -3/2*w^3 + 2*w^2 + 13/2*w], [643, 643, w^2 - 3*w - 5], [653, 653, 2*w^2 - 2*w - 11], [659, 659, 5*w^3 - 3*w^2 - 36*w - 9], [659, 659, -w^3 + 7*w - 1], [673, 673, 5/2*w^3 - 2*w^2 - 31/2*w - 4], [677, 677, 13/2*w^3 - 6*w^2 - 87/2*w + 2], [691, 691, -3/2*w^3 + 2*w^2 + 13/2*w - 2], [701, 701, -w^3 + 3*w^2 + 4*w - 13], [701, 701, w^3 - 2*w^2 - 9*w + 3], [709, 709, -w^2 - 5*w - 1], [739, 739, 3/2*w^3 - w^2 - 15/2*w], [743, 743, -2*w^3 + w^2 + 15*w + 1], [751, 751, -2*w^3 + 4*w^2 + 12*w - 11], [757, 757, -6*w^3 + 9*w^2 + 39*w - 23], [757, 757, -w^3 - w^2 + 8*w + 3], [757, 757, 13/2*w^3 - 8*w^2 - 83/2*w + 16], [757, 757, 3*w^3 - 4*w^2 - 17*w + 3], [761, 761, 7/2*w^3 - 5*w^2 - 47/2*w + 12], [769, 769, -3*w^2 + 7*w + 3], [773, 773, -3/2*w^3 - w^2 + 19/2*w + 4], [787, 787, -4*w^3 + 5*w^2 + 23*w - 17], [787, 787, 1/2*w^3 - w^2 - 9/2*w - 4], [809, 809, -2*w^3 + 2*w^2 + 16*w + 3], [809, 809, -4*w^3 + 5*w^2 + 25*w - 9], [811, 811, 3*w^3 - 3*w^2 - 20*w + 5], [811, 811, w^3 - 3*w^2 - 6*w + 17], [823, 823, 5/2*w^3 - w^2 - 33/2*w - 10], [829, 829, 5/2*w^3 - 3*w^2 - 33/2*w + 2], [829, 829, -3*w^3 + 2*w^2 + 23*w - 1], [839, 839, -11/2*w^3 + 7*w^2 + 67/2*w - 18], [853, 853, -w^3 + 2*w^2 + 3*w - 5], [853, 853, -5*w^3 + 7*w^2 + 32*w - 15], [857, 857, 3/2*w^3 - 5*w^2 - 15/2*w + 26], [859, 859, -w^3 + 3*w^2 + 4*w - 17], [863, 863, 4*w^3 - 4*w^2 - 26*w + 3], [877, 877, w^3 - 5*w - 5], [887, 887, 3/2*w^3 - w^2 - 11/2*w + 8], [907, 907, -9/2*w^3 + 7*w^2 + 53/2*w - 24], [919, 919, 7/2*w^3 - 6*w^2 - 41/2*w + 22], [929, 929, -2*w^3 + 3*w^2 + 11*w - 1], [937, 937, 2*w^3 - 4*w^2 - 12*w + 7], [947, 947, 9/2*w^3 - 5*w^2 - 61/2*w + 10], [953, 953, -1/2*w^3 + 2*w^2 + 3/2*w + 2], [961, 31, -1/2*w^3 + 2*w^2 - 1/2*w - 6], [977, 977, 5/2*w^3 - 2*w^2 - 31/2*w], [983, 983, -5/2*w^3 + 6*w^2 + 27/2*w - 24], [983, 983, -3/2*w^3 + 2*w^2 + 21/2*w - 10], [983, 983, -w^3 + 4*w^2 + 7*w - 15], [983, 983, -1/2*w^3 + 3*w^2 - 3/2*w - 4], [997, 997, 1/2*w^3 - w^2 - 5/2*w - 4], [997, 997, 1/2*w^3 + w^2 - 17/2*w - 2]]; primes := [ideal : I in primesArray]; heckePol := x^16 - 28*x^14 + 317*x^12 - 1861*x^10 + 6043*x^8 - 10716*x^6 + 9516*x^4 - 3356*x^2 + 256; K := NumberField(heckePol); heckeEigenvaluesArray := [545/98144*e^15 - 3159/24536*e^13 + 112557/98144*e^11 - 485285/98144*e^9 + 1067243/98144*e^7 - 317283/24536*e^5 + 278095/24536*e^3 - 180343/24536*e, e, -1, -659/12268*e^15 + 4028/3067*e^13 - 154987/12268*e^11 + 744815/12268*e^9 - 1871941/12268*e^7 + 582831/3067*e^5 - 295100/3067*e^3 + 25965/3067*e, 127/24536*e^15 - 995/6134*e^13 + 51035/24536*e^11 - 343363/24536*e^9 + 1285917/24536*e^7 - 650149/6134*e^5 + 619565/6134*e^3 - 173889/6134*e, -1177/24536*e^15 + 7869/6134*e^13 - 332717/24536*e^11 + 1756205/24536*e^9 - 4804779/24536*e^7 + 1598967/6134*e^5 - 865133/6134*e^3 + 124711/6134*e, 77/6134*e^14 - 313/3067*e^12 - 7069/6134*e^10 + 97505/6134*e^8 - 387983/6134*e^6 + 309399/3067*e^4 - 187629/3067*e^2 + 45080/3067, -3847/49072*e^15 + 22895/12268*e^13 - 850603/49072*e^11 + 3892123/49072*e^9 - 9057253/49072*e^7 + 2444351/12268*e^5 - 916837/12268*e^3 + 88689/12268*e, -31/49072*e^15 - 1351/12268*e^13 + 112541/49072*e^11 - 798517/49072*e^9 + 2394155/49072*e^7 - 713103/12268*e^5 + 253515/12268*e^3 - 58983/12268*e, 3053/49072*e^15 - 22277/12268*e^13 + 1021481/49072*e^11 - 5826905/49072*e^9 + 17298439/49072*e^7 - 6386297/12268*e^5 + 4056575/12268*e^3 - 758859/12268*e, -293/3067*e^14 + 6405/3067*e^12 - 54277/3067*e^10 + 229748/3067*e^8 - 525245/3067*e^6 + 633806/3067*e^4 - 324084/3067*e^2 + 30540/3067, -2293/24536*e^15 + 14439/6134*e^13 - 575041/24536*e^11 + 2869905/24536*e^9 - 7484591/24536*e^7 + 2395469/6134*e^5 - 1234025/6134*e^3 + 160491/6134*e, -511/6134*e^14 + 5423/3067*e^12 - 85805/6134*e^10 + 310383/6134*e^8 - 493319/6134*e^6 + 103932/3067*e^4 + 84733/3067*e^2 - 22022/3067, 1661/6134*e^14 - 19458/3067*e^12 + 354881/6134*e^10 - 1592851/6134*e^8 + 3655045/6134*e^6 - 1997545/3067*e^4 + 843125/3067*e^2 - 94876/3067, -173/3067*e^14 + 3876/3067*e^12 - 33827/3067*e^10 + 148811/3067*e^8 - 363104/3067*e^6 + 506673/3067*e^4 - 350628/3067*e^2 + 50042/3067, 3979/49072*e^15 - 22287/12268*e^13 + 771887/49072*e^11 - 3293839/49072*e^9 + 7428225/49072*e^7 - 2286935/12268*e^5 + 1666473/12268*e^3 - 651181/12268*e, 275/6134*e^14 - 1556/3067*e^12 - 8597/6134*e^10 + 234315/6134*e^8 - 1040397/6134*e^6 + 861389/3067*e^4 - 490465/3067*e^2 + 62856/3067, -61/24536*e^15 + 1299/6134*e^13 - 90393/24536*e^11 + 642505/24536*e^9 - 2100431/24536*e^7 + 728857/6134*e^5 - 244747/6134*e^3 - 131893/6134*e, -13/49072*e^15 + 3193/12268*e^13 - 266233/49072*e^11 + 2095785/49072*e^9 - 7682535/49072*e^7 + 3304385/12268*e^5 - 2323147/12268*e^3 + 401875/12268*e, 1113/6134*e^14 - 12610/3067*e^12 + 219577/6134*e^10 - 919299/6134*e^8 + 1864767/6134*e^6 - 752552/3067*e^4 + 37613/3067*e^2 + 48806/3067, -219/12268*e^15 + 943/3067*e^13 - 16619/12268*e^11 - 45741/12268*e^9 + 553575/12268*e^7 - 373810/3067*e^5 + 343040/3067*e^3 - 47657/3067*e, -1491/49072*e^15 + 9025/12268*e^13 - 374471/49072*e^11 + 2233439/49072*e^9 - 8244369/49072*e^7 + 4372365/12268*e^5 - 4284649/12268*e^3 + 1062749/12268*e, -27/3067*e^14 + 339/3067*e^12 + 766/3067*e^10 - 29251/3067*e^8 + 160343/3067*e^6 - 354080/3067*e^4 + 315126/3067*e^2 - 78456/3067, 3851/49072*e^15 - 25293/12268*e^13 + 1060863/49072*e^11 - 5656199/49072*e^9 + 16122585/49072*e^7 - 5865141/12268*e^5 + 3651153/12268*e^3 - 487901/12268*e, 2069/49072*e^15 - 15099/12268*e^13 + 675905/49072*e^11 - 3624201/49072*e^9 + 9431879/49072*e^7 - 2610739/12268*e^5 + 720115/12268*e^3 + 162053/12268*e, -496/3067*e^14 + 11680/3067*e^12 - 107018/3067*e^10 + 482369/3067*e^8 - 1110604/3067*e^6 + 1215967/3067*e^4 - 514726/3067*e^2 + 83374/3067, -3425/49072*e^15 + 18333/12268*e^13 - 603357/49072*e^11 + 2476845/49072*e^9 - 5743779/49072*e^7 + 2073789/12268*e^5 - 1713671/12268*e^3 + 468175/12268*e, 510/3067*e^14 - 11515/3067*e^12 + 100714/3067*e^10 - 433692/3067*e^8 + 965896/3067*e^6 - 1047137/3067*e^4 + 426980/3067*e^2 - 29062/3067, -179/49072*e^15 + 1499/12268*e^13 - 60919/49072*e^11 + 187975/49072*e^9 + 322391/49072*e^7 - 577881/12268*e^5 + 601523/12268*e^3 + 14797/12268*e, -1285/12268*e^15 + 8208/3067*e^13 - 329653/12268*e^11 + 1639201/12268*e^9 - 4151139/12268*e^7 + 1205016/3067*e^5 - 402791/3067*e^3 - 100961/3067*e, -2295/49072*e^15 + 12571/12268*e^13 - 434811/49072*e^11 + 1973083/49072*e^9 - 5386245/49072*e^7 + 2293267/12268*e^5 - 2042989/12268*e^3 + 556385/12268*e, 68/3067*e^14 - 513/3067*e^12 - 7836/3067*e^10 + 107179/3067*e^8 - 462327/3067*e^6 + 837119/3067*e^4 - 570782/3067*e^2 + 76276/3067, 226/3067*e^14 - 5223/3067*e^12 + 47204/3067*e^10 - 213618/3067*e^8 + 518062/3067*e^6 - 671523/3067*e^4 + 433368/3067*e^2 - 95050/3067, -6843/49072*e^15 + 40137/12268*e^13 - 1452159/49072*e^11 + 6325975/49072*e^9 - 13384777/49072*e^7 + 2934981/12268*e^5 - 580633/12268*e^3 + 151853/12268*e, -696/3067*e^14 + 15895/3067*e^12 - 140079/3067*e^10 + 601929/3067*e^8 - 1316432/3067*e^6 + 1402297/3067*e^4 - 654506/3067*e^2 + 91764/3067, -1121/49072*e^15 + 4967/12268*e^13 - 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7387/3067*e^13 + 131303/6134*e^11 - 614743/6134*e^9 + 1711939/6134*e^7 - 1448911/3067*e^5 + 1275760/3067*e^3 - 365266/3067*e, 1837/6134*e^14 - 21926/3067*e^12 + 411455/6134*e^10 - 1923795/6134*e^8 + 4658375/6134*e^6 - 2712559/3067*e^4 + 1237091/3067*e^2 - 114516/3067, -751/3067*e^14 + 22038/3067*e^12 - 249385/3067*e^10 + 1370888/3067*e^8 - 3783390/3067*e^6 + 4884744/3067*e^4 - 2372128/3067*e^2 + 223652/3067, 8095/24536*e^15 - 48497/6134*e^13 + 1826027/24536*e^11 - 8580931/24536*e^9 + 21108317/24536*e^7 - 6498003/6134*e^5 + 3599253/6134*e^3 - 830369/6134*e, 6071/49072*e^15 - 37373/12268*e^13 + 1454523/49072*e^11 - 7130531/49072*e^9 + 18565533/49072*e^7 - 6120745/12268*e^5 + 3338233/12268*e^3 - 564089/12268*e, -680/3067*e^15 + 17398/3067*e^13 - 173134/3067*e^11 + 835884/3067*e^9 - 1967713/3067*e^7 + 1894059/3067*e^5 - 211490/3067*e^3 - 235236/3067*e, 201/3067*e^14 - 479/3067*e^12 - 40121/3067*e^10 + 393258/3067*e^8 - 1334065/3067*e^6 + 1643584/3067*e^4 - 395326/3067*e^2 - 57964/3067, 535/6134*e^14 - 6596/3067*e^12 + 126699/6134*e^10 - 589719/6134*e^8 + 1315193/6134*e^6 - 530997/3067*e^4 - 95975/3067*e^2 + 71204/3067, 462/3067*e^15 - 9890/3067*e^13 + 77199/3067*e^11 - 252261/3067*e^9 + 193176/3067*e^7 + 679525/3067*e^5 - 1297711/3067*e^3 + 412146/3067*e, 7545/24536*e^15 - 46941/6134*e^13 + 1867757/24536*e^11 - 9528013/24536*e^9 + 26489203/24536*e^7 - 9711781/6134*e^5 + 6497313/6134*e^3 - 1377811/6134*e, 11623/49072*e^15 - 71839/12268*e^13 + 2838235/49072*e^11 - 14409627/49072*e^9 + 40265013/49072*e^7 - 15226751/12268*e^5 + 10934253/12268*e^3 - 2046017/12268*e, -951/3067*e^14 + 20119/3067*e^12 - 159766/3067*e^10 + 591817/3067*e^8 - 1020362/3067*e^6 + 626991/3067*e^4 + 88908/3067*e^2 - 13318/3067, -8055/49072*e^15 + 42919/12268*e^13 - 1367339/49072*e^11 + 5023835/49072*e^9 - 8458101/49072*e^7 + 1352995/12268*e^5 - 43745/12268*e^3 - 315575/12268*e, -507/3067*e^14 + 10455/3067*e^12 - 81034/3067*e^10 + 300631/3067*e^8 - 583298/3067*e^6 + 611435/3067*e^4 - 228902/3067*e^2 - 52186/3067, 170/3067*e^14 - 2816/3067*e^12 + 14147/3067*e^10 - 15750/3067*e^8 - 22561/3067*e^6 - 43368/3067*e^4 + 115746/3067*e^2 + 61876/3067, 2605/12268*e^15 - 17463/3067*e^13 + 732489/12268*e^11 - 3777777/12268*e^9 + 9881919/12268*e^7 - 3053628/3067*e^5 + 1528992/3067*e^3 - 383667/3067*e, 831/3067*e^14 - 17590/3067*e^12 + 142383/3067*e^10 - 566086/3067*e^8 + 1204792/3067*e^6 - 1395422/3067*e^4 + 784128/3067*e^2 - 122730/3067, -4243/6134*e^14 + 51343/3067*e^12 - 976361/6134*e^10 + 4630613/6134*e^8 - 11463495/6134*e^6 + 7014273/3067*e^4 - 3519051/3067*e^2 + 454660/3067, 907/3067*e^14 - 18885/3067*e^12 + 144089/3067*e^10 - 481478/3067*e^8 + 599311/3067*e^6 + 135360/3067*e^4 - 577256/3067*e^2 + 102880/3067, 6143/6134*e^14 - 72131/3067*e^12 + 1319577/6134*e^10 - 5950453/6134*e^8 + 13768499/6134*e^6 - 7652447/3067*e^4 + 3327313/3067*e^2 - 354964/3067, -3863/24536*e^15 + 26353/6134*e^13 - 1127315/24536*e^11 + 5943083/24536*e^9 - 15923189/24536*e^7 + 5031105/6134*e^5 - 2438411/6134*e^3 + 250181/6134*e, -2561/12268*e^15 + 15621/3067*e^13 - 603333/12268*e^11 + 2946693/12268*e^9 - 7721879/12268*e^7 + 2649117/3067*e^5 - 1710538/3067*e^3 + 390413/3067*e, -43/12268*e^15 - 291/3067*e^13 + 33821/12268*e^11 - 266273/12268*e^9 + 863763/12268*e^7 - 274334/3067*e^5 + 3298/3067*e^3 + 279893/3067*e, -1414/3067*e^14 + 29340/3067*e^12 - 225123/3067*e^10 + 782109/3067*e^8 - 1182609/3067*e^6 + 488343/3067*e^4 + 262478/3067*e^2 - 118262/3067, 2199/3067*e^14 - 50101/3067*e^12 + 439920/3067*e^10 - 1870916/3067*e^8 + 3935882/3067*e^6 - 3585112/3067*e^4 + 815216/3067*e^2 + 12490/3067, -6371/49072*e^15 + 39337/12268*e^13 - 1520983/49072*e^11 + 7211727/49072*e^9 - 16892993/49072*e^7 + 3954009/12268*e^5 + 413963/12268*e^3 - 1759851/12268*e, 1145/3067*e^14 - 28348/3067*e^12 + 276147/3067*e^10 - 1337752/3067*e^8 + 3334773/3067*e^6 - 3947419/3067*e^4 + 1746410/3067*e^2 - 176080/3067, -4689/6134*e^14 + 55506/3067*e^12 - 1019141/6134*e^10 + 4552501/6134*e^8 - 10078243/6134*e^6 + 4854767/3067*e^4 - 1227133/3067*e^2 - 16124/3067, 233/6134*e^14 - 270/3067*e^12 - 44891/6134*e^10 + 428721/6134*e^8 - 1422095/6134*e^6 + 899509/3067*e^4 - 361913/3067*e^2 + 116336/3067, -133/3067*e^14 - 34/3067*e^12 + 32285/3067*e^10 - 289146/3067*e^8 + 914676/3067*e^6 - 981284/3067*e^4 + 8564/3067*e^2 + 134240/3067, -1155/3067*e^14 + 27792/3067*e^12 - 262005/3067*e^10 + 1224117/3067*e^8 - 2942674/3067*e^6 + 3393941/3067*e^4 - 1517240/3067*e^2 + 64554/3067, -1145/24536*e^15 + 13221/6134*e^13 - 809805/24536*e^11 + 5751165/24536*e^9 - 20761467/24536*e^7 + 9429539/6134*e^5 - 7611849/6134*e^3 + 1654195/6134*e]; 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;