/* 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, -1, -6, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [3, 3, w + 1], [5, 5, -w^3 + 5*w + 2], [5, 5, w - 1], [11, 11, -w^3 + 5*w], [13, 13, -w^3 + w^2 + 6*w - 3], [16, 2, 2], [19, 19, 2*w^3 - w^2 - 11*w + 2], [25, 5, -w^2 + w + 3], [27, 3, w^3 - w^2 - 5*w + 4], [31, 31, -w - 3], [37, 37, -w^3 - w^2 + 6*w + 4], [41, 41, w^3 + w^2 - 6*w - 5], [43, 43, w^3 - 7*w - 2], [47, 47, -2*w^3 + 11*w + 2], [61, 61, w^2 - 3], [67, 67, w^2 + w - 4], [79, 79, -4*w^3 + 2*w^2 + 22*w - 9], [97, 97, -3*w^3 + 2*w^2 + 19*w - 6], [97, 97, -w^3 + 6*w - 3], [97, 97, -3*w^3 + 16*w], [97, 97, -w^3 + w^2 + 4*w - 3], [101, 101, -2*w^3 + w^2 + 11*w - 6], [103, 103, w^3 - w^2 - 7*w], [109, 109, -w^3 - w^2 + 6*w + 2], [121, 11, 2*w^3 - 11*w + 1], [127, 127, -2*w^3 + w^2 + 13*w - 1], [127, 127, w^3 - w^2 - 6*w - 2], [157, 157, -2*w^3 + 2*w^2 + 10*w - 7], [157, 157, w - 4], [163, 163, 2*w^3 - w^2 - 13*w + 3], [173, 173, -2*w^3 + w^2 + 13*w - 6], [173, 173, 2*w^3 - 12*w + 1], [173, 173, 2*w^3 - 11*w + 2], [173, 173, w^2 + w - 5], [179, 179, -2*w^3 + 10*w + 3], [181, 181, -w^3 - w^2 + 7*w + 8], [191, 191, w^3 - 5*w - 5], [191, 191, 2*w^3 - w^2 - 9*w - 1], [193, 193, -3*w^3 + 3*w^2 + 19*w - 7], [197, 197, -w^3 + w^2 + 6*w - 7], [223, 223, w^3 + w^2 - 6*w + 1], [227, 227, 6*w^3 - 3*w^2 - 34*w + 9], [227, 227, w^3 + w^2 - 8*w], [233, 233, 2*w^3 - 13*w - 3], [233, 233, -w^3 + 2*w^2 + 4*w - 6], [239, 239, -2*w^3 + 10*w + 1], [239, 239, w^3 - 8*w - 2], [251, 251, -3*w^3 + w^2 + 18*w], [251, 251, -5*w^3 + 2*w^2 + 30*w - 11], [263, 263, -2*w^3 + w^2 + 10*w + 2], [269, 269, 3*w^3 - 18*w - 8], [271, 271, 2*w^3 + w^2 - 10*w - 2], [271, 271, -3*w^3 + 19*w], [277, 277, -w^3 + w^2 + 4*w - 8], [277, 277, 2*w^3 + w^2 - 14*w - 8], [281, 281, w^3 - 8*w], [307, 307, 3*w^3 - w^2 - 16*w + 2], [307, 307, 2*w^3 - w^2 - 9*w + 1], [307, 307, -2*w^3 + 2*w^2 + 13*w - 7], [307, 307, -w^3 + w^2 + 7*w - 6], [313, 313, -3*w^3 + w^2 + 15*w + 4], [317, 317, -4*w^3 + w^2 + 22*w - 3], [317, 317, 2*w^3 - w^2 - 10*w - 3], [337, 337, w^3 + w^2 - 4*w - 5], [349, 349, -2*w^3 - w^2 + 11*w + 3], [349, 349, -4*w^3 + 2*w^2 + 24*w - 11], [353, 353, 3*w^3 - w^2 - 17*w + 7], [353, 353, -3*w^3 - 2*w^2 + 17*w + 11], [367, 367, 2*w^3 - 11*w - 7], [373, 373, -w^3 + w^2 + 4*w - 5], [383, 383, -4*w^3 + 3*w^2 + 23*w - 9], [383, 383, 4*w^3 - w^2 - 22*w], [389, 389, w^2 - 2*w - 5], [401, 401, -3*w^3 + 2*w^2 + 16*w - 9], [401, 401, 2*w^3 - w^2 - 14*w - 1], [409, 409, 2*w^3 + 2*w^2 - 13*w - 9], [419, 419, -4*w^3 + 24*w + 7], [419, 419, 2*w^2 + 2*w - 7], [419, 419, -w^3 + 2*w^2 + 8*w - 6], [419, 419, 6*w^3 - 3*w^2 - 33*w + 13], [431, 431, 2*w^3 - w^2 - 12*w - 5], [439, 439, -3*w^3 + 2*w^2 + 18*w - 4], [439, 439, w^3 + 2*w^2 - 7*w - 3], [443, 443, 6*w^3 - 3*w^2 - 33*w + 7], [449, 449, -3*w^3 + w^2 + 14*w - 4], [457, 457, 2*w^3 - 9*w], [457, 457, 3*w^3 - w^2 - 16*w - 2], [479, 479, 3*w^3 - w^2 - 19*w + 4], [487, 487, -w^3 + w^2 + 4*w - 6], [491, 491, 3*w^3 + w^2 - 17*w - 5], [499, 499, 5*w^3 - 3*w^2 - 28*w + 8], [499, 499, 2*w^3 - w^2 - 12*w - 1], [503, 503, -2*w^3 - w^2 + 12*w + 3], [503, 503, w^3 - 3*w - 4], [509, 509, -3*w - 5], [509, 509, -4*w^3 + 3*w^2 + 25*w - 10], [523, 523, -2*w^3 + w^2 + 14*w - 6], [529, 23, 4*w^3 - 2*w^2 - 21*w + 5], [529, 23, -3*w^3 - w^2 + 15*w + 6], [557, 557, -2*w^3 + w^2 + 13*w + 2], [557, 557, 3*w^3 - 2*w^2 - 16*w + 3], [569, 569, 2*w^2 - w - 8], [587, 587, w^3 + 2*w^2 - 5*w - 16], [587, 587, -4*w^3 + w^2 + 25*w - 5], [599, 599, w^3 + 2*w^2 - 6*w - 5], [599, 599, -5*w^3 + 3*w^2 + 30*w - 15], [601, 601, w^3 - 6*w - 6], [607, 607, -w^3 + w^2 + 3*w - 4], [607, 607, -w^3 + 2*w - 3], [613, 613, 6*w^3 - 2*w^2 - 31*w + 11], [613, 613, w^3 + w^2 - 4*w - 6], [617, 617, 5*w^3 - w^2 - 30*w + 2], [617, 617, -2*w^3 + 2*w^2 + 9*w + 3], [619, 619, -w^3 + 2*w^2 + 6*w - 5], [619, 619, 2*w^3 - w^2 - 11*w - 3], [631, 631, -w^3 + 2*w^2 + 5*w - 5], [631, 631, -3*w^3 - w^2 + 18*w + 5], [647, 647, 6*w^3 - w^2 - 36*w + 2], [659, 659, 4*w^3 + w^2 - 24*w - 14], [661, 661, 2*w^2 - w - 4], [673, 673, 2*w^3 + w^2 - 14*w - 12], [677, 677, 3*w^3 + 2*w^2 - 16*w - 7], [677, 677, -3*w^3 + w^2 + 15*w + 1], [683, 683, 2*w^2 - 2*w - 9], [683, 683, 2*w^3 - 14*w + 1], [691, 691, 5*w^3 - 2*w^2 - 30*w + 6], [701, 701, -2*w^3 + w^2 + 12*w + 2], [701, 701, -5*w^3 + 2*w^2 + 26*w], [719, 719, 3*w^3 - 18*w + 1], [719, 719, 3*w^3 - w^2 - 19*w + 1], [727, 727, -3*w^2 - 2*w + 11], [727, 727, 2*w^3 + w^2 - 13*w - 2], [733, 733, 2*w^3 - 3*w^2 - 8*w + 5], [739, 739, -3*w^3 + 16*w - 6], [743, 743, -4*w^3 - w^2 + 25*w + 8], [751, 751, w^2 - 2*w - 7], [751, 751, -3*w^3 + 15*w + 5], [757, 757, 3*w^3 - 19*w - 2], [761, 761, 2*w^2 + w - 5], [769, 769, -2*w^3 + w^2 + 14*w - 9], [769, 769, 2*w^2 - w - 7], [773, 773, w^3 - w^2 - 5*w - 4], [773, 773, w^3 - 3*w - 6], [787, 787, 3*w^3 - 19*w - 3], [797, 797, 8*w^3 - 3*w^2 - 47*w + 10], [809, 809, 4*w^3 - 25*w - 1], [811, 811, w^3 - 9*w - 1], [821, 821, 6*w^3 - 3*w^2 - 35*w + 8], [823, 823, 2*w^3 + w^2 - 10*w + 1], [823, 823, -2*w^3 + 2*w^2 + 7*w - 4], [829, 829, -5*w^3 + w^2 + 27*w - 4], [829, 829, 5*w^3 - 2*w^2 - 26*w + 10], [839, 839, w^3 + 2*w^2 - 5*w - 7], [839, 839, 3*w^3 - w^2 - 15*w + 2], [841, 29, 3*w^3 + w^2 - 14*w - 2], [841, 29, 4*w^3 - 2*w^2 - 21*w + 10], [857, 857, -8*w^3 + 4*w^2 + 44*w - 11], [857, 857, -3*w^3 + 3*w^2 + 20*w - 9], [859, 859, w^3 + 2*w^2 - 6*w - 6], [877, 877, w^3 - 3*w^2 - 4*w + 14], [883, 883, 2*w^3 + w^2 - 10*w - 11], [887, 887, 3*w^3 + w^2 - 20*w - 2], [919, 919, 7*w^3 - 3*w^2 - 39*w + 9], [929, 929, 3*w^3 - 17*w + 2], [929, 929, -6*w^3 + 2*w^2 + 35*w - 10], [937, 937, w^3 + w^2 - 3*w - 5], [937, 937, -2*w^3 + 3*w^2 + 12*w - 12], [937, 937, -3*w^3 - 2*w^2 + 19*w + 8], [937, 937, 3*w^3 - 2*w^2 - 13*w + 3], [941, 941, w^3 + 2*w^2 - 6*w - 8], [941, 941, 5*w^3 - w^2 - 28*w - 3], [947, 947, -4*w^3 + 2*w^2 + 25*w - 6], [953, 953, w^3 - 7*w - 7], [953, 953, -3*w^3 + 2*w^2 + 18*w - 3], [967, 967, w^3 - 2*w^2 - 5*w - 3], [971, 971, w^2 + 2*w - 6], [977, 977, 2*w^3 - 11*w - 8], [997, 997, -3*w^3 + w^2 + 17*w + 2], [997, 997, -2*w^3 + 2*w^2 + 14*w - 7]]; primes := [ideal : I in primesArray]; heckePol := x^8 - 98*x^6 + 2825*x^4 - 20232*x^2 + 20736; K := NumberField(heckePol); heckeEigenvaluesArray := [0, -1/47520*e^7 - 19/4752*e^5 + 2747/9504*e^3 - 623/220*e, -2/495*e^6 + 23/99*e^4 - 248/99*e^2 + 346/55, -1/47520*e^7 - 19/4752*e^5 + 2747/9504*e^3 - 843/220*e, -13/7920*e^7 + 83/792*e^5 - 2437/1584*e^3 + 1469/330*e, -1/990*e^6 + 23/396*e^4 - 149/396*e^2 - 271/55, 7/9504*e^7 - 127/4752*e^5 - 2689/9504*e^3 + 235/132*e, 0, -1/23760*e^7 - 19/2376*e^5 + 2747/4752*e^3 - 623/110*e, 0, 19/11880*e^7 - 67/594*e^5 + 5029/2376*e^3 - 1999/165*e, 1/660*e^7 - 1/22*e^5 - 151/132*e^3 + 1522/165*e, -4/495*e^6 + 46/99*e^4 - 496/99*e^2 + 692/55, -31/9504*e^7 + 1015/4752*e^5 - 31991/9504*e^3 + 1681/132*e, -1/660*e^7 + 1/22*e^5 + 151/132*e^3 - 1522/165*e, 0, 13/15840*e^7 - 17/1584*e^5 - 4559/3168*e^3 + 8651/660*e, 7/396*e^6 - 353/396*e^4 + 883/198*e^2 + 162/11, 31/1980*e^6 - 307/396*e^4 + 367/99*e^2 + 268/55, -17/5280*e^7 + 39/176*e^5 - 4165/1056*e^3 + 13463/660*e, 13/7920*e^7 - 83/792*e^5 + 2437/1584*e^3 - 1469/330*e, -7/396*e^6 + 353/396*e^4 - 883/198*e^2 - 162/11, 31/9504*e^7 - 1015/4752*e^5 + 31991/9504*e^3 - 1681/132*e, 3/220*e^6 - 29/44*e^4 + 65/22*e^2 - 274/55, 7/396*e^6 - 353/396*e^4 + 883/198*e^2 + 162/11, -2/495*e^6 + 23/99*e^4 - 149/99*e^2 - 1084/55, 0, 1/1188*e^7 - 2/297*e^5 - 2053/1188*e^3 + 526/33*e, -31/1980*e^6 + 307/396*e^4 - 367/99*e^2 - 268/55, 4, 7/396*e^6 - 353/396*e^4 + 883/198*e^2 + 162/11, -2/495*e^6 + 23/99*e^4 - 248/99*e^2 + 456/55, 2/55*e^6 - 23/11*e^4 + 248/11*e^2 - 2454/55, -2/495*e^6 + 23/99*e^4 - 248/99*e^2 + 786/55, 1/720*e^7 - 5/72*e^5 + 85/144*e^3 - 233/30*e, 29/9504*e^7 - 413/4752*e^5 - 24491/9504*e^3 + 2809/132*e, -16/495*e^6 + 184/99*e^4 - 1984/99*e^2 + 2218/55, -31/47520*e^7 + 203/4752*e^5 - 8299/9504*e^3 + 6301/660*e, 109/47520*e^7 - 305/4752*e^5 - 19055/9504*e^3 + 3667/220*e, 1/30*e^6 - 5/3*e^4 + 49/6*e^2 + 98/5, 31/990*e^6 - 307/198*e^4 + 734/99*e^2 + 536/55, -109/47520*e^7 + 305/4752*e^5 + 19055/9504*e^3 - 3667/220*e, 8, 7/396*e^6 - 353/396*e^4 + 883/198*e^2 + 162/11, -23/4752*e^7 + 785/2376*e^5 - 27427/4752*e^3 + 631/22*e, -1/23760*e^7 - 19/2376*e^5 + 2747/4752*e^3 - 843/110*e, 1/30*e^6 - 5/3*e^4 + 49/6*e^2 + 98/5, -1/47520*e^7 - 19/4752*e^5 + 2747/9504*e^3 - 843/220*e, -4/165*e^6 + 46/33*e^4 - 496/33*e^2 + 2076/55, 103/15840*e^7 - 683/1584*e^5 + 22243/3168*e^3 - 19339/660*e, 13/23760*e^7 - 149/2376*e^5 + 11017/4752*e^3 - 7823/330*e, -31/990*e^6 + 307/198*e^4 - 734/99*e^2 - 536/55, 8/495*e^6 - 92/99*e^4 + 992/99*e^2 - 1714/55, 7/7920*e^7 + 1/792*e^5 - 3653/1584*e^3 + 7129/330*e, -79/47520*e^7 + 83/4752*e^5 + 30101/9504*e^3 - 19171/660*e, -2/165*e^6 + 23/33*e^4 - 248/33*e^2 + 1368/55, -1/23760*e^7 - 19/2376*e^5 + 2747/4752*e^3 - 623/110*e, 19/5940*e^7 - 67/297*e^5 + 5029/1188*e^3 - 3998/165*e, -1/495*e^6 + 23/198*e^4 - 149/198*e^2 - 542/55, 8/495*e^6 - 92/99*e^4 + 992/99*e^2 - 284/55, 19/23760*e^7 - 35/2376*e^5 - 5465/4752*e^3 + 3391/330*e, 1/9504*e^7 + 95/4752*e^5 - 13735/9504*e^3 + 623/44*e, 49/23760*e^7 - 257/2376*e^5 + 5581/4752*e^3 - 1593/110*e, -3/220*e^6 + 29/44*e^4 - 65/22*e^2 + 274/55, 3/220*e^6 - 29/44*e^4 + 65/22*e^2 - 274/55, 29/990*e^6 - 142/99*e^4 + 1319/198*e^2 - 6/55, 3/220*e^6 - 29/44*e^4 + 65/22*e^2 - 274/55, 389/47520*e^7 - 2509/4752*e^5 + 75857/9504*e^3 - 17219/660*e, 8/165*e^6 - 92/33*e^4 + 992/33*e^2 - 3052/55, 73/47520*e^7 - 197/4752*e^5 - 13619/9504*e^3 + 7957/660*e, -7/2160*e^7 + 47/216*e^5 - 1579/432*e^3 + 497/30*e, -2/1485*e^7 + 23/297*e^5 - 347/297*e^3 + 2216/165*e, 47/9504*e^7 - 1475/4752*e^5 + 41119/9504*e^3 - 419/44*e, -19/11880*e^7 + 67/594*e^5 - 5029/2376*e^3 + 1999/165*e, -2/99*e^6 + 115/99*e^4 - 1240/99*e^2 + 434/11, 19/7920*e^7 - 35/792*e^5 - 5465/1584*e^3 + 3391/110*e, -1/720*e^7 + 5/72*e^5 - 85/144*e^3 + 233/30*e, -1/495*e^6 + 23/198*e^4 - 149/198*e^2 - 542/55, -32/495*e^6 + 368/99*e^4 - 3968/99*e^2 + 4326/55, 37/23760*e^7 - 89/2376*e^5 - 8183/4752*e^3 + 4913/330*e, -1/11880*e^7 - 19/1188*e^5 + 2747/2376*e^3 - 623/55*e, 151/47520*e^7 - 1091/4752*e^5 + 42979/9504*e^3 - 18521/660*e, -8/495*e^6 + 92/99*e^4 - 992/99*e^2 + 1494/55, -7/3168*e^7 + 127/1584*e^5 + 2689/3168*e^3 - 235/44*e, -1/1584*e^7 + 37/792*e^5 - 1841/1584*e^3 + 817/66*e, -157/23760*e^7 + 977/2376*e^5 - 26497/4752*e^3 + 3347/330*e, 7/47520*e^7 + 133/4752*e^5 - 19229/9504*e^3 + 4361/220*e, 4/165*e^6 - 46/33*e^4 + 496/33*e^2 - 1856/55, -467/47520*e^7 + 3007/4752*e^5 - 90479/9504*e^3 + 6719/220*e, -7/198*e^6 + 353/198*e^4 - 883/99*e^2 - 324/11, -7/3960*e^7 - 1/396*e^5 + 3653/792*e^3 - 7129/165*e, -31/4752*e^7 + 1015/2376*e^5 - 31991/4752*e^3 + 1681/66*e, -4/165*e^6 + 46/33*e^4 - 496/33*e^2 + 1196/55, 10, -1/11880*e^7 - 19/1188*e^5 + 2747/2376*e^3 - 843/55*e, 31/660*e^6 - 307/132*e^4 + 367/33*e^2 + 804/55, -8/495*e^6 + 92/99*e^4 - 992/99*e^2 + 1054/55, 79/47520*e^7 - 83/4752*e^5 - 30101/9504*e^3 + 19171/660*e, -23/15840*e^7 + 223/1584*e^5 - 13115/3168*e^3 + 25699/660*e, -233/47520*e^7 + 1513/4752*e^5 - 46613/9504*e^3 + 3781/220*e, -1/7920*e^7 - 19/792*e^5 + 2747/1584*e^3 - 2529/110*e, 17/9504*e^7 + 31/4752*e^5 - 46583/9504*e^3 + 6077/132*e, 16/495*e^6 - 184/99*e^4 + 1984/99*e^2 - 2988/55, -8/495*e^6 + 92/99*e^4 - 992/99*e^2 - 46/55, -31/2376*e^7 + 1015/1188*e^5 - 31991/2376*e^3 + 1681/33*e, 1/495*e^6 - 23/198*e^4 + 149/198*e^2 + 542/55, -28/495*e^6 + 322/99*e^4 - 3472/99*e^2 + 3084/55, -469/47520*e^7 + 2969/4752*e^5 - 84985/9504*e^3 + 5033/220*e, -5/1584*e^7 + 53/792*e^5 + 6371/1584*e^3 - 2339/66*e, -49/9504*e^7 + 889/4752*e^5 + 18823/9504*e^3 - 1645/132*e, 14/495*e^6 - 161/99*e^4 + 1736/99*e^2 - 1212/55, -5/198*e^6 + 119/99*e^4 - 1021/198*e^2 + 218/11, 467/47520*e^7 - 3007/4752*e^5 + 90479/9504*e^3 - 6719/220*e, 1/23760*e^7 + 19/2376*e^5 - 2747/4752*e^3 + 623/110*e, -61/9504*e^7 + 2125/4752*e^5 - 77717/9504*e^3 + 5891/132*e, 467/47520*e^7 - 3007/4752*e^5 + 90479/9504*e^3 - 6719/220*e, -4/495*e^6 + 46/99*e^4 - 496/99*e^2 + 912/55, 4/55*e^6 - 46/11*e^4 + 496/11*e^2 - 4358/55, -7/198*e^6 + 353/198*e^4 - 883/99*e^2 - 324/11, 389/47520*e^7 - 2509/4752*e^5 + 75857/9504*e^3 - 17219/660*e, -5/3168*e^7 + 185/1584*e^5 - 7621/3168*e^3 + 2105/132*e, 16/495*e^6 - 184/99*e^4 + 1984/99*e^2 - 3758/55, -7/1485*e^7 + 421/1188*e^5 - 8917/1188*e^3 + 2842/55*e, 7/1056*e^7 - 215/528*e^5 + 1861/352*e^3 - 833/132*e, 16/495*e^6 - 184/99*e^4 + 1984/99*e^2 - 3868/55, -31/495*e^6 + 307/99*e^4 - 1468/99*e^2 - 1072/55, 17/23760*e^7 - 73/2376*e^5 + 29/4752*e^3 - 347/330*e, -7/3960*e^7 + 8/99*e^5 + 155/792*e^3 - 3059/165*e, 31/495*e^6 - 307/99*e^4 + 1468/99*e^2 + 1072/55, 8/495*e^6 - 92/99*e^4 + 992/99*e^2 - 944/55, -19/2970*e^7 + 134/297*e^5 - 5029/594*e^3 + 7996/165*e, 31/990*e^6 - 307/198*e^4 + 734/99*e^2 + 536/55, 35/9504*e^7 - 635/4752*e^5 - 13445/9504*e^3 + 1175/132*e, -4/165*e^6 + 46/33*e^4 - 496/33*e^2 + 3286/55, -1/99*e^6 + 115/198*e^4 - 745/198*e^2 - 542/11, -179/23760*e^7 + 559/2376*e^5 + 24433/4752*e^3 - 13351/330*e, 1/297*e^7 - 115/594*e^5 + 1141/594*e^3 + 212/33*e, -31/1980*e^6 + 307/396*e^4 - 367/99*e^2 - 268/55, -79/47520*e^7 + 479/4752*e^5 - 11875/9504*e^3 + 409/660*e, -26/495*e^6 + 299/99*e^4 - 3224/99*e^2 + 4058/55, -31/9504*e^7 + 1015/4752*e^5 - 31991/9504*e^3 + 1681/132*e, 2/45*e^6 - 23/9*e^4 + 248/9*e^2 - 266/5, -3/220*e^6 + 29/44*e^4 - 65/22*e^2 + 274/55, 32/495*e^6 - 368/99*e^4 + 3968/99*e^2 - 4326/55, 8/165*e^6 - 92/33*e^4 + 992/33*e^2 - 2502/55, 31/660*e^6 - 307/132*e^4 + 367/33*e^2 + 804/55, -32/495*e^6 + 368/99*e^4 - 3968/99*e^2 + 3226/55, 1/120*e^7 - 1/4*e^5 - 151/24*e^3 + 761/15*e, 4/495*e^6 - 46/99*e^4 + 298/99*e^2 + 2168/55, 17/47520*e^7 - 469/4752*e^5 + 46757/9504*e^3 - 32467/660*e, 197/23760*e^7 - 1207/2376*e^5 + 31061/4752*e^3 - 2287/330*e, 389/47520*e^7 - 2509/4752*e^5 + 75857/9504*e^3 - 17219/660*e, -7/47520*e^7 - 133/4752*e^5 + 19229/9504*e^3 - 5901/220*e, -7/1056*e^7 + 215/528*e^5 - 1861/352*e^3 + 833/132*e, -383/47520*e^7 + 2623/4752*e^5 - 92339/9504*e^3 + 32393/660*e, -1/5280*e^7 - 19/528*e^5 + 2747/1056*e^3 - 5607/220*e, -229/47520*e^7 + 1589/4752*e^5 - 57601/9504*e^3 + 7153/220*e, -2/165*e^6 + 23/33*e^4 - 248/33*e^2 + 1698/55, 29/4752*e^7 - 413/2376*e^5 - 24491/4752*e^3 + 2809/66*e, -17/5280*e^7 + 39/176*e^5 - 4165/1056*e^3 + 13463/660*e, 4/165*e^6 - 46/33*e^4 + 496/33*e^2 + 564/55, 31/495*e^6 - 307/99*e^4 + 1468/99*e^2 + 1072/55, 1/660*e^7 - 1/22*e^5 - 151/132*e^3 + 1522/165*e, 7/396*e^6 - 353/396*e^4 + 883/198*e^2 + 162/11, 19/1980*e^6 - 169/396*e^4 + 287/198*e^2 - 1358/55, -8/165*e^6 + 92/33*e^4 - 992/33*e^2 + 1842/55, -163/47520*e^7 + 863/4752*e^5 - 10015/9504*e^3 - 11827/660*e, -31/1980*e^6 + 307/396*e^4 - 367/99*e^2 - 268/55, 23/5940*e^7 - 29/297*e^5 - 4771/1188*e^3 + 5674/165*e, -13/1584*e^7 + 415/792*e^5 - 12185/1584*e^3 + 1469/66*e, -11/1440*e^7 + 31/144*e^5 + 1897/288*e^3 - 3277/60*e, 16/495*e^6 - 184/99*e^4 + 1984/99*e^2 - 1008/55, -2/165*e^6 + 23/33*e^4 - 248/33*e^2 - 1272/55, 41/9504*e^7 - 857/4752*e^5 - 2399/9504*e^3 - 153/44*e, -4/165*e^6 + 46/33*e^4 - 496/33*e^2 + 2296/55, -37/990*e^6 + 188/99*e^4 - 1915/198*e^2 - 2162/55, 3/220*e^6 - 29/44*e^4 + 65/22*e^2 - 274/55, 773/47520*e^7 - 5113/4752*e^5 + 165449/9504*e^3 - 47083/660*e, -14/495*e^6 + 161/99*e^4 - 1736/99*e^2 + 1102/55]; 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;