/* 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![5, -1, -6, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [5, 5, w], [5, 5, -w^2 + w + 2], [7, 7, -w^2 + 2], [13, 13, -w^2 + 3], [13, 13, w^2 - w - 4], [16, 2, 2], [19, 19, -w^2 + w + 1], [23, 23, -w^3 + 4*w + 2], [25, 5, -w^3 + w^2 + 3*w - 1], [29, 29, w^3 - w^2 - 4*w + 1], [29, 29, -w + 3], [31, 31, -w^3 + w^2 + 5*w - 2], [37, 37, w^3 - 4*w - 1], [37, 37, w^3 - 3*w + 1], [53, 53, -w^3 + 2*w^2 + 4*w - 6], [59, 59, w^2 + w - 4], [61, 61, -2*w^2 + w + 8], [73, 73, -w^3 + 5*w - 1], [79, 79, 2*w^2 + w - 6], [79, 79, 2*w^3 - w^2 - 9*w + 3], [81, 3, -3], [83, 83, -w^3 + w^2 + 3*w - 4], [97, 97, -w^3 + 6*w + 2], [97, 97, -2*w^3 + w^2 + 10*w - 1], [97, 97, 3*w^3 - 4*w^2 - 15*w + 13], [97, 97, w^3 - w^2 - 6*w + 2], [103, 103, -2*w^3 + 3*w^2 + 11*w - 9], [109, 109, w^2 + 3*w - 3], [137, 137, -w^2 + 7], [149, 149, w^3 - 2*w^2 - 5*w + 4], [149, 149, w^3 + 2*w^2 - 6*w - 7], [151, 151, w^3 + w^2 - 5*w - 3], [163, 163, w^3 - 3*w - 4], [163, 163, -w^3 + 6*w - 3], [167, 167, w^3 - 2*w^2 - 6*w + 4], [169, 13, w^2 + 2*w - 4], [191, 191, w^3 - w^2 - 5*w - 2], [191, 191, 2*w^3 - 9*w - 2], [197, 197, w^3 - 4*w + 4], [211, 211, -w^3 + 2*w^2 + 6*w - 8], [223, 223, -w^3 + 3*w^2 + 4*w - 6], [229, 229, w^3 - w^2 - 5*w - 1], [251, 251, 2*w^3 - 2*w^2 - 7*w + 3], [257, 257, 3*w^2 - 14], [263, 263, w^2 - 3*w - 1], [263, 263, w^2 - 2*w - 7], [269, 269, -w^3 + w^2 + 2*w - 4], [269, 269, 2*w^3 - 11*w - 1], [271, 271, -w^3 + w^2 + 4*w - 7], [277, 277, w^3 - w^2 - 3*w + 8], [311, 311, w^3 + 2*w^2 - 7*w - 11], [311, 311, -3*w^2 - 2*w + 8], [317, 317, -2*w^3 + 2*w^2 + 7*w - 4], [331, 331, w^2 - w - 8], [331, 331, 3*w^3 - 2*w^2 - 12*w + 9], [337, 337, -2*w^3 + 2*w^2 + 11*w - 8], [343, 7, w^3 - 3*w^2 - 4*w + 11], [347, 347, -2*w^3 + 2*w^2 + 9*w - 4], [349, 349, -w^3 + 2*w^2 + 2*w - 6], [353, 353, 2*w^2 - w - 12], [353, 353, 2*w^3 - w^2 - 9*w - 1], [367, 367, -2*w^3 + w^2 + 10*w - 6], [367, 367, 2*w^3 - 11*w - 3], [379, 379, -3*w - 1], [379, 379, w^3 - 2*w - 3], [383, 383, -w^3 + 2*w^2 + 4*w - 11], [383, 383, -w^3 - w^2 + 7*w + 4], [389, 389, 2*w^3 - w^2 - 11*w + 4], [401, 401, w^3 + 2*w^2 - 7*w - 6], [401, 401, 2*w^3 - 11*w - 6], [419, 419, -2*w^3 + w^2 + 8*w - 1], [421, 421, -w^3 + 3*w^2 + 3*w - 12], [421, 421, -w^3 - 2*w^2 + 2*w + 6], [433, 433, w^3 - 2*w^2 - 6*w + 11], [433, 433, w^2 - 8], [439, 439, -w^3 + 3*w^2 + 3*w - 7], [443, 443, w^3 - w^2 - 2*w - 4], [443, 443, 2*w^3 + w^2 - 7*w - 2], [443, 443, 2*w^3 - w^2 - 8*w + 2], [443, 443, -w^2 - 2], [467, 467, 3*w^3 - 5*w^2 - 15*w + 19], [467, 467, 2*w^3 - 3*w^2 - 8*w + 13], [467, 467, -w^3 + 2*w^2 + 3*w - 9], [467, 467, -w^3 + 3*w^2 + 3*w - 8], [479, 479, -w^3 - 3*w^2 + 5*w + 13], [479, 479, -w^3 + w^2 + w - 4], [487, 487, 3*w^3 - w^2 - 13*w - 2], [491, 491, w^2 + w - 9], [499, 499, -3*w^3 + 3*w^2 + 13*w - 7], [503, 503, 2*w^2 + w - 9], [509, 509, -2*w^2 - 3*w + 6], [509, 509, -w^3 + w^2 - 4], [521, 521, -3*w^2 + w + 12], [523, 523, 2*w^3 - 8*w - 3], [529, 23, 2*w^3 - w^2 - 6*w - 1], [541, 541, -w^3 + w^2 + 7*w - 4], [547, 547, 2*w^3 - 3*w^2 - 9*w + 7], [557, 557, 3*w^2 - 2*w - 9], [557, 557, w^3 - 6*w - 7], [563, 563, -2*w^3 + w^2 + 6*w - 6], [577, 577, -w^3 + 4*w^2 + 2*w - 13], [587, 587, w^2 - 2*w + 3], [587, 587, w^2 + 2*w - 6], [593, 593, -2*w^3 + 2*w^2 + 11*w - 3], [599, 599, w^3 + 3*w^2 - 6*w - 12], [601, 601, -2*w^3 + w^2 + 9*w + 2], [607, 607, -w^3 + 3*w^2 + 4*w - 9], [617, 617, -2*w^3 + w^2 + 7*w - 3], [619, 619, -2*w^3 + 4*w^2 + 9*w - 11], [631, 631, 3*w^2 - w - 7], [631, 631, -w^3 - w^2 + 6*w + 1], [641, 641, -w^3 + 2*w^2 + 5*w - 1], [641, 641, -w^3 + 2*w^2 + 2*w - 7], [643, 643, -3*w^3 + w^2 + 14*w - 4], [647, 647, -3*w^3 + 5*w^2 + 14*w - 18], [647, 647, -w^3 + 7*w + 2], [647, 647, -2*w^3 + 7*w - 1], [647, 647, 2*w^3 - w^2 - 10*w - 1], [653, 653, -2*w^3 - w^2 + 11*w + 4], [659, 659, -w^3 - w^2 + 5*w - 1], [673, 673, -w^3 + w^2 + 7*w - 2], [673, 673, -w^3 + w^2 + 2*w - 8], [683, 683, 2*w^3 - 11*w - 8], [683, 683, 2*w^3 - 7*w - 3], [701, 701, 2*w^3 - 4*w^2 - 10*w + 17], [727, 727, -w^3 + w^2 + 4*w - 8], [727, 727, 3*w^2 - 11], [739, 739, 2*w^3 + 3*w^2 - 8*w - 13], [743, 743, 2*w^2 + w - 12], [743, 743, 3*w^2 - 2*w - 14], [751, 751, w^3 + w^2 - 3*w - 7], [757, 757, -2*w^3 + 10*w - 1], [761, 761, -2*w^3 + w^2 + 7*w - 1], [761, 761, 3*w^2 + w - 7], [773, 773, -2*w^3 + 2*w^2 + 8*w - 1], [797, 797, -2*w^3 + w^2 + 9*w + 3], [797, 797, 3*w^3 - 14*w - 8], [809, 809, 5*w^3 - 5*w^2 - 24*w + 16], [821, 821, -4*w^3 + 3*w^2 + 18*w - 9], [821, 821, w^3 - w - 3], [823, 823, 3*w^2 + w - 8], [829, 829, -2*w^3 + 3*w^2 + 10*w - 7], [829, 829, 3*w^2 - 2*w - 16], [841, 29, 2*w^2 + w - 11], [853, 853, 3*w^3 - 2*w^2 - 14*w + 3], [853, 853, 3*w^3 - 5*w^2 - 14*w + 17], [857, 857, 3*w^2 - w - 8], [857, 857, 3*w^3 - w^2 - 13*w + 3], [859, 859, w^3 - 3*w^2 - 6*w + 12], [877, 877, w^3 + 2*w^2 - 5*w - 4], [877, 877, w^2 - w - 9], [881, 881, w^3 + 2*w^2 - 4*w - 12], [881, 881, w^2 - 4*w - 4], [883, 883, -3*w^3 + 2*w^2 + 11*w - 8], [887, 887, -2*w^3 + w^2 + 6*w - 1], [911, 911, 3*w^2 - w - 17], [911, 911, 2*w^3 - 2*w^2 - 12*w + 3], [919, 919, 5*w^3 - 6*w^2 - 22*w + 24], [919, 919, -2*w^3 + w^2 + 6*w - 7], [929, 929, 2*w^3 - 7*w - 1], [937, 937, -w^3 - w^2 + 8*w - 2], [947, 947, 3*w^3 + w^2 - 15*w - 6], [947, 947, -2*w^3 + 9*w + 9], [953, 953, w^3 - 2*w^2 - 6*w + 1], [977, 977, w^3 - w^2 - 3*w + 9], [977, 977, w^3 + w^2 - 3*w - 8], [983, 983, w^3 - 5*w - 7], [991, 991, -w^3 + 2*w^2 + 2*w - 8], [991, 991, w^2 + 4*w - 1], [997, 997, 2*w^3 + w^2 - 12*w - 8]]; primes := [ideal : I in primesArray]; heckePol := x^7 - 3*x^6 - 30*x^5 + 94*x^4 + 119*x^3 - 345*x^2 + 22*x + 134; K := NumberField(heckePol); heckeEigenvaluesArray := [-1, 1, e, 57/1012*e^6 - 27/253*e^5 - 464/253*e^4 + 820/253*e^3 + 10861/1012*e^2 - 2055/253*e - 2371/506, -101/3542*e^6 + 109/1771*e^5 + 3231/3542*e^4 - 3554/1771*e^3 - 9132/1771*e^2 + 12569/1771*e + 1436/1771, 197/7084*e^6 - 80/1771*e^5 - 3265/3542*e^4 + 2186/1771*e^3 + 39499/7084*e^2 - 45/1771*e - 10183/3542, -85/3542*e^6 + 201/3542*e^5 + 1193/1771*e^4 - 6245/3542*e^3 - 6165/3542*e^2 + 7860/1771*e - 5630/1771, -569/7084*e^6 + 579/3542*e^5 + 4441/1771*e^4 - 17725/3542*e^3 - 88357/7084*e^2 + 20474/1771*e + 8879/3542, 181/1771*e^6 - 303/1771*e^5 - 5685/1771*e^4 + 9652/1771*e^3 + 30942/1771*e^2 - 22640/1771*e - 23944/1771, 181/1771*e^6 - 303/1771*e^5 - 5685/1771*e^4 + 9652/1771*e^3 + 30942/1771*e^2 - 24411/1771*e - 18631/1771, 31/1012*e^6 - 28/253*e^5 - 447/506*e^4 + 841/253*e^3 + 2933/1012*e^2 - 2609/253*e - 819/506, -8/161*e^6 + 17/322*e^5 + 523/322*e^4 - 541/322*e^3 - 3405/322*e^2 + 201/161*e + 1270/161, -601/7084*e^6 + 298/1771*e^5 + 9727/3542*e^4 - 9294/1771*e^3 - 112555/7084*e^2 + 23412/1771*e + 30095/3542, -383/3542*e^6 + 739/3542*e^5 + 12147/3542*e^4 - 22097/3542*e^3 - 33735/1771*e^2 + 18748/1771*e + 25470/1771, -27/7084*e^6 - 41/3542*e^5 + 879/3542*e^4 + 1873/3542*e^3 - 27169/7084*e^2 - 7815/1771*e + 7275/3542, -135/7084*e^6 - 205/3542*e^5 + 1312/1771*e^4 + 5823/3542*e^3 - 54379/7084*e^2 - 12510/1771*e + 22207/3542, -63/1012*e^6 + 73/506*e^5 + 1039/506*e^4 - 2039/506*e^3 - 12457/1012*e^2 + 1499/253*e + 4325/506, -643/7084*e^6 + 729/3542*e^5 + 5252/1771*e^4 - 21381/3542*e^3 - 123727/7084*e^2 + 24833/1771*e + 47315/3542, -13/1012*e^6 - 1/506*e^5 + 57/253*e^4 + 21/506*e^3 + 1855/1012*e^2 - 1289/253*e - 4537/506, -171/1012*e^6 + 81/253*e^5 + 1392/253*e^4 - 2460/253*e^3 - 33595/1012*e^2 + 5659/253*e + 14703/506, 32/1771*e^6 - 34/1771*e^5 - 1609/3542*e^4 + 1726/1771*e^3 + 4121/3542*e^2 - 11752/1771*e - 8783/1771, 411/7084*e^6 - 163/3542*e^5 - 7477/3542*e^4 + 4941/3542*e^3 + 119193/7084*e^2 + 895/1771*e - 45805/3542, 103/7084*e^6 + 111/1771*e^5 - 1779/3542*e^4 - 3343/1771*e^3 + 37265/7084*e^2 + 22532/1771*e - 47037/3542, -81/644*e^6 + 19/161*e^5 + 1349/322*e^4 - 652/161*e^3 - 18395/644*e^2 + 1349/161*e + 8301/322, 271/3542*e^6 - 310/1771*e^5 - 8003/3542*e^4 + 9799/1771*e^3 + 15297/1771*e^2 - 28289/1771*e + 2740/1771, 249/7084*e^6 - 409/3542*e^5 - 1987/1771*e^4 + 12637/3542*e^3 + 44729/7084*e^2 - 28285/1771*e - 34033/3542, 6/161*e^6 - 53/322*e^5 - 176/161*e^4 + 1573/322*e^3 + 492/161*e^2 - 2767/161*e + 1060/161, -85/3542*e^6 + 201/3542*e^5 + 1193/1771*e^4 - 6245/3542*e^3 - 6165/3542*e^2 + 13173/1771*e - 5630/1771, -215/1771*e^6 + 1121/3542*e^5 + 13633/3542*e^4 - 33155/3542*e^3 - 72129/3542*e^2 + 39554/1771*e + 22982/1771, -9/308*e^6 + 6/77*e^5 + 139/154*e^4 - 214/77*e^3 - 1459/308*e^2 + 1168/77*e - 809/154, 85/1771*e^6 - 201/1771*e^5 - 2386/1771*e^4 + 6245/1771*e^3 + 6165/1771*e^2 - 19262/1771*e + 13031/1771, 37/3542*e^6 - 75/1771*e^5 - 811/1771*e^4 + 1828/1771*e^3 + 17685/3542*e^2 - 817/1771*e - 33386/1771, -393/3542*e^6 + 971/3542*e^5 + 12011/3542*e^4 - 29499/3542*e^3 - 28993/1771*e^2 + 35195/1771*e + 23245/1771, 2525/7084*e^6 - 2725/3542*e^5 - 19751/1771*e^4 + 83537/3542*e^3 + 405241/7084*e^2 - 106639/1771*e - 101427/3542, 279/7084*e^6 - 757/3542*e^5 - 1885/1771*e^4 + 21969/3542*e^3 + 5651/7084*e^2 - 32589/1771*e - 14961/3542, -1/23*e^6 + 5/46*e^5 + 51/46*e^4 - 151/46*e^3 - 49/46*e^2 + 120/23*e - 100/23, -327/1771*e^6 + 1359/3542*e^5 + 10075/1771*e^4 - 41695/3542*e^3 - 47261/1771*e^2 + 50579/1771*e + 15646/1771, -16/253*e^6 + 17/253*e^5 + 931/506*e^4 - 610/253*e^3 - 4211/506*e^2 + 1069/253*e + 1735/253, -57/322*e^6 + 131/322*e^5 + 1833/322*e^4 - 4131/322*e^3 - 5235/161*e^2 + 6985/161*e + 3015/161, -97/3542*e^6 - 229/3542*e^5 + 2174/1771*e^4 + 6833/3542*e^3 - 52873/3542*e^2 - 13845/1771*e + 32433/1771, 35/253*e^6 - 53/253*e^5 - 2337/506*e^4 + 1619/253*e^3 + 15331/506*e^2 - 3556/253*e - 8207/253, 13/506*e^6 + 1/253*e^5 - 481/506*e^4 - 21/253*e^3 + 1982/253*e^2 + 1566/253*e - 270/253, 628/1771*e^6 - 1110/1771*e^5 - 40653/3542*e^4 + 35201/1771*e^3 + 238715/3542*e^2 - 96037/1771*e - 69427/1771, 997/7084*e^6 - 585/3542*e^5 - 8653/1771*e^4 + 18863/3542*e^3 + 254829/7084*e^2 - 18594/1771*e - 147421/3542, 79/644*e^6 - 47/322*e^5 - 633/161*e^4 + 1723/322*e^3 + 15011/644*e^2 - 3037/161*e - 3433/322, 59/7084*e^6 + 12/1771*e^5 - 862/1771*e^4 - 505/1771*e^3 + 51367/7084*e^2 + 3106/1771*e - 63911/3542, -442/1771*e^6 + 691/1771*e^5 + 13824/1771*e^4 - 21848/1771*e^3 - 72791/1771*e^2 + 41011/1771*e + 42395/1771, -387/3542*e^6 + 593/1771*e^5 + 12801/3542*e^4 - 18019/1771*e^3 - 38568/1771*e^2 + 62872/1771*e + 31664/1771, -361/1771*e^6 + 937/1771*e^5 + 22413/3542*e^4 - 27773/1771*e^3 - 104767/3542*e^2 + 62180/1771*e + 9371/1771, -3/322*e^6 - 27/322*e^5 + 44/161*e^4 + 613/322*e^3 - 1373/322*e^2 - 234/161*e + 3599/161, 1175/7084*e^6 - 1233/3542*e^5 - 18575/3542*e^4 + 35507/3542*e^3 + 197941/7084*e^2 - 28074/1771*e - 84793/3542, -1653/7084*e^6 + 1819/3542*e^5 + 26659/3542*e^4 - 56921/3542*e^3 - 299283/7084*e^2 + 89449/1771*e + 79385/3542, 52/161*e^6 - 191/322*e^5 - 3319/322*e^4 + 5851/322*e^3 + 18349/322*e^2 - 6217/161*e - 6001/161, -13/322*e^6 + 22/161*e^5 + 435/322*e^4 - 738/161*e^3 - 1338/161*e^2 + 3816/161*e + 2018/161, -502/1771*e^6 + 2395/3542*e^5 + 31329/3542*e^4 - 70535/3542*e^3 - 153973/3542*e^2 + 75443/1771*e + 26321/1771, -975/7084*e^6 + 342/1771*e^5 + 16393/3542*e^4 - 11736/1771*e^3 - 224689/7084*e^2 + 44246/1771*e + 111583/3542, 809/3542*e^6 - 1413/3542*e^5 - 12563/1771*e^4 + 44853/3542*e^3 + 129933/3542*e^2 - 51369/1771*e - 59968/1771, 5/44*e^6 - 3/22*e^5 - 87/22*e^4 + 85/22*e^3 + 1275/44*e^2 + 27/11*e - 961/22, 709/7084*e^6 - 216/1771*e^5 - 5736/1771*e^4 + 7319/1771*e^3 + 146849/7084*e^2 - 16946/1771*e - 41485/3542, 367/3542*e^6 - 361/1771*e^5 - 5651/1771*e^4 + 12388/1771*e^3 + 58913/3542*e^2 - 49459/1771*e - 20175/1771, -1181/3542*e^6 + 2251/3542*e^5 + 18180/1771*e^4 - 71559/3542*e^3 - 178791/3542*e^2 + 95769/1771*e + 48038/1771, 516/1771*e^6 - 991/1771*e^5 - 32365/3542*e^4 + 30931/1771*e^3 + 168505/3542*e^2 - 88554/1771*e - 20091/1771, 767/3542*e^6 - 1147/3542*e^5 - 25343/3542*e^4 + 35725/3542*e^3 + 79747/1771*e^2 - 37901/1771*e - 71084/1771, 3/1771*e^6 + 215/3542*e^5 - 981/3542*e^4 - 6539/3542*e^3 + 21583/3542*e^2 + 13509/1771*e - 35856/1771, -72/1771*e^6 + 153/3542*e^5 + 4063/3542*e^4 - 7767/3542*e^3 - 20341/3542*e^2 + 29984/1771*e + 6922/1771, 1553/3542*e^6 - 3089/3542*e^5 - 49365/3542*e^4 + 94723/3542*e^3 + 134191/1771*e^2 - 111833/1771*e - 85696/1771, 1615/7084*e^6 - 512/1771*e^5 - 26209/3542*e^4 + 16824/1771*e^3 + 326113/7084*e^2 - 35708/1771*e - 110863/3542, -67/506*e^6 + 87/506*e^5 + 1973/506*e^4 - 2839/506*e^3 - 4104/253*e^2 + 1329/253*e - 360/253, -111/7084*e^6 + 225/3542*e^5 + 2433/3542*e^4 - 3713/3542*e^3 - 35345/7084*e^2 - 17370/1771*e - 7873/3542, -19/506*e^6 + 18/253*e^5 + 225/253*e^4 - 631/253*e^3 + 1271/506*e^2 + 2129/253*e - 6631/253, 2055/7084*e^6 - 1293/1771*e^5 - 16036/1771*e^4 + 39803/1771*e^3 + 316147/7084*e^2 - 121266/1771*e - 44841/3542, 619/7084*e^6 - 1159/3542*e^5 - 4271/1771*e^4 + 34459/3542*e^3 + 30311/7084*e^2 - 55393/1771*e + 475/3542, 52/253*e^6 - 237/506*e^5 - 1671/253*e^4 + 7507/506*e^3 + 9278/253*e^2 - 12266/253*e - 6461/253, 52/253*e^6 - 237/506*e^5 - 3089/506*e^4 + 7001/506*e^3 + 11725/506*e^2 - 6953/253*e + 4165/253, 111/644*e^6 - 32/161*e^5 - 975/161*e^4 + 971/161*e^3 + 29227/644*e^2 - 1467/161*e - 15955/322, -307/3542*e^6 + 1101/3542*e^5 + 4288/1771*e^4 - 35265/3542*e^3 - 20183/3542*e^2 + 71205/1771*e - 16063/1771, -41/7084*e^6 - 587/3542*e^5 + 2909/3542*e^4 + 16881/3542*e^3 - 133611/7084*e^2 - 29774/1771*e + 98023/3542, 1/14*e^6 - 5/14*e^5 - 27/14*e^4 + 155/14*e^3 + 13/7*e^2 - 286/7*e - 19/7, -87/506*e^6 + 149/253*e^5 + 2713/506*e^4 - 4394/253*e^3 - 6005/253*e^2 + 11959/253*e + 2021/253, -381/1771*e^6 + 1031/3542*e^5 + 25411/3542*e^4 - 30253/3542*e^3 - 169549/3542*e^2 + 11082/1771*e + 78395/1771, -2727/7084*e^6 + 2943/3542*e^5 + 42733/3542*e^4 - 90645/3542*e^3 - 441769/7084*e^2 + 117437/1771*e + 97215/3542, 233/3542*e^6 - 801/3542*e^5 - 7103/3542*e^4 + 27953/3542*e^3 + 16315/1771*e^2 - 76655/1771*e - 11028/1771, 2971/7084*e^6 - 1647/1771*e^5 - 46741/3542*e^4 + 50273/1771*e^3 + 496553/7084*e^2 - 126114/1771*e - 135017/3542, 471/7084*e^6 - 859/3542*e^5 - 7069/3542*e^4 + 23605/3542*e^3 + 48121/7084*e^2 - 11255/1771*e + 24217/3542, -250/1771*e^6 + 487/1771*e^5 + 16223/3542*e^4 - 15034/1771*e^3 - 94291/3542*e^2 + 30713/1771*e + 46369/1771, -221/7084*e^6 - 135/1771*e^5 + 1728/1771*e^4 + 4353/1771*e^3 - 51449/7084*e^2 - 42912/1771*e + 61515/3542, -101/1771*e^6 + 218/1771*e^5 + 3231/1771*e^4 - 7108/1771*e^3 - 16493/1771*e^2 + 28680/1771*e + 40063/1771, -116/1771*e^6 + 566/1771*e^5 + 3027/1771*e^4 - 16440/1771*e^3 - 2267/1771*e^2 + 44372/1771*e - 26826/1771, -117/506*e^6 + 235/506*e^5 + 1785/253*e^4 - 7465/506*e^3 - 16701/506*e^2 + 10700/253*e + 3189/253, -255/3542*e^6 + 603/3542*e^5 + 3579/1771*e^4 - 18735/3542*e^3 - 18495/3542*e^2 + 28893/1771*e - 27516/1771, -47/644*e^6 + 15/161*e^5 + 452/161*e^4 - 430/161*e^3 - 16251/644*e^2 + 904/161*e + 4435/322, -145/506*e^6 + 164/253*e^5 + 2303/253*e^4 - 5215/253*e^3 - 25161/506*e^2 + 16221/253*e + 6320/253, -1893/7084*e^6 + 1061/3542*e^5 + 32111/3542*e^4 - 35943/3542*e^3 - 454203/7084*e^2 + 53041/1771*e + 259757/3542, 1203/3542*e^6 - 2053/3542*e^5 - 39957/3542*e^4 + 62341/3542*e^3 + 128374/1771*e^2 - 65717/1771*e - 96273/1771, -313/3542*e^6 + 443/1771*e^5 + 9557/3542*e^4 - 14363/1771*e^3 - 20883/1771*e^2 + 57696/1771*e - 20940/1771, -4/23*e^6 + 10/23*e^5 + 125/23*e^4 - 302/23*e^3 - 627/23*e^2 + 779/23*e + 566/23, -3/1012*e^6 + 68/253*e^5 - 71/506*e^4 - 2187/253*e^3 + 8563/1012*e^2 + 12625/253*e - 9649/506, -1207/7084*e^6 + 625/1771*e^5 + 9710/1771*e^4 - 18185/1771*e^3 - 215055/7084*e^2 + 34554/1771*e + 74131/3542, -1559/7084*e^6 + 1437/3542*e^5 + 11701/1771*e^4 - 49405/3542*e^3 - 208499/7084*e^2 + 86353/1771*e + 20605/3542, 494/1771*e^6 - 1189/1771*e^5 - 15242/1771*e^4 + 36607/1771*e^3 + 70937/1771*e^2 - 102612/1771*e - 13942/1771, -7/22*e^6 + 15/22*e^5 + 113/11*e^4 - 469/22*e^3 - 1279/22*e^2 + 690/11*e + 318/11, 34/1771*e^6 - 515/3542*e^5 - 2263/3542*e^4 + 10309/3542*e^3 + 3161/3542*e^2 + 22048/1771*e + 8046/1771, 1515/7084*e^6 - 1635/3542*e^5 - 23347/3542*e^4 + 47997/3542*e^3 + 222601/7084*e^2 - 36710/1771*e - 58731/3542, -18/1771*e^6 + 481/3542*e^5 + 573/3542*e^4 - 15667/3542*e^3 + 10411/3542*e^2 + 34061/1771*e + 845/1771, 59/644*e^6 - 137/322*e^5 - 379/161*e^4 + 3981/322*e^3 - 153/644*e^2 - 4461/161*e + 7895/322, -45/644*e^6 + 39/322*e^5 + 330/161*e^4 - 1279/322*e^3 - 4173/644*e^2 + 1787/161*e - 5585/322, -258/1771*e^6 + 991/3542*e^5 + 15297/3542*e^4 - 30931/3542*e^3 - 65657/3542*e^2 + 24796/1771*e + 14473/1771, 127/1012*e^6 - 107/506*e^5 - 985/253*e^4 + 3259/506*e^3 + 20879/1012*e^2 - 2821/253*e - 12855/506, 2053/3542*e^6 - 4063/3542*e^5 - 32794/1771*e^4 + 124791/3542*e^3 + 362673/3542*e^2 - 151401/1771*e - 121439/1771, 10/1771*e^6 - 232/1771*e^5 + 136/1771*e^4 + 7402/1771*e^3 - 13026/1771*e^2 - 29352/1771*e + 50496/1771, 773/3542*e^6 - 466/1771*e^5 - 24553/3542*e^4 + 14593/1771*e^3 + 71943/1771*e^2 - 1369/1771*e - 78604/1771, 785/1771*e^6 - 2775/3542*e^5 - 51259/3542*e^4 + 87117/3542*e^3 + 315661/3542*e^2 - 120489/1771*e - 104051/1771, 815/3542*e^6 - 599/1771*e^5 - 26107/3542*e^4 + 19157/1771*e^3 + 73987/1771*e^2 - 37860/1771*e - 3732/1771, 1685/3542*e^6 - 1836/1771*e^5 - 53237/3542*e^4 + 57784/1771*e^3 + 139603/1771*e^2 - 177690/1771*e - 66952/1771, 799/7084*e^6 - 738/1771*e^5 - 5430/1771*e^4 + 23088/1771*e^3 + 36699/7084*e^2 - 75904/1771*e + 36983/3542, 9/1771*e^6 - 563/1771*e^5 + 599/3542*e^4 + 15871/1771*e^3 - 34427/3542*e^2 - 63962/1771*e + 4005/1771, 155/308*e^6 - 52/77*e^5 - 2565/154*e^4 + 1675/77*e^3 + 33101/308*e^2 - 3783/77*e - 15139/154, -1033/3542*e^6 + 1651/3542*e^5 + 16707/1771*e^4 - 53393/3542*e^3 - 203685/3542*e^2 + 78333/1771*e + 73884/1771, 327/3542*e^6 - 1565/3542*e^5 - 4152/1771*e^4 + 46527/3542*e^3 - 5869/3542*e^2 - 84618/1771*e + 41765/1771, 3173/7084*e^6 - 1756/1771*e^5 - 24986/1771*e^4 + 53827/1771*e^3 + 525997/7084*e^2 - 142225/1771*e - 159141/3542, -155/644*e^6 + 94/161*e^5 + 1244/161*e^4 - 2802/161*e^3 - 27039/644*e^2 + 6352/161*e + 7775/322, 1115/7084*e^6 - 537/3542*e^5 - 18983/3542*e^4 + 16843/3542*e^3 + 276097/7084*e^2 - 10611/1771*e - 140647/3542, -213/1771*e^6 + 337/1771*e^5 + 7375/1771*e^4 - 9607/1771*e^3 - 55140/1771*e^2 + 13140/1771*e + 91170/1771, 653/7084*e^6 - 845/3542*e^5 - 12207/3542*e^4 + 23311/3542*e^3 + 199251/7084*e^2 - 25087/1771*e - 131869/3542, -107/322*e^6 + 82/161*e^5 + 1784/161*e^4 - 2619/161*e^3 - 23425/322*e^2 + 6627/161*e + 10566/161, 446/1771*e^6 - 1138/1771*e^5 - 14478/1771*e^4 + 34018/1771*e^3 + 84228/1771*e^2 - 102694/1771*e - 81348/1771, 157/1771*e^6 - 555/3542*e^5 - 8835/3542*e^4 + 16715/3542*e^3 + 29129/3542*e^2 - 15597/1771*e - 11601/1771, 1867/7084*e^6 - 911/1771*e^5 - 14550/1771*e^4 + 28745/1771*e^3 + 304595/7084*e^2 - 90280/1771*e - 118549/3542, 337/1771*e^6 - 26/1771*e^5 - 11710/1771*e^4 + 2570/1771*e^3 + 89136/1771*e^2 + 6848/1771*e - 97975/1771, -648/1771*e^6 + 1574/1771*e^5 + 20940/1771*e^4 - 48234/1771*e^3 - 117214/1771*e^2 + 149428/1771*e + 69382/1771, -639/7084*e^6 - 190/1771*e^5 + 10177/3542*e^4 + 4749/1771*e^3 - 135313/7084*e^2 - 39733/1771*e + 30495/3542, -477/3542*e^6 + 1503/3542*e^5 + 15119/3542*e^4 - 47755/3542*e^3 - 40165/1771*e^2 + 101093/1771*e + 27578/1771, 93/1771*e^6 - 419/3542*e^5 - 1923/1771*e^4 + 13353/3542*e^3 - 11694/1771*e^2 - 8032/1771*e + 57324/1771, -199/3542*e^6 - 171/1771*e^5 + 2543/1771*e^4 + 4097/1771*e^3 - 10683/3542*e^2 - 34520/1771*e - 50476/1771, 3/14*e^6 - 4/7*e^5 - 95/14*e^4 + 124/7*e^3 + 242/7*e^2 - 382/7*e + 13/7, 335/1012*e^6 - 172/253*e^5 - 5565/506*e^4 + 5130/253*e^3 + 68617/1012*e^2 - 11798/253*e - 21223/506, 1033/7084*e^6 + 30/1771*e^5 - 16707/3542*e^4 + 1394/1771*e^3 + 228479/7084*e^2 - 4632/1771*e - 43777/3542, 383/1012*e^6 - 243/506*e^5 - 3100/253*e^4 + 8139/506*e^3 + 76831/1012*e^2 - 9121/253*e - 35337/506, 380/1771*e^6 - 1693/3542*e^5 - 23313/3542*e^4 + 50733/3542*e^3 + 109815/3542*e^2 - 43921/1771*e - 22168/1771, -625/3542*e^6 + 166/1771*e^5 + 21607/3542*e^4 - 5510/1771*e^3 - 82619/1771*e^2 + 1643/1771*e + 116847/1771, 213/3542*e^6 - 337/3542*e^5 - 7375/3542*e^4 + 9607/3542*e^3 + 24028/1771*e^2 - 17196/1771*e + 9316/1771, 113/644*e^6 - 55/322*e^5 - 2033/322*e^4 + 1845/322*e^3 + 33255/644*e^2 - 1067/161*e - 20179/322, 410/1771*e^6 - 657/1771*e^5 - 13905/1771*e^4 + 21893/1771*e^3 + 96410/1771*e^2 - 73534/1771*e - 106223/1771, -117/506*e^6 + 235/506*e^5 + 3823/506*e^4 - 6959/506*e^3 - 11260/253*e^2 + 8423/253*e + 1924/253, 15/253*e^6 + 63/506*e^5 - 857/506*e^4 - 1829/506*e^3 + 3173/506*e^2 + 7584/253*e + 3386/253, -130/1771*e^6 + 719/3542*e^5 + 8861/3542*e^4 - 20665/3542*e^3 - 52715/3542*e^2 + 16750/1771*e + 9448/1771, -1033/3542*e^6 + 1651/3542*e^5 + 31643/3542*e^4 - 53393/3542*e^3 - 74392/1771*e^2 + 60623/1771*e + 3044/1771, 786/1771*e^6 - 1942/1771*e^5 - 24022/1771*e^4 + 58998/1771*e^3 + 108888/1771*e^2 - 154948/1771*e - 34537/1771, -821/1771*e^6 + 3737/3542*e^5 + 25317/1771*e^4 - 114909/3542*e^3 - 119969/1771*e^2 + 153191/1771*e + 63237/1771, 1/7084*e^6 - 360/1771*e^5 + 1066/1771*e^4 + 9837/1771*e^3 - 129523/7084*e^2 - 24111/1771*e + 133933/3542, -191/1012*e^6 + 141/506*e^5 + 1577/253*e^4 - 4479/506*e^3 - 39927/1012*e^2 + 4902/253*e + 9747/506, 1321/7084*e^6 - 932/1771*e^5 - 10385/1771*e^4 + 27415/1771*e^3 + 212489/7084*e^2 - 57639/1771*e - 139087/3542, 101/506*e^6 - 109/253*e^5 - 1489/253*e^4 + 3301/253*e^3 + 12445/506*e^2 - 5738/253*e - 4978/253, 41/1771*e^6 + 577/3542*e^5 - 505/1771*e^4 - 16165/3542*e^3 - 8069/1771*e^2 + 53569/1771*e + 25329/1771, 199/3542*e^6 - 1429/3542*e^5 - 2543/1771*e^4 + 43165/3542*e^3 - 7027/3542*e^2 - 87679/1771*e - 18593/1771, -1765/7084*e^6 + 993/3542*e^5 + 32273/3542*e^4 - 28949/3542*e^3 - 534511/7084*e^2 + 18266/1771*e + 266985/3542, -691/1771*e^6 + 1509/1771*e^5 + 45315/3542*e^4 - 45351/1771*e^3 - 272231/3542*e^2 + 113418/1771*e + 103377/1771, 405/1012*e^6 - 397/506*e^5 - 3177/253*e^4 + 12385/506*e^3 + 65985/1012*e^2 - 13829/253*e - 9443/506, -89/322*e^6 + 165/322*e^5 + 1359/161*e^4 - 5535/322*e^3 - 13255/322*e^2 + 10446/161*e + 725/161, 137/1771*e^6 + 373/3542*e^5 - 9379/3542*e^4 - 12893/3542*e^3 + 74149/3542*e^2 + 55504/1771*e - 70089/1771, -1140/1771*e^6 + 5079/3542*e^5 + 35855/1771*e^4 - 152199/3542*e^3 - 183318/1771*e^2 + 167183/1771*e + 100153/1771, 85/7084*e^6 - 493/1771*e^5 + 289/1771*e^4 + 12630/1771*e^3 - 107179/7084*e^2 - 19869/1771*e + 177417/3542, -2075/3542*e^6 + 3865/3542*e^5 + 65643/3542*e^4 - 119115/3542*e^3 - 181353/1771*e^2 + 133746/1771*e + 151964/1771, 1033/7084*e^6 - 1711/3542*e^5 - 7468/1771*e^4 + 54147/3542*e^3 + 104509/7084*e^2 - 98495/1771*e - 29609/3542, 745/3542*e^6 - 1345/3542*e^5 - 12644/1771*e^4 + 41401/3542*e^3 + 159461/3542*e^2 - 53785/1771*e - 19307/1771, 261/1771*e^6 - 388/1771*e^5 - 8139/1771*e^4 + 13967/1771*e^3 + 40078/1771*e^2 - 53791/1771*e + 22282/1771, 2389/7084*e^6 - 1105/1771*e^5 - 19505/1771*e^4 + 34843/1771*e^3 + 459133/7084*e^2 - 98580/1771*e - 121061/3542, 775/3542*e^6 - 2041/3542*e^5 - 12440/1771*e^4 + 60065/3542*e^3 + 131009/3542*e^2 - 65935/1771*e - 40968/1771, -227/506*e^6 + 255/253*e^5 + 3567/253*e^4 - 7885/253*e^3 - 36853/506*e^2 + 23119/253*e + 5279/253, 5/44*e^6 - 3/22*e^5 - 87/22*e^4 + 107/22*e^3 + 1319/44*e^2 - 193/11*e - 389/22]; heckeEigenvalues := AssociativeArray(); for i := 1 to #heckeEigenvaluesArray do heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i]; end for; ALEigenvalues := AssociativeArray(); ALEigenvalues[ideal] := 1; 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;