/* 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![41, 13, -12, -2, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [9, 3, -w^3 + 3*w^2 + 5*w - 15], [9, 3, -w^3 + 8*w + 8], [16, 2, 2], [19, 19, w + 1], [19, 19, -w^2 + 6], [19, 19, -w^2 + 2*w + 5], [19, 19, -w + 2], [25, 5, 2*w^2 - 2*w - 13], [29, 29, -w^2 + 9], [29, 29, -w^2 + 2*w + 6], [29, 29, w^2 - 7], [29, 29, -w^2 + 2*w + 8], [31, 31, -2*w^2 + w + 12], [31, 31, 2*w^2 - 3*w - 11], [41, 41, -w], [41, 41, -w + 1], [49, 7, w^3 + 2*w^2 - 10*w - 20], [49, 7, w^3 - 5*w^2 - 3*w + 27], [61, 61, 2*w^2 - 3*w - 14], [61, 61, 2*w^2 - w - 15], [71, 71, -w^3 + w^2 + 7*w - 5], [71, 71, w^3 - 2*w^2 - 6*w + 2], [79, 79, w^3 - 2*w^2 - 6*w + 4], [79, 79, 3*w^2 - 2*w - 18], [89, 89, 2*w^3 - 6*w^2 - 11*w + 33], [89, 89, -4*w^2 + 3*w + 25], [101, 101, w^3 + w^2 - 9*w - 12], [101, 101, w^3 - 4*w^2 - 4*w + 19], [109, 109, -w^3 + 3*w^2 + 6*w - 19], [109, 109, -2*w^2 + w + 18], [121, 11, -3*w^2 + 3*w + 19], [121, 11, 3*w^2 - 3*w - 20], [131, 131, 2*w^2 - 3*w - 15], [131, 131, 2*w^2 - w - 16], [139, 139, 3*w^2 - 4*w - 21], [139, 139, -w^3 - w^2 + 7*w + 13], [151, 151, 4*w^2 - 3*w - 27], [151, 151, 2*w^3 - w^2 - 13*w - 8], [151, 151, w^3 + 3*w^2 - 13*w - 27], [151, 151, 4*w^2 - 5*w - 26], [179, 179, w^3 - 3*w^2 - 5*w + 11], [179, 179, 3*w^3 - 11*w^2 - 14*w + 61], [179, 179, 2*w^3 - 2*w^2 - 12*w + 3], [179, 179, w^3 - 8*w - 4], [181, 181, -w - 4], [181, 181, w^3 - 5*w^2 - 3*w + 25], [181, 181, w^3 + 2*w^2 - 10*w - 18], [181, 181, w - 5], [229, 229, w^2 - 2*w - 11], [229, 229, w^2 - 12], [239, 239, 6*w^2 - 7*w - 42], [239, 239, 2*w^3 - 2*w^2 - 14*w - 3], [241, 241, w^3 - 4*w^2 - 4*w + 18], [241, 241, 3*w^2 - 2*w - 16], [269, 269, 3*w^2 - 4*w - 22], [269, 269, -w^3 + 6*w + 8], [281, 281, w^3 - w^2 - 9*w + 4], [281, 281, 2*w^3 - 9*w^2 - 8*w + 49], [311, 311, -2*w^3 - w^2 + 17*w + 19], [311, 311, 2*w^3 - 7*w^2 - 9*w + 33], [331, 331, w^3 + 2*w^2 - 9*w - 21], [331, 331, w^3 - 5*w^2 - 2*w + 27], [349, 349, 2*w^3 - w^2 - 15*w - 4], [349, 349, -2*w^3 + 5*w^2 + 11*w - 18], [359, 359, -w^3 + 6*w^2 - 31], [359, 359, -w^3 - 3*w^2 + 9*w + 26], [379, 379, 2*w^3 - 5*w^2 - 11*w + 20], [379, 379, 2*w^3 - 14*w - 11], [401, 401, w^3 - 3*w^2 - 4*w + 16], [401, 401, -2*w^3 + 4*w^2 + 11*w - 16], [401, 401, 2*w^3 - 2*w^2 - 13*w - 3], [401, 401, w^3 - 7*w - 10], [409, 409, w^3 - 4*w^2 - 4*w + 17], [409, 409, 2*w^3 - 6*w^2 - 9*w + 22], [409, 409, -2*w^3 + 15*w + 9], [409, 409, -w^3 + 3*w^2 + 6*w - 12], [419, 419, -w^3 + 5*w^2 + 3*w - 24], [419, 419, w^3 + 2*w^2 - 10*w - 17], [421, 421, w^3 + 3*w^2 - 12*w - 25], [421, 421, -w^3 - w^2 + 9*w + 7], [431, 431, -w^3 - 2*w^2 + 12*w + 20], [431, 431, w^3 + 3*w^2 - 10*w - 28], [431, 431, -w^3 + 6*w^2 + w - 34], [431, 431, -w^3 + 5*w^2 + 5*w - 29], [439, 439, w^3 - 7*w^2 - 2*w + 39], [439, 439, w^3 + 4*w^2 - 13*w - 31], [449, 449, -w^3 - 4*w^2 + 13*w + 34], [449, 449, 6*w^2 - 5*w - 36], [461, 461, -w^3 + 5*w^2 + w - 25], [461, 461, 2*w^3 - 15*w - 12], [461, 461, -2*w^3 + 6*w^2 + 9*w - 25], [461, 461, w^3 + 2*w^2 - 8*w - 20], [479, 479, -w^3 - 4*w^2 + 12*w + 31], [479, 479, 2*w^3 - 15*w - 16], [479, 479, -2*w^3 + 6*w^2 + 9*w - 29], [479, 479, w^3 - 7*w^2 - w + 38], [491, 491, w^2 - 3*w - 7], [491, 491, -2*w^3 + 3*w^2 + 11*w - 2], [491, 491, 4*w^2 - 2*w - 29], [491, 491, w^2 + w - 9], [499, 499, 5*w^2 - 6*w - 33], [499, 499, 5*w^2 - 4*w - 34], [509, 509, 2*w - 3], [509, 509, -2*w - 1], [541, 541, -w^3 - 4*w^2 + 12*w + 29], [541, 541, -w^3 + 7*w^2 + w - 36], [599, 599, -w^3 - w^2 + 8*w + 7], [599, 599, w^3 - 4*w^2 - 3*w + 13], [619, 619, w^2 - 3*w - 6], [619, 619, w^2 + w - 8], [641, 641, -2*w^3 + 16*w + 11], [641, 641, -2*w^3 + 6*w^2 + 10*w - 25], [659, 659, 2*w^3 + w^2 - 16*w - 16], [659, 659, 2*w^3 - 7*w^2 - 8*w + 29], [691, 691, -w^3 + 5*w^2 + 3*w - 23], [691, 691, -w^3 + 3*w^2 + 6*w - 11], [701, 701, w^3 - w^2 - 9*w + 3], [701, 701, 2*w^3 - w^2 - 14*w - 5], [709, 709, 2*w^3 - 3*w^2 - 11*w + 6], [719, 719, w^2 - 2*w - 12], [719, 719, -w^2 + 13], [739, 739, -2*w^3 + 8*w^2 + 7*w - 36], [739, 739, w^2 + 2*w - 13], [751, 751, -2*w^3 + 5*w^2 + 10*w - 20], [751, 751, 2*w^3 - w^2 - 14*w - 7], [761, 761, w^3 + 4*w^2 - 12*w - 32], [761, 761, -w^3 + 7*w^2 + w - 39], [769, 769, -2*w^3 - 4*w^2 + 23*w + 45], [769, 769, 2*w^3 - 5*w^2 - 9*w + 21], [769, 769, 2*w^3 + 2*w^2 - 18*w - 31], [769, 769, 2*w^3 - 10*w^2 - 9*w + 62], [811, 811, -w^3 + 10*w + 5], [811, 811, -5*w^2 + 8*w + 30], [811, 811, w^3 - 8*w^2 + 2*w + 43], [811, 811, 2*w^3 - 2*w^2 - 13*w + 2], [829, 829, -2*w^3 - 2*w^2 + 18*w + 25], [829, 829, -2*w^3 + 8*w^2 + 8*w - 39], [859, 859, -w^3 - 4*w^2 + 12*w + 33], [859, 859, -w^3 + 7*w^2 + w - 40], [919, 919, w^3 - 4*w^2 - 3*w + 24], [919, 919, -w^3 + 5*w^2 + w - 26], [929, 929, 5*w^2 - 7*w - 28], [929, 929, -w^3 + 2*w^2 + 6*w - 13], [929, 929, w^3 - w^2 - 7*w - 6], [929, 929, -2*w^3 + 2*w^2 + 14*w - 5], [941, 941, -2*w^3 + 3*w^2 + 11*w - 1], [941, 941, 2*w^3 - 3*w^2 - 11*w + 11], [961, 31, 5*w^2 - 5*w - 33], [971, 971, -w^3 - 3*w^2 + 11*w + 22], [971, 971, 7*w^2 - 5*w - 45], [971, 971, 7*w^2 - 9*w - 43], [971, 971, w^3 - 6*w^2 - 2*w + 29], [991, 991, w^3 - 3*w^2 - 3*w + 15], [991, 991, -w^3 + 5*w^2 + 2*w - 29]]; primes := [ideal : I in primesArray]; heckePol := x^6 - 15*x^5 + 79*x^4 - 151*x^3 - 46*x^2 + 453*x - 364; K := NumberField(heckePol); heckeEigenvaluesArray := [9/29*e^5 - 98/29*e^4 + 321/29*e^3 - 136/29*e^2 - 30*e + 858/29, e, -14/29*e^5 + 175/29*e^4 - 683/29*e^3 + 537/29*e^2 + 60*e - 2369/29, -1, -19/29*e^5 + 223/29*e^4 - 813/29*e^3 + 561/29*e^2 + 70*e - 2691/29, 5, -11/29*e^5 + 123/29*e^4 - 402/29*e^3 + 134/29*e^2 + 36*e - 923/29, 1/29*e^5 - 27/29*e^4 + 171/29*e^3 - 260/29*e^2 - 12*e + 685/29, -25/29*e^5 + 298/29*e^4 - 1085/29*e^3 + 671/29*e^2 + 96*e - 3292/29, 24/29*e^5 - 300/29*e^4 + 1175/29*e^3 - 933/29*e^2 - 104*e + 4144/29, -2/29*e^5 + 25/29*e^4 - 81/29*e^3 - 2/29*e^2 + 7*e + 22/29, 20/29*e^5 - 221/29*e^4 + 723/29*e^3 - 299/29*e^2 - 63*e + 1955/29, 9/29*e^5 - 98/29*e^4 + 350/29*e^3 - 310/29*e^2 - 27*e + 1235/29, 11/29*e^5 - 181/29*e^4 + 895/29*e^3 - 1033/29*e^2 - 78*e + 3678/29, -2/29*e^5 + 54/29*e^4 - 342/29*e^3 + 491/29*e^2 + 33*e - 1631/29, -20/29*e^5 + 250/29*e^4 - 955/29*e^3 + 647/29*e^2 + 86*e - 2970/29, 17/29*e^5 - 169/29*e^4 + 500/29*e^3 - 186/29*e^2 - 46*e + 1698/29, -27/29*e^5 + 265/29*e^4 - 673/29*e^3 - 230/29*e^2 + 64*e - 834/29, e^4 - 8*e^3 + 13*e^2 + 19*e - 42, -4/29*e^5 + 50/29*e^4 - 191/29*e^3 + 112/29*e^2 + 18*e - 246/29, 5/29*e^5 - 135/29*e^4 + 855/29*e^3 - 1300/29*e^2 - 70*e + 4034/29, 12/29*e^5 - 150/29*e^4 + 573/29*e^3 - 423/29*e^2 - 48*e + 2014/29, 14/29*e^5 - 204/29*e^4 + 944/29*e^3 - 1088/29*e^2 - 76*e + 3935/29, 42/29*e^5 - 467/29*e^4 + 1527/29*e^3 - 538/29*e^2 - 144*e + 4120/29, -56/29*e^5 + 671/29*e^4 - 2500/29*e^3 + 1800/29*e^2 + 215*e - 8258/29, -21/29*e^5 + 248/29*e^4 - 952/29*e^3 + 878/29*e^2 + 78*e - 3684/29, -10/29*e^5 + 96/29*e^4 - 231/29*e^3 - 126/29*e^2 + 25*e - 470/29, -7/29*e^5 + 131/29*e^4 - 733/29*e^3 + 1037/29*e^2 + 64*e - 3258/29, 26/29*e^5 - 296/29*e^4 + 995/29*e^3 - 467/29*e^2 - 84*e + 2817/29, -28/29*e^5 + 321/29*e^4 - 1105/29*e^3 + 610/29*e^2 + 93*e - 3375/29, 59/29*e^5 - 665/29*e^4 + 2259/29*e^3 - 1159/29*e^2 - 198*e + 6862/29, -14/29*e^5 + 117/29*e^4 - 248/29*e^3 - 72/29*e^2 + 24*e - 774/29, -45/29*e^5 + 548/29*e^4 - 2098/29*e^3 + 1666/29*e^2 + 182*e - 7538/29, 60/29*e^5 - 721/29*e^4 + 2720/29*e^3 - 2057/29*e^2 - 236*e + 9316/29, 22/29*e^5 - 304/29*e^4 + 1384/29*e^3 - 1602/29*e^2 - 119*e + 5790/29, -44/29*e^5 + 550/29*e^4 - 2130/29*e^3 + 1609/29*e^2 + 182*e - 6911/29, 9/29*e^5 - 156/29*e^4 + 843/29*e^3 - 1209/29*e^2 - 63*e + 3584/29, 23/29*e^5 - 244/29*e^4 + 743/29*e^3 - 151/29*e^2 - 67*e + 1487/29, 39/29*e^5 - 531/29*e^4 + 2261/29*e^3 - 2078/29*e^2 - 200*e + 8358/29, -75/29*e^5 + 894/29*e^4 - 3313/29*e^3 + 2332/29*e^2 + 284*e - 10514/29, -4/29*e^5 + 50/29*e^4 - 191/29*e^3 + 170/29*e^2 + 10*e - 333/29, -6/29*e^5 + 133/29*e^4 - 823/29*e^3 + 1299/29*e^2 + 67*e - 3704/29, 42/29*e^5 - 467/29*e^4 + 1527/29*e^3 - 567/29*e^2 - 140*e + 4207/29, 2*e^4 - 19*e^3 + 41*e^2 + 43*e - 117, 89/29*e^5 - 1069/29*e^4 + 3996/29*e^3 - 2927/29*e^2 - 340*e + 13318/29, 7/29*e^5 - 102/29*e^4 + 501/29*e^3 - 689/29*e^2 - 36*e + 2098/29, 20/29*e^5 - 105/29*e^4 - 292/29*e^3 + 1644/29*e^2 + 18*e - 3497/29, -45/29*e^5 + 490/29*e^4 - 1663/29*e^3 + 1028/29*e^2 + 145*e - 5363/29, 69/29*e^5 - 819/29*e^4 + 3012/29*e^3 - 2019/29*e^2 - 268*e + 9768/29, 49/29*e^5 - 598/29*e^4 + 2231/29*e^3 - 1546/29*e^2 - 190*e + 7233/29, 23/29*e^5 - 360/29*e^4 + 1700/29*e^3 - 1862/29*e^2 - 144*e + 7084/29, -46/29*e^5 + 604/29*e^4 - 2588/29*e^3 + 2767/29*e^2 + 210*e - 10108/29, 16/29*e^5 - 84/29*e^4 - 135/29*e^3 + 799/29*e^2 + 8*e - 814/29, -e^5 + 11*e^4 - 38*e^3 + 24*e^2 + 100*e - 126, -7/29*e^5 - 43/29*e^4 + 746/29*e^3 - 1631/29*e^2 - 56*e + 3934/29, -51/29*e^5 + 623/29*e^4 - 2399/29*e^3 + 1892/29*e^2 + 204*e - 7907/29, 4/29*e^5 - 50/29*e^4 + 278/29*e^3 - 634/29*e^2 - 12*e + 1957/29, 20/29*e^5 - 250/29*e^4 + 955/29*e^3 - 589/29*e^2 - 93*e + 3144/29, -49/29*e^5 + 656/29*e^4 - 2898/29*e^3 + 3315/29*e^2 + 232*e - 11757/29, 26/29*e^5 - 238/29*e^4 + 560/29*e^3 + 229/29*e^2 - 54*e + 700/29, -41/29*e^5 + 469/29*e^4 - 1588/29*e^3 + 771/29*e^2 + 132*e - 4450/29, -43/29*e^5 + 436/29*e^4 - 1234/29*e^3 + 102/29*e^2 + 105*e - 2282/29, -56/29*e^5 + 729/29*e^4 - 2993/29*e^3 + 2670/29*e^2 + 259*e - 11129/29, -27/29*e^5 + 410/29*e^4 - 1978/29*e^3 + 2438/29*e^2 + 157*e - 8142/29, 111/29*e^5 - 1286/29*e^4 + 4539/29*e^3 - 2528/29*e^2 - 416*e + 13975/29, -22/29*e^5 + 362/29*e^4 - 1819/29*e^3 + 2182/29*e^2 + 162*e - 7588/29, 24/29*e^5 - 213/29*e^4 + 392/29*e^3 + 633/29*e^2 - 50*e - 32/29, 35/29*e^5 - 394/29*e^4 + 1403/29*e^3 - 1096/29*e^2 - 110*e + 5299/29, 123/29*e^5 - 1436/29*e^4 + 5141/29*e^3 - 3241/29*e^2 - 442*e + 15931/29, 8/29*e^5 - 303/29*e^4 + 2180/29*e^3 - 3791/29*e^2 - 174*e + 11222/29, -5/29*e^5 - 10/29*e^4 + 421/29*e^3 - 1165/29*e^2 - 32*e + 3216/29, -129/29*e^5 + 1598/29*e^4 - 6254/29*e^3 + 5207/29*e^2 + 535*e - 22042/29, -81/29*e^5 + 969/29*e^4 - 3585/29*e^3 + 2413/29*e^2 + 313*e - 11173/29, -39/29*e^5 + 444/29*e^4 - 1594/29*e^3 + 1150/29*e^2 + 140*e - 5661/29, 111/29*e^5 - 1315/29*e^4 + 4800/29*e^3 - 3224/29*e^2 - 412*e + 15193/29, -5/29*e^5 + 77/29*e^4 - 362/29*e^3 + 343/29*e^2 + 31*e - 1250/29, -17/29*e^5 + 53/29*e^4 + 573/29*e^3 - 2105/29*e^2 - 33*e + 5117/29, 92/29*e^5 - 1005/29*e^4 + 3204/29*e^3 - 923/29*e^2 - 301*e + 8065/29, -46/29*e^5 + 691/29*e^4 - 3284/29*e^3 + 3869/29*e^2 + 274*e - 13762/29, -15/29*e^5 + 202/29*e^4 - 883/29*e^3 + 971/29*e^2 + 73*e - 3054/29, 2/29*e^5 + 4/29*e^4 - 209/29*e^3 + 727/29*e^2 + 8*e - 1907/29, -65/29*e^5 + 682/29*e^4 - 2038/29*e^3 + 399/29*e^2 + 174*e - 4418/29, -10/29*e^5 - 20/29*e^4 + 755/29*e^3 - 1953/29*e^2 - 50*e + 4518/29, 45/29*e^5 - 374/29*e^4 + 590/29*e^3 + 1147/29*e^2 - 51*e - 640/29, 37/29*e^5 - 535/29*e^4 + 2441/29*e^3 - 2689/29*e^2 - 209*e + 10294/29, -77/29*e^5 + 977/29*e^4 - 3887/29*e^3 + 3345/29*e^2 + 321*e - 13885/29, 30/29*e^5 - 491/29*e^4 + 2346/29*e^3 - 2348/29*e^2 - 213*e + 8747/29, 17/29*e^5 - 169/29*e^4 + 384/29*e^3 + 452/29*e^2 - 35*e - 854/29, 65/29*e^5 - 711/29*e^4 + 2299/29*e^3 - 805/29*e^2 - 218*e + 5897/29, 45/29*e^5 - 548/29*e^4 + 2040/29*e^3 - 1289/29*e^2 - 180*e + 6030/29, 2/29*e^5 + 33/29*e^4 - 441/29*e^3 + 1046/29*e^2 + 38*e - 3038/29, -54/29*e^5 + 762/29*e^4 - 3347/29*e^3 + 3368/29*e^2 + 280*e - 13094/29, -90/29*e^5 + 980/29*e^4 - 3297/29*e^3 + 1940/29*e^2 + 289*e - 10958/29, -40/29*e^5 + 355/29*e^4 - 634/29*e^3 - 1171/29*e^2 + 74*e + 962/29, -122/29*e^5 + 1380/29*e^4 - 4709/29*e^3 + 2488/29*e^2 + 408*e - 14144/29, 89/29*e^5 - 1127/29*e^4 + 4518/29*e^3 - 3913/29*e^2 - 393*e + 16856/29, -14/29*e^5 + 146/29*e^4 - 509/29*e^3 + 363/29*e^2 + 52*e - 1528/29, -7/29*e^5 + 15/29*e^4 + 282/29*e^3 - 906/29*e^2 - 14*e + 1759/29, 44/29*e^5 - 550/29*e^4 + 2130/29*e^3 - 1522/29*e^2 - 194*e + 6795/29, -14/29*e^5 + 146/29*e^4 - 422/29*e^3 + 189/29*e^2 + 17*e - 1325/29, -1/29*e^5 + 114/29*e^4 - 954/29*e^3 + 1855/29*e^2 + 76*e - 5354/29, 4/29*e^5 - 137/29*e^4 + 974/29*e^3 - 1649/29*e^2 - 79*e + 4973/29, -59/29*e^5 + 810/29*e^4 - 3506/29*e^3 + 3421/29*e^2 + 298*e - 12894/29, -26/29*e^5 + 267/29*e^4 - 734/29*e^3 - 84/29*e^2 + 73*e - 1396/29, -164/29*e^5 + 1992/29*e^4 - 7657/29*e^3 + 6216/29*e^2 + 654*e - 26674/29, 13/29*e^5 - 206/29*e^4 + 1063/29*e^3 - 1466/29*e^2 - 87*e + 4990/29, -70/29*e^5 + 730/29*e^4 - 2081/29*e^3 - 99/29*e^2 + 206*e - 4276/29, 56/29*e^5 - 613/29*e^4 + 1891/29*e^3 - 263/29*e^2 - 173*e + 3589/29, 31/29*e^5 - 402/29*e^4 + 1734/29*e^3 - 1970/29*e^2 - 141*e + 6880/29, 59/29*e^5 - 607/29*e^4 + 1824/29*e^3 - 405/29*e^2 - 183*e + 4919/29, -84/29*e^5 + 760/29*e^4 - 1604/29*e^3 - 1389/29*e^2 + 152*e - 700/29, -9/29*e^5 + 127/29*e^4 - 553/29*e^3 + 658/29*e^2 + 33*e - 2482/29, -1/29*e^5 - 31/29*e^4 + 380/29*e^3 - 813/29*e^2 - 39*e + 1664/29, -57/29*e^5 + 814/29*e^4 - 3686/29*e^3 + 4061/29*e^2 + 306*e - 14540/29, 69/29*e^5 - 761/29*e^4 + 2519/29*e^3 - 1149/29*e^2 - 229*e + 8231/29, 82/29*e^5 - 967/29*e^4 + 3640/29*e^3 - 2905/29*e^2 - 310*e + 11974/29, -46/29*e^5 + 633/29*e^4 - 2617/29*e^3 + 1984/29*e^2 + 247*e - 9006/29, 142/29*e^5 - 1775/29*e^4 + 6969/29*e^3 - 5629/29*e^2 - 608*e + 24393/29, 65/29*e^5 - 769/29*e^4 + 2821/29*e^3 - 1994/29*e^2 - 244*e + 10189/29, 82/29*e^5 - 1170/29*e^4 + 5351/29*e^3 - 6095/29*e^2 - 438*e + 21312/29, -32/29*e^5 + 342/29*e^4 - 1035/29*e^3 + 287/29*e^2 + 86*e - 3360/29, 38/29*e^5 - 388/29*e^4 + 1104/29*e^3 - 223/29*e^2 - 88*e + 3468/29, -187/29*e^5 + 2323/29*e^4 - 9096/29*e^3 + 7440/29*e^2 + 792*e - 32366/29, 42/29*e^5 - 612/29*e^4 + 2861/29*e^3 - 3293/29*e^2 - 247*e + 12095/29, -38/29*e^5 + 475/29*e^4 - 1945/29*e^3 + 2050/29*e^2 + 152*e - 7673/29, -50/29*e^5 + 770/29*e^4 - 3707/29*e^3 + 4242/29*e^2 + 329*e - 15371/29, -54/29*e^5 + 762/29*e^4 - 3347/29*e^3 + 3194/29*e^2 + 300*e - 12282/29, 13/29*e^5 - 61/29*e^4 - 329/29*e^3 + 1463/29*e^2 + 32*e - 3449/29, 104/29*e^5 - 1300/29*e^4 + 5111/29*e^3 - 4159/29*e^2 - 453*e + 18837/29, -155/29*e^5 + 1865/29*e^4 - 6988/29*e^3 + 5152/29*e^2 + 604*e - 23844/29, 180/29*e^5 - 2250/29*e^4 + 8798/29*e^3 - 7070/29*e^2 - 764*e + 30732/29, -68/29*e^5 + 879/29*e^4 - 3537/29*e^3 + 2977/29*e^2 + 309*e - 12650/29, 84/29*e^5 - 963/29*e^4 + 3460/29*e^3 - 2352/29*e^2 - 314*e + 12068/29, 10/29*e^5 - 415/29*e^4 + 3015/29*e^3 - 5181/29*e^2 - 251*e + 16072/29, 44/29*e^5 - 637/29*e^4 + 2942/29*e^3 - 3291/29*e^2 - 256*e + 12218/29, 6/29*e^5 - 220/29*e^4 + 1606/29*e^3 - 2865/29*e^2 - 117*e + 7474/29, 18/29*e^5 - 283/29*e^4 + 1338/29*e^3 - 1316/29*e^2 - 130*e + 5370/29, 140/29*e^5 - 1692/29*e^4 + 6279/29*e^3 - 4297/29*e^2 - 536*e + 19746/29, -72/29*e^5 + 784/29*e^4 - 2597/29*e^3 + 1262/29*e^2 + 241*e - 7850/29, -117/29*e^5 + 1245/29*e^4 - 3825/29*e^3 + 840/29*e^2 + 362*e - 8776/29, -20/29*e^5 + 250/29*e^4 - 1129/29*e^3 + 1546/29*e^2 + 99*e - 6102/29, e^5 - 14*e^4 + 62*e^3 - 66*e^2 - 156*e + 266, 131/29*e^5 - 1797/29*e^4 + 7785/29*e^3 - 7757/29*e^2 - 651*e + 29154/29, 49/29*e^5 - 598/29*e^4 + 2376/29*e^3 - 2300/29*e^2 - 195*e + 9176/29, -45/29*e^5 + 519/29*e^4 - 1924/29*e^3 + 1695/29*e^2 + 155*e - 7538/29, 10/29*e^5 + 20/29*e^4 - 813/29*e^3 + 2156/29*e^2 + 68*e - 5736/29, 17/29*e^5 - 24/29*e^4 - 834/29*e^3 + 2540/29*e^2 + 64*e - 6074/29, -66/29*e^5 + 767/29*e^4 - 2789/29*e^3 + 1935/29*e^2 + 245*e - 9163/29, -19/29*e^5 + 223/29*e^4 - 668/29*e^3 - 135/29*e^2 + 57*e + 354/29, 21/29*e^5 - 74/29*e^4 - 672/29*e^3 + 2544/29*e^2 + 51*e - 5944/29, -14/29*e^5 + 175/29*e^4 - 654/29*e^3 + 537/29*e^2 + 33*e - 1818/29, 43/29*e^5 - 581/29*e^4 + 2510/29*e^3 - 2567/29*e^2 - 222*e + 10895/29, 122/29*e^5 - 1177/29*e^4 + 2998/29*e^3 + 557/29*e^2 - 268*e + 6024/29, 11/29*e^5 - 239/29*e^4 + 1243/29*e^3 - 1149/29*e^2 - 113*e + 3359/29]; 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;