/* 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![-55, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, w + 1], [3, 3, w + 1], [3, 3, w + 2], [5, 5, -2*w + 15], [11, 11, 3*w - 22], [13, 13, w + 4], [13, 13, w + 9], [17, 17, w + 2], [17, 17, w + 15], [19, 19, -w - 6], [19, 19, w - 6], [23, 23, w + 3], [23, 23, w + 20], [47, 47, w + 14], [47, 47, w + 33], [49, 7, -7], [67, 67, w + 16], [67, 67, w + 51], [73, 73, w + 36], [73, 73, w + 37], [79, 79, 5*w + 36], [79, 79, 5*w - 36], [89, 89, -w - 12], [89, 89, w - 12], [103, 103, w + 40], [103, 103, w + 63], [131, 131, -6*w - 43], [131, 131, -6*w + 43], [139, 139, 2*w - 9], [139, 139, -2*w - 9], [151, 151, 4*w - 27], [151, 151, 4*w + 27], [163, 163, w + 50], [163, 163, w + 113], [173, 173, w + 48], [173, 173, w + 125], [181, 181, -3*w - 26], [181, 181, 3*w - 26], [193, 193, w + 21], [193, 193, w + 172], [197, 197, w + 45], [197, 197, w + 152], [211, 211, 2*w - 3], [211, 211, -2*w - 3], [223, 223, w + 72], [223, 223, w + 151], [229, 229, 6*w + 47], [229, 229, 6*w - 47], [233, 233, w + 88], [233, 233, w + 145], [239, 239, -3*w - 16], [239, 239, 3*w - 16], [269, 269, -w - 18], [269, 269, w - 18], [271, 271, -8*w - 57], [271, 271, -8*w + 57], [277, 277, w + 71], [277, 277, w + 206], [293, 293, w + 73], [293, 293, w + 220], [337, 337, w + 27], [337, 337, w + 310], [359, 359, 9*w + 64], [359, 359, 9*w - 64], [367, 367, w + 34], [367, 367, w + 333], [373, 373, w + 146], [373, 373, w + 227], [383, 383, w + 173], [383, 383, w + 210], [389, 389, -5*w - 42], [389, 389, 5*w - 42], [401, 401, 29*w - 216], [401, 401, 11*w - 84], [421, 421, -6*w + 49], [421, 421, -6*w - 49], [431, 431, 3*w - 8], [431, 431, -3*w - 8], [439, 439, -4*w - 21], [439, 439, 4*w - 21], [443, 443, w + 99], [443, 443, w + 344], [449, 449, -8*w - 63], [449, 449, -8*w + 63], [457, 457, w + 91], [457, 457, w + 366], [463, 463, w + 38], [463, 463, w + 425], [467, 467, w + 191], [467, 467, w + 276], [479, 479, -3*w - 4], [479, 479, 3*w - 4], [487, 487, w + 223], [487, 487, w + 264], [491, 491, 3*w - 2], [491, 491, -3*w - 2], [509, 509, 2*w - 27], [509, 509, -2*w - 27], [521, 521, -w - 24], [521, 521, w - 24], [557, 557, w + 75], [557, 557, w + 482], [571, 571, 11*w + 78], [571, 571, 11*w - 78], [587, 587, w + 246], [587, 587, w + 341], [593, 593, w + 184], [593, 593, w + 409], [613, 613, w + 235], [613, 613, w + 378], [641, 641, -4*w - 39], [641, 641, 4*w - 39], [643, 643, w + 174], [643, 643, w + 469], [647, 647, w + 257], [647, 647, w + 390], [659, 659, -18*w + 131], [659, 659, 30*w - 221], [661, 661, 3*w - 34], [661, 661, -3*w - 34], [673, 673, w + 196], [673, 673, w + 477], [677, 677, w + 240], [677, 677, w + 437], [683, 683, w + 136], [683, 683, w + 547], [709, 709, 21*w - 158], [709, 709, 27*w - 202], [727, 727, w + 339], [727, 727, w + 388], [733, 733, w + 39], [733, 733, w + 694], [739, 739, 10*w - 69], [739, 739, 10*w + 69], [811, 811, 35*w - 258], [811, 811, -19*w + 138], [823, 823, w + 177], [823, 823, w + 646], [829, 829, -6*w - 53], [829, 829, 6*w - 53], [841, 29, -29], [853, 853, w + 316], [853, 853, w + 537], [857, 857, w + 270], [857, 857, w + 587], [863, 863, w + 405], [863, 863, w + 458], [877, 877, w + 377], [877, 877, w + 500], [881, 881, -19*w + 144], [881, 881, -37*w + 276], [883, 883, w + 52], [883, 883, w + 831], [907, 907, w + 350], [907, 907, w + 557], [919, 919, -8*w - 51], [919, 919, 8*w - 51], [929, 929, 5*w - 48], [929, 929, -5*w - 48], [937, 937, w + 176], [937, 937, w + 761], [947, 947, w + 252], [947, 947, w + 695], [953, 953, w + 334], [953, 953, w + 619], [961, 31, -31], [983, 983, w + 277], [983, 983, w + 706], [997, 997, w + 371], [997, 997, w + 626]]; primes := [ideal : I in primesArray]; heckePol := x^8 + 26*x^6 + 157*x^4 + 264*x^2 + 36; K := NumberField(heckePol); heckeEigenvaluesArray := [1/396*e^7 + 19/198*e^5 + 415/396*e^3 + 181/66*e, -1/396*e^7 - 19/198*e^5 - 415/396*e^3 - 115/66*e, -1/396*e^7 - 19/198*e^5 - 415/396*e^3 - 115/66*e, -1, -2/33*e^6 - 43/33*e^4 - 137/33*e^2 - 20/11, -1/198*e^7 - 19/99*e^5 - 415/198*e^3 - 181/33*e, -1/198*e^7 - 19/99*e^5 - 415/198*e^3 - 181/33*e, -17/396*e^7 - 112/99*e^5 - 2897/396*e^3 - 1031/66*e, -17/396*e^7 - 112/99*e^5 - 2897/396*e^3 - 1031/66*e, 3/22*e^6 + 35/11*e^4 + 299/22*e^2 + 100/11, 3/22*e^6 + 35/11*e^4 + 299/22*e^2 + 100/11, 23/198*e^7 + 577/198*e^5 + 1555/99*e^3 + 632/33*e, 23/198*e^7 + 577/198*e^5 + 1555/99*e^3 + 632/33*e, 23/198*e^7 + 577/198*e^5 + 1555/99*e^3 + 632/33*e, 23/198*e^7 + 577/198*e^5 + 1555/99*e^3 + 632/33*e, 2/33*e^6 + 43/33*e^4 + 104/33*e^2 - 145/11, -4/33*e^7 - 205/66*e^5 - 1175/66*e^3 - 249/11*e, -4/33*e^7 - 205/66*e^5 - 1175/66*e^3 - 249/11*e, 1/198*e^7 + 19/99*e^5 + 415/198*e^3 + 181/33*e, 1/198*e^7 + 19/99*e^5 + 415/198*e^3 + 181/33*e, -4/33*e^6 - 86/33*e^4 - 274/33*e^2 - 40/11, -4/33*e^6 - 86/33*e^4 - 274/33*e^2 - 40/11, -2/33*e^6 - 43/33*e^4 - 104/33*e^2 + 13/11, -2/33*e^6 - 43/33*e^4 - 104/33*e^2 + 13/11, 23/198*e^7 + 577/198*e^5 + 1555/99*e^3 + 632/33*e, 23/198*e^7 + 577/198*e^5 + 1555/99*e^3 + 632/33*e, -5/22*e^6 - 62/11*e^4 - 623/22*e^2 - 240/11, -5/22*e^6 - 62/11*e^4 - 623/22*e^2 - 240/11, -3/11*e^6 - 70/11*e^4 - 299/11*e^2 - 200/11, -3/11*e^6 - 70/11*e^4 - 299/11*e^2 - 200/11, -2/33*e^6 - 43/33*e^4 - 137/33*e^2 - 20/11, -2/33*e^6 - 43/33*e^4 - 137/33*e^2 - 20/11, -59/132*e^7 - 725/66*e^5 - 7325/132*e^3 - 1417/22*e, -59/132*e^7 - 725/66*e^5 - 7325/132*e^3 - 1417/22*e, -2/33*e^7 - 43/33*e^5 - 137/33*e^3 - 42/11*e, -2/33*e^7 - 43/33*e^5 - 137/33*e^3 - 42/11*e, -6, -6, 25/396*e^7 + 188/99*e^5 + 6217/396*e^3 + 2479/66*e, 25/396*e^7 + 188/99*e^5 + 6217/396*e^3 + 2479/66*e, -1/6*e^7 - 13/3*e^5 - 163/6*e^3 - 57*e, -1/6*e^7 - 13/3*e^5 - 163/6*e^3 - 57*e, 5/66*e^6 + 62/33*e^4 + 623/66*e^2 + 80/11, 5/66*e^6 + 62/33*e^4 + 623/66*e^2 + 80/11, 34/99*e^7 + 1693/198*e^5 + 8915/198*e^3 + 1781/33*e, 34/99*e^7 + 1693/198*e^5 + 8915/198*e^3 + 1781/33*e, 8/33*e^6 + 172/33*e^4 + 416/33*e^2 - 162/11, 8/33*e^6 + 172/33*e^4 + 416/33*e^2 - 162/11, -7/132*e^7 - 50/33*e^5 - 1519/132*e^3 - 585/22*e, -7/132*e^7 - 50/33*e^5 - 1519/132*e^3 - 585/22*e, 14/33*e^6 + 334/33*e^4 + 1520/33*e^2 + 360/11, 14/33*e^6 + 334/33*e^4 + 1520/33*e^2 + 360/11, -8/33*e^6 - 172/33*e^4 - 416/33*e^2 + 206/11, -8/33*e^6 - 172/33*e^4 - 416/33*e^2 + 206/11, -8/33*e^6 - 205/33*e^4 - 1109/33*e^2 - 300/11, -8/33*e^6 - 205/33*e^4 - 1109/33*e^2 - 300/11, 31/198*e^7 + 391/99*e^5 + 4549/198*e^3 + 1519/33*e, 31/198*e^7 + 391/99*e^5 + 4549/198*e^3 + 1519/33*e, 37/198*e^7 + 505/99*e^5 + 7039/198*e^3 + 2605/33*e, 37/198*e^7 + 505/99*e^5 + 7039/198*e^3 + 2605/33*e, 17/132*e^7 + 112/33*e^5 + 2897/132*e^3 + 1031/22*e, 17/132*e^7 + 112/33*e^5 + 2897/132*e^3 + 1031/22*e, 3/11*e^6 + 70/11*e^4 + 299/11*e^2 + 200/11, 3/11*e^6 + 70/11*e^4 + 299/11*e^2 + 200/11, -1/9*e^7 - 49/18*e^5 - 245/18*e^3 - 47/3*e, -1/9*e^7 - 49/18*e^5 - 245/18*e^3 - 47/3*e, -31/198*e^7 - 391/99*e^5 - 4549/198*e^3 - 1519/33*e, -31/198*e^7 - 391/99*e^5 - 4549/198*e^3 - 1519/33*e, 7/66*e^7 + 167/66*e^5 + 380/33*e^3 + 134/11*e, 7/66*e^7 + 167/66*e^5 + 380/33*e^3 + 134/11*e, -4/33*e^6 - 86/33*e^4 - 208/33*e^2 - 128/11, -4/33*e^6 - 86/33*e^4 - 208/33*e^2 - 128/11, -2/33*e^6 - 43/33*e^4 - 104/33*e^2 + 189/11, -2/33*e^6 - 43/33*e^4 - 104/33*e^2 + 189/11, -4/33*e^6 - 86/33*e^4 - 208/33*e^2 - 172/11, -4/33*e^6 - 86/33*e^4 - 208/33*e^2 - 172/11, 4/11*e^6 + 97/11*e^4 + 461/11*e^2 + 340/11, 4/11*e^6 + 97/11*e^4 + 461/11*e^2 + 340/11, -8/33*e^6 - 172/33*e^4 - 548/33*e^2 - 80/11, -8/33*e^6 - 172/33*e^4 - 548/33*e^2 - 80/11, 1/3*e^7 + 49/6*e^5 + 245/6*e^3 + 47*e, 1/3*e^7 + 49/6*e^5 + 245/6*e^3 + 47*e, -8/33*e^6 - 172/33*e^4 - 416/33*e^2 + 294/11, -8/33*e^6 - 172/33*e^4 - 416/33*e^2 + 294/11, -13/132*e^7 - 74/33*e^5 - 1237/132*e^3 - 307/22*e, -13/132*e^7 - 74/33*e^5 - 1237/132*e^3 - 307/22*e, 37/99*e^7 + 1921/198*e^5 + 11405/198*e^3 + 2471/33*e, 37/99*e^7 + 1921/198*e^5 + 11405/198*e^3 + 2471/33*e, 277/396*e^7 + 3481/198*e^5 + 37735/396*e^3 + 7699/66*e, 277/396*e^7 + 3481/198*e^5 + 37735/396*e^3 + 7699/66*e, -7/33*e^6 - 200/33*e^4 - 1321/33*e^2 - 400/11, -7/33*e^6 - 200/33*e^4 - 1321/33*e^2 - 400/11, -53/66*e^7 - 1321/66*e^5 - 3490/33*e^3 - 1398/11*e, -53/66*e^7 - 1321/66*e^5 - 3490/33*e^3 - 1398/11*e, -49/66*e^6 - 568/33*e^4 - 4759/66*e^2 - 520/11, -49/66*e^6 - 568/33*e^4 - 4759/66*e^2 - 520/11, 4/33*e^6 + 86/33*e^4 + 208/33*e^2 - 4/11, 4/33*e^6 + 86/33*e^4 + 208/33*e^2 - 4/11, -8/33*e^6 - 172/33*e^4 - 416/33*e^2 + 294/11, -8/33*e^6 - 172/33*e^4 - 416/33*e^2 + 294/11, 4/33*e^7 + 119/33*e^5 + 967/33*e^3 + 766/11*e, 4/33*e^7 + 119/33*e^5 + 967/33*e^3 + 766/11*e, -1/66*e^6 + 14/33*e^4 + 773/66*e^2 + 160/11, -1/66*e^6 + 14/33*e^4 + 773/66*e^2 + 160/11, 19/44*e^7 + 229/22*e^5 + 2165/44*e^3 + 1187/22*e, 19/44*e^7 + 229/22*e^5 + 2165/44*e^3 + 1187/22*e, 35/198*e^7 + 467/99*e^5 + 6209/198*e^3 + 2243/33*e, 35/198*e^7 + 467/99*e^5 + 6209/198*e^3 + 2243/33*e, -35/198*e^7 - 467/99*e^5 - 6209/198*e^3 - 2243/33*e, -35/198*e^7 - 467/99*e^5 - 6209/198*e^3 - 2243/33*e, 2/11*e^6 + 43/11*e^4 + 104/11*e^2 - 303/11, 2/11*e^6 + 43/11*e^4 + 104/11*e^2 - 303/11, -59/132*e^7 - 725/66*e^5 - 7325/132*e^3 - 1417/22*e, -59/132*e^7 - 725/66*e^5 - 7325/132*e^3 - 1417/22*e, -59/99*e^7 - 2999/198*e^5 - 16795/198*e^3 - 3505/33*e, -59/99*e^7 - 2999/198*e^5 - 16795/198*e^3 - 3505/33*e, -7/66*e^6 - 67/33*e^4 - 199/66*e^2 + 20/11, -7/66*e^6 - 67/33*e^4 - 199/66*e^2 + 20/11, -4/11*e^6 - 86/11*e^4 - 208/11*e^2 + 452/11, -4/11*e^6 - 86/11*e^4 - 208/11*e^2 + 452/11, -67/396*e^7 - 488/99*e^5 - 15331/396*e^3 - 5989/66*e, -67/396*e^7 - 488/99*e^5 - 15331/396*e^3 - 5989/66*e, -5/66*e^7 - 95/33*e^5 - 2075/66*e^3 - 905/11*e, -5/66*e^7 - 95/33*e^5 - 2075/66*e^3 - 905/11*e, 31/132*e^7 + 391/66*e^5 + 4285/132*e^3 + 881/22*e, 31/132*e^7 + 391/66*e^5 + 4285/132*e^3 + 881/22*e, -4/33*e^6 - 86/33*e^4 - 208/33*e^2 - 172/11, -4/33*e^6 - 86/33*e^4 - 208/33*e^2 - 172/11, 19/198*e^7 + 425/198*e^5 + 725/99*e^3 + 172/33*e, 19/198*e^7 + 425/198*e^5 + 725/99*e^3 + 172/33*e, 1/33*e^7 + 5/33*e^5 - 278/33*e^3 - 320/11*e, 1/33*e^7 + 5/33*e^5 - 278/33*e^3 - 320/11*e, 7/33*e^6 + 134/33*e^4 + 199/33*e^2 - 40/11, 7/33*e^6 + 134/33*e^4 + 199/33*e^2 - 40/11, -37/66*e^6 - 472/33*e^4 - 5059/66*e^2 - 680/11, -37/66*e^6 - 472/33*e^4 - 5059/66*e^2 - 680/11, -4/33*e^7 - 205/66*e^5 - 1175/66*e^3 - 249/11*e, -4/33*e^7 - 205/66*e^5 - 1175/66*e^3 - 249/11*e, 8/33*e^6 + 172/33*e^4 + 416/33*e^2 - 118/11, 8/33*e^6 + 172/33*e^4 + 416/33*e^2 - 118/11, 10/33*e^6 + 215/33*e^4 + 520/33*e^2 + 23/11, -2/33*e^7 - 43/33*e^5 - 137/33*e^3 - 42/11*e, -2/33*e^7 - 43/33*e^5 - 137/33*e^3 - 42/11*e, 37/396*e^7 + 302/99*e^5 + 11197/396*e^3 + 4651/66*e, 37/396*e^7 + 302/99*e^5 + 11197/396*e^3 + 4651/66*e, -14/99*e^7 - 767/198*e^5 - 5185/198*e^3 - 1207/33*e, -14/99*e^7 - 767/198*e^5 - 5185/198*e^3 - 1207/33*e, -19/99*e^7 - 623/99*e^5 - 5806/99*e^3 - 4832/33*e, -19/99*e^7 - 623/99*e^5 - 5806/99*e^3 - 4832/33*e, 8/33*e^6 + 172/33*e^4 + 416/33*e^2 - 470/11, 8/33*e^6 + 172/33*e^4 + 416/33*e^2 - 470/11, -181/396*e^7 - 2251/198*e^5 - 23635/396*e^3 - 4711/66*e, -181/396*e^7 - 2251/198*e^5 - 23635/396*e^3 - 4711/66*e, 7/396*e^7 + 133/198*e^5 + 2905/396*e^3 + 805/66*e, 7/396*e^7 + 133/198*e^5 + 2905/396*e^3 + 805/66*e, -4/33*e^6 - 86/33*e^4 - 274/33*e^2 - 40/11, -4/33*e^6 - 86/33*e^4 - 274/33*e^2 - 40/11, -2/11*e^6 - 43/11*e^4 - 104/11*e^2 + 215/11, -2/11*e^6 - 43/11*e^4 - 104/11*e^2 + 215/11, -1/198*e^7 - 19/99*e^5 - 415/198*e^3 - 181/33*e, -1/198*e^7 - 19/99*e^5 - 415/198*e^3 - 181/33*e, 101/396*e^7 + 1325/198*e^5 + 16175/396*e^3 + 3563/66*e, 101/396*e^7 + 1325/198*e^5 + 16175/396*e^3 + 3563/66*e, 107/396*e^7 + 670/99*e^5 + 15299/396*e^3 + 5045/66*e, 107/396*e^7 + 670/99*e^5 + 15299/396*e^3 + 5045/66*e, -2/33*e^6 - 43/33*e^4 - 104/33*e^2 + 673/11, -6/11*e^7 - 291/22*e^5 - 1405/22*e^3 - 785/11*e, -6/11*e^7 - 291/22*e^5 - 1405/22*e^3 - 785/11*e, -2/11*e^7 - 43/11*e^5 - 137/11*e^3 - 126/11*e, -2/11*e^7 - 43/11*e^5 - 137/11*e^3 - 126/11*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;