/* 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![-89, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [3, 3, w + 1], [4, 2, 2], [7, 7, w + 3], [11, 11, w + 3], [11, 11, w + 7], [17, 17, w + 8], [23, 23, w + 4], [23, 23, w + 18], [25, 5, 5], [29, 29, w + 1], [29, 29, w + 27], [31, 31, w + 13], [31, 31, w + 17], [43, 43, -w - 11], [43, 43, w - 12], [47, 47, -w - 6], [47, 47, w - 7], [59, 59, -w - 5], [59, 59, w - 6], [61, 61, w + 16], [61, 61, w + 44], [67, 67, -w - 12], [67, 67, w - 13], [71, 71, w + 29], [71, 71, w + 41], [73, 73, w + 24], [73, 73, w + 48], [83, 83, -w - 2], [83, 83, w - 3], [89, 89, -w], [89, 89, w - 1], [97, 97, w + 19], [97, 97, w + 77], [101, 101, 2*w - 17], [101, 101, -2*w - 15], [107, 107, w + 50], [107, 107, w + 56], [113, 113, w + 36], [113, 113, w + 76], [127, 127, 2*w - 23], [127, 127, 2*w + 21], [139, 139, w + 22], [139, 139, w + 116], [151, 151, -w - 15], [151, 151, w - 16], [169, 13, -13], [181, 181, w + 75], [181, 181, w + 105], [197, 197, w + 47], [197, 197, w + 149], [199, 199, w + 38], [199, 199, w + 160], [233, 233, w + 51], [233, 233, w + 181], [241, 241, w + 28], [241, 241, w + 212], [251, 251, -3*w - 22], [251, 251, 3*w - 25], [257, 257, 2*w - 11], [257, 257, -2*w - 9], [283, 283, w + 45], [283, 283, w + 237], [293, 293, 2*w - 9], [293, 293, -2*w - 7], [313, 313, w + 138], [313, 313, w + 174], [317, 317, w + 127], [317, 317, w + 189], [331, 331, -w - 20], [331, 331, w - 21], [347, 347, w + 62], [347, 347, w + 284], [353, 353, 2*w - 3], [353, 353, -2*w - 1], [361, 19, -19], [367, 367, w + 34], [367, 367, w + 332], [373, 373, -w - 21], [373, 373, w - 22], [383, 383, -3*w - 19], [383, 383, 3*w - 22], [397, 397, w + 170], [397, 397, w + 226], [401, 401, w + 96], [401, 401, w + 304], [421, 421, -5*w - 49], [421, 421, 5*w - 54], [431, 431, w + 69], [431, 431, w + 361], [439, 439, w + 37], [439, 439, w + 401], [449, 449, w + 114], [449, 449, w + 334], [457, 457, 3*w - 37], [457, 457, -3*w - 34], [461, 461, 3*w - 20], [461, 461, -3*w - 17], [463, 463, -w - 23], [463, 463, w - 24], [467, 467, -4*w - 29], [467, 467, 4*w - 33], [509, 509, -6*w - 49], [509, 509, 6*w - 55], [563, 563, 3*w - 17], [563, 563, -3*w - 14], [587, 587, 4*w - 31], [587, 587, -4*w - 27], [593, 593, -3*w - 13], [593, 593, 3*w - 16], [601, 601, w + 43], [601, 601, w + 557], [607, 607, w + 137], [607, 607, w + 469], [613, 613, -w - 26], [613, 613, w - 27], [617, 617, w + 119], [617, 617, w + 497], [619, 619, w + 66], [619, 619, w + 552], [631, 631, -6*w - 59], [631, 631, 6*w - 65], [641, 641, w + 84], [641, 641, w + 556], [643, 643, w + 141], [643, 643, w + 501], [647, 647, 3*w - 14], [647, 647, -3*w - 11], [653, 653, w + 264], [653, 653, w + 388], [683, 683, w + 270], [683, 683, w + 412], [691, 691, w + 46], [691, 691, w + 644], [739, 739, -5*w - 52], [739, 739, 5*w - 57], [743, 743, w + 297], [743, 743, w + 445], [757, 757, -3*w - 38], [757, 757, 3*w - 41], [761, 761, 3*w - 8], [761, 761, -3*w - 5], [773, 773, -3*w - 4], [773, 773, 3*w - 7], [787, 787, w + 49], [787, 787, w + 737], [797, 797, -3*w - 1], [797, 797, 3*w - 4], [809, 809, w + 302], [809, 809, w + 506], [811, 811, w + 243], [811, 811, w + 567], [821, 821, w + 95], [821, 821, w + 725], [827, 827, w + 242], [827, 827, w + 584], [853, 853, w + 399], [853, 853, w + 453], [883, 883, -7*w - 69], [883, 883, 7*w - 76], [911, 911, w + 254], [911, 911, w + 656], [919, 919, 3*w - 43], [919, 919, -3*w - 40], [947, 947, w + 102], [947, 947, w + 844], [967, 967, -w - 32], [967, 967, w - 33], [971, 971, -5*w - 33], [971, 971, 5*w - 38], [997, 997, w + 55], [997, 997, w + 941]]; primes := [ideal : I in primesArray]; heckePol := x^12 - 140*x^10 + 7246*x^8 - 168516*x^6 + 1618137*x^4 - 3479624*x^2 + 1763584; K := NumberField(heckePol); heckeEigenvaluesArray := [15311139/1393457988512*e^11 - 935243319/696728994256*e^9 + 2502959962/43545562141*e^7 - 715165736465/696728994256*e^5 + 9282077472085/1393457988512*e^3 - 1239955667803/174182248564*e, -121809/8394325232*e^10 + 13386039/8394325232*e^8 - 491046079/8394325232*e^6 + 7008535345/8394325232*e^4 - 4000351657/1049290654*e^2 + 3262213561/524645327, -1398525/348364497128*e^11 + 683924177/1393457988512*e^9 - 29655804789/1393457988512*e^7 + 552511262739/1393457988512*e^5 - 3952164556715/1393457988512*e^3 + 454877787181/174182248564*e, -15311139/1393457988512*e^11 + 935243319/696728994256*e^9 - 2502959962/43545562141*e^7 + 715165736465/696728994256*e^5 - 9282077472085/1393457988512*e^3 + 1414137916367/174182248564*e, -15311139/1393457988512*e^11 + 935243319/696728994256*e^9 - 2502959962/43545562141*e^7 + 715165736465/696728994256*e^5 - 9282077472085/1393457988512*e^3 + 1414137916367/174182248564*e, -565797/8394325232*e^10 + 64348299/8394325232*e^8 - 2407411435/8394325232*e^6 + 31990528597/8394325232*e^4 - 4860115326/524645327*e^2 + 6474916558/524645327, 48066725/1393457988512*e^11 - 2692974409/696728994256*e^9 + 50022642981/348364497128*e^7 - 1383643827579/696728994256*e^5 + 9979603023479/1393457988512*e^3 - 2101982506069/174182248564*e, 48066725/1393457988512*e^11 - 2692974409/696728994256*e^9 + 50022642981/348364497128*e^7 - 1383643827579/696728994256*e^5 + 9979603023479/1393457988512*e^3 - 2101982506069/174182248564*e, 121809/8394325232*e^10 - 13386039/8394325232*e^8 + 491046079/8394325232*e^6 - 7008535345/8394325232*e^4 + 4000351657/1049290654*e^2 + 1459594382/524645327, -16377793/696728994256*e^11 + 878865545/348364497128*e^9 - 29998963285/348364497128*e^7 + 334239045557/348364497128*e^5 - 348762775697/696728994256*e^3 + 343922294851/87091124282*e, -16377793/696728994256*e^11 + 878865545/348364497128*e^9 - 29998963285/348364497128*e^7 + 334239045557/348364497128*e^5 - 348762775697/696728994256*e^3 + 343922294851/87091124282*e, 411639/43545562141*e^11 - 1142172793/1393457988512*e^9 + 18138195659/1393457988512*e^7 + 553464848105/1393457988512*e^5 - 13565024592835/1393457988512*e^3 + 2861083941765/174182248564*e, 411639/43545562141*e^11 - 1142172793/1393457988512*e^9 + 18138195659/1393457988512*e^7 + 553464848105/1393457988512*e^5 - 13565024592835/1393457988512*e^3 + 2861083941765/174182248564*e, -36999/1049290654*e^10 + 4246855/1049290654*e^8 - 159697113/1049290654*e^6 + 2081832771/1049290654*e^4 - 1731744556/524645327*e^2 - 1006069964/524645327, -36999/1049290654*e^10 + 4246855/1049290654*e^8 - 159697113/1049290654*e^6 + 2081832771/1049290654*e^4 - 1731744556/524645327*e^2 - 1006069964/524645327, -1382715/8394325232*e^10 + 38178499/2098581308*e^8 - 5468646107/8394325232*e^6 + 33571330019/4197162616*e^4 - 6029545128/524645327*e^2 + 715286628/524645327, -1382715/8394325232*e^10 + 38178499/2098581308*e^8 - 5468646107/8394325232*e^6 + 33571330019/4197162616*e^4 - 6029545128/524645327*e^2 + 715286628/524645327, -671177/16788650464*e^10 + 64816539/16788650464*e^8 - 1784806263/16788650464*e^6 + 9061375245/16788650464*e^4 + 17344197603/2098581308*e^2 - 7835408576/524645327, -671177/16788650464*e^10 + 64816539/16788650464*e^8 - 1784806263/16788650464*e^6 + 9061375245/16788650464*e^4 + 17344197603/2098581308*e^2 - 7835408576/524645327, -50859987/1393457988512*e^11 + 5748356139/1393457988512*e^9 - 216856133599/1393457988512*e^7 + 3086911675781/1393457988512*e^5 - 5762855553055/696728994256*e^3 + 99191437381/87091124282*e, -50859987/1393457988512*e^11 + 5748356139/1393457988512*e^9 - 216856133599/1393457988512*e^7 + 3086911675781/1393457988512*e^5 - 5762855553055/696728994256*e^3 + 99191437381/87091124282*e, 3063477/8394325232*e^10 - 343892707/8394325232*e^8 + 12771483547/8394325232*e^6 - 174384270325/8394325232*e^4 + 69322017101/1049290654*e^2 - 25311583952/524645327, 3063477/8394325232*e^10 - 343892707/8394325232*e^8 + 12771483547/8394325232*e^6 - 174384270325/8394325232*e^4 + 69322017101/1049290654*e^2 - 25311583952/524645327, 33227827/1393457988512*e^11 - 257635139/87091124282*e^9 + 91571090061/696728994256*e^7 - 443492297383/174182248564*e^5 + 27191169478051/1393457988512*e^3 - 4753785118313/174182248564*e, 33227827/1393457988512*e^11 - 257635139/87091124282*e^9 + 91571090061/696728994256*e^7 - 443492297383/174182248564*e^5 + 27191169478051/1393457988512*e^3 - 4753785118313/174182248564*e, -12020645/696728994256*e^11 + 1156969349/696728994256*e^9 - 28905101771/696728994256*e^7 - 76369828673/696728994256*e^5 + 1816678022525/174182248564*e^3 - 822848333818/43545562141*e, -12020645/696728994256*e^11 + 1156969349/696728994256*e^9 - 28905101771/696728994256*e^7 - 76369828673/696728994256*e^5 + 1816678022525/174182248564*e^3 - 822848333818/43545562141*e, 1495/148572128*e^10 - 148511/148572128*e^8 + 4222653/148572128*e^6 - 24231589/148572128*e^4 - 22844739/18571516*e^2 - 38930424/4642879, 1495/148572128*e^10 - 148511/148572128*e^8 + 4222653/148572128*e^6 - 24231589/148572128*e^4 - 22844739/18571516*e^2 - 38930424/4642879, 3436607/16788650464*e^10 - 370244531/16788650464*e^8 + 12722098477/16788650464*e^6 - 143346695321/16788650464*e^4 + 6773982909/2098581308*e^2 + 7120121948/524645327, 3436607/16788650464*e^10 - 370244531/16788650464*e^8 + 12722098477/16788650464*e^6 - 143346695321/16788650464*e^4 + 6773982909/2098581308*e^2 + 7120121948/524645327, 44946529/1393457988512*e^11 - 4868349513/1393457988512*e^9 + 167560727115/1393457988512*e^7 - 1830103078323/1393457988512*e^5 - 162397768925/174182248564*e^3 + 399139970072/43545562141*e, 44946529/1393457988512*e^11 - 4868349513/1393457988512*e^9 + 167560727115/1393457988512*e^7 - 1830103078323/1393457988512*e^5 - 162397768925/174182248564*e^3 + 399139970072/43545562141*e, 2934365/16788650464*e^10 - 322209735/16788650464*e^8 + 11414452003/16788650464*e^6 - 137023489633/16788650464*e^4 + 21536725005/2098581308*e^2 - 5114424540/524645327, 2934365/16788650464*e^10 - 322209735/16788650464*e^8 + 11414452003/16788650464*e^6 - 137023489633/16788650464*e^4 + 21536725005/2098581308*e^2 - 5114424540/524645327, -48066725/1393457988512*e^11 + 2692974409/696728994256*e^9 - 50022642981/348364497128*e^7 + 1383643827579/696728994256*e^5 - 9979603023479/1393457988512*e^3 + 2101982506069/174182248564*e, -48066725/1393457988512*e^11 + 2692974409/696728994256*e^9 - 50022642981/348364497128*e^7 + 1383643827579/696728994256*e^5 - 9979603023479/1393457988512*e^3 + 2101982506069/174182248564*e, -7419449/696728994256*e^11 + 631893297/696728994256*e^9 - 8474195901/696728994256*e^7 - 390325361953/696728994256*e^5 + 4302891613643/348364497128*e^3 - 662950653061/43545562141*e, -7419449/696728994256*e^11 + 631893297/696728994256*e^9 - 8474195901/696728994256*e^7 - 390325361953/696728994256*e^5 + 4302891613643/348364497128*e^3 - 662950653061/43545562141*e, -609045/8394325232*e^10 + 66930195/8394325232*e^8 - 2455230395/8394325232*e^6 + 35042676725/8394325232*e^4 - 20001758285/1049290654*e^2 + 12638550516/524645327, -609045/8394325232*e^10 + 66930195/8394325232*e^8 - 2455230395/8394325232*e^6 + 35042676725/8394325232*e^4 - 20001758285/1049290654*e^2 + 12638550516/524645327, -181499/4197162616*e^11 + 81190797/16788650464*e^9 - 2983800553/16788650464*e^7 + 39282343991/16788650464*e^5 - 96179461695/16788650464*e^3 - 4296436295/2098581308*e, -181499/4197162616*e^11 + 81190797/16788650464*e^9 - 2983800553/16788650464*e^7 + 39282343991/16788650464*e^5 - 96179461695/16788650464*e^3 - 4296436295/2098581308*e, -3307095/8394325232*e^10 + 370664785/8394325232*e^8 - 13753575705/8394325232*e^6 + 188401341015/8394325232*e^4 - 77322720415/1049290654*e^2 + 26589557804/524645327, -3307095/8394325232*e^10 + 370664785/8394325232*e^8 - 13753575705/8394325232*e^6 + 188401341015/8394325232*e^4 - 77322720415/1049290654*e^2 + 26589557804/524645327, -3846705/8394325232*e^10 + 431411703/8394325232*e^8 - 16013244767/8394325232*e^6 + 219073073873/8394325232*e^4 - 88786912841/1049290654*e^2 + 36305077578/524645327, 1398525/174182248564*e^11 - 683924177/696728994256*e^9 + 29655804789/696728994256*e^7 - 552511262739/696728994256*e^5 + 3952164556715/696728994256*e^3 - 454877787181/87091124282*e, 1398525/174182248564*e^11 - 683924177/696728994256*e^9 + 29655804789/696728994256*e^7 - 552511262739/696728994256*e^5 + 3952164556715/696728994256*e^3 - 454877787181/87091124282*e, 27743087/348364497128*e^11 - 6017842115/696728994256*e^9 + 208564767825/696728994256*e^7 - 2376962293205/696728994256*e^5 + 1373819796193/696728994256*e^3 - 776081199947/87091124282*e, 27743087/348364497128*e^11 - 6017842115/696728994256*e^9 + 208564767825/696728994256*e^7 - 2376962293205/696728994256*e^5 + 1373819796193/696728994256*e^3 - 776081199947/87091124282*e, -67003571/1393457988512*e^11 + 1851990943/348364497128*e^9 - 8329510768/43545562141*e^7 + 104245337330/43545562141*e^5 - 5636747374665/1393457988512*e^3 - 1117113699433/174182248564*e, -67003571/1393457988512*e^11 + 1851990943/348364497128*e^9 - 8329510768/43545562141*e^7 + 104245337330/43545562141*e^5 - 5636747374665/1393457988512*e^3 - 1117113699433/174182248564*e, -39108381/696728994256*e^11 + 4260111025/696728994256*e^9 - 148566841255/696728994256*e^7 + 1708484202091/696728994256*e^5 - 64066063781/43545562141*e^3 + 216079452548/43545562141*e, -39108381/696728994256*e^11 + 4260111025/696728994256*e^9 - 148566841255/696728994256*e^7 + 1708484202091/696728994256*e^5 - 64066063781/43545562141*e^3 + 216079452548/43545562141*e, 40669147/1393457988512*e^11 - 3411721823/1393457988512*e^9 + 43647680865/1393457988512*e^7 + 2137489563747/1393457988512*e^5 - 5874173285155/174182248564*e^3 + 2449659774856/43545562141*e, 40669147/1393457988512*e^11 - 3411721823/1393457988512*e^9 + 43647680865/1393457988512*e^7 + 2137489563747/1393457988512*e^5 - 5874173285155/174182248564*e^3 + 2449659774856/43545562141*e, -3938849/16788650464*e^10 + 418279327/16788650464*e^8 - 14029744951/16788650464*e^6 + 149669901009/16788650464*e^4 + 7988759187/2098581308*e^2 - 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73889299013/16788650464*e^4 - 7503567381/2098581308*e^2 + 22393314330/524645327, 1967143/16788650464*e^10 - 207984447/16788650464*e^8 + 6952956029/16788650464*e^6 - 73889299013/16788650464*e^4 - 7503567381/2098581308*e^2 + 22393314330/524645327, 7419449/696728994256*e^11 - 631893297/696728994256*e^9 + 8474195901/696728994256*e^7 + 390325361953/696728994256*e^5 - 4302891613643/348364497128*e^3 + 662950653061/43545562141*e, 7419449/696728994256*e^11 - 631893297/696728994256*e^9 + 8474195901/696728994256*e^7 + 390325361953/696728994256*e^5 - 4302891613643/348364497128*e^3 + 662950653061/43545562141*e, -154506581/1393457988512*e^11 + 4177416687/348364497128*e^9 - 290892869287/696728994256*e^7 + 1726562245285/348364497128*e^5 - 8053322681525/1393457988512*e^3 - 5768154852021/174182248564*e, -154506581/1393457988512*e^11 + 4177416687/348364497128*e^9 - 290892869287/696728994256*e^7 + 1726562245285/348364497128*e^5 - 8053322681525/1393457988512*e^3 - 5768154852021/174182248564*e, 1706259/174182248564*e^11 - 1351348889/696728994256*e^9 + 91717315097/696728994256*e^7 - 2582510216699/696728994256*e^5 + 26821175395759/696728994256*e^3 - 3810254844005/87091124282*e, 1706259/174182248564*e^11 - 1351348889/696728994256*e^9 + 91717315097/696728994256*e^7 - 2582510216699/696728994256*e^5 + 26821175395759/696728994256*e^3 - 3810254844005/87091124282*e, -110500107/1393457988512*e^11 + 5714492093/696728994256*e^9 - 88495802167/348364497128*e^7 + 1271471194787/696728994256*e^5 + 23746004334271/1393457988512*e^3 - 2513778128717/174182248564*e, -110500107/1393457988512*e^11 + 5714492093/696728994256*e^9 - 88495802167/348364497128*e^7 + 1271471194787/696728994256*e^5 + 23746004334271/1393457988512*e^3 - 2513778128717/174182248564*e, -109980259/696728994256*e^11 + 12579420373/696728994256*e^9 - 241090398843/348364497128*e^7 + 6998440914621/696728994256*e^5 - 27070533162165/696728994256*e^3 + 1705281926451/87091124282*e, -109980259/696728994256*e^11 + 12579420373/696728994256*e^9 - 241090398843/348364497128*e^7 + 6998440914621/696728994256*e^5 - 27070533162165/696728994256*e^3 + 1705281926451/87091124282*e, -470175/8394325232*e^10 + 54563641/8394325232*e^8 - 2064107729/8394325232*e^6 + 26300788991/8394325232*e^4 - 2926626567/1049290654*e^2 + 10989651648/524645327, -470175/8394325232*e^10 + 54563641/8394325232*e^8 - 2064107729/8394325232*e^6 + 26300788991/8394325232*e^4 - 2926626567/1049290654*e^2 + 10989651648/524645327, -364011563/1393457988512*e^11 + 20227363005/696728994256*e^9 - 368681092875/348364497128*e^7 + 9666709450963/696728994256*e^5 - 53300717647985/1393457988512*e^3 + 11550703561027/174182248564*e, -364011563/1393457988512*e^11 + 20227363005/696728994256*e^9 - 368681092875/348364497128*e^7 + 9666709450963/696728994256*e^5 - 53300717647985/1393457988512*e^3 + 11550703561027/174182248564*e, -831039/2098581308*e^10 + 92411325/2098581308*e^8 - 3413413073/2098581308*e^6 + 47533673351/2098581308*e^4 - 45198750238/524645327*e^2 + 53831736096/524645327, -831039/2098581308*e^10 + 92411325/2098581308*e^8 - 3413413073/2098581308*e^6 + 47533673351/2098581308*e^4 - 45198750238/524645327*e^2 + 53831736096/524645327, 229761343/1393457988512*e^11 - 3151866913/174182248564*e^9 + 451558649481/696728994256*e^7 - 723104717027/87091124282*e^5 + 31376322786415/1393457988512*e^3 - 8880852656525/174182248564*e, 229761343/1393457988512*e^11 - 3151866913/174182248564*e^9 + 451558649481/696728994256*e^7 - 723104717027/87091124282*e^5 + 31376322786415/1393457988512*e^3 - 8880852656525/174182248564*e, -3393591/8394325232*e^10 + 375828577/8394325232*e^8 - 13849213625/8394325232*e^6 + 194505637271/8394325232*e^4 - 98935066335/1049290654*e^2 + 47311150952/524645327, -3393591/8394325232*e^10 + 375828577/8394325232*e^8 - 13849213625/8394325232*e^6 + 194505637271/8394325232*e^4 - 98935066335/1049290654*e^2 + 47311150952/524645327, 3436607/8394325232*e^10 - 370244531/8394325232*e^8 + 12722098477/8394325232*e^6 - 143346695321/8394325232*e^4 + 6773982909/1049290654*e^2 + 14240243896/524645327, 3436607/8394325232*e^10 - 370244531/8394325232*e^8 + 12722098477/8394325232*e^6 - 143346695321/8394325232*e^4 + 6773982909/1049290654*e^2 + 14240243896/524645327, 226376591/1393457988512*e^11 - 27343782281/1393457988512*e^9 + 1150231164747/1393457988512*e^7 - 19948270792347/1393457988512*e^5 + 61341186259545/696728994256*e^3 - 7856841340803/87091124282*e, 226376591/1393457988512*e^11 - 27343782281/1393457988512*e^9 + 1150231164747/1393457988512*e^7 - 19948270792347/1393457988512*e^5 + 61341186259545/696728994256*e^3 - 7856841340803/87091124282*e]; heckeEigenvalues := AssociativeArray(); for i := 1 to #heckeEigenvaluesArray do heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i]; end for; ALEigenvalues := AssociativeArray(); // 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;