/* 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![29, 3, -12, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [4, 2, 1/2*w^3 - 2*w^2 - 2*w + 15/2], [4, 2, -w^2 + w + 8], [5, 5, -1/4*w^3 + 1/2*w^2 + 1/2*w - 9/4], [5, 5, 1/2*w^3 - w^2 - 4*w + 9/2], [19, 19, 1/4*w^3 - 3/2*w^2 - 1/2*w + 21/4], [19, 19, 1/4*w^3 - 3/2*w^2 - 1/2*w + 41/4], [29, 29, w], [29, 29, 1/4*w^3 - 1/2*w^2 - 1/2*w + 1/4], [29, 29, 1/4*w^3 - 1/2*w^2 - 5/2*w + 9/4], [29, 29, -1/2*w^3 + w^2 + 4*w - 5/2], [41, 41, 3/4*w^3 - 5/2*w^2 - 7/2*w + 47/4], [41, 41, 1/4*w^3 + 1/2*w^2 - 5/2*w - 15/4], [61, 61, -3/2*w^3 + 3*w^2 + 11*w - 21/2], [61, 61, w^3 - 2*w^2 - 4*w + 6], [79, 79, 3/4*w^3 - 5/2*w^2 - 7/2*w + 27/4], [79, 79, 1/4*w^3 + 1/2*w^2 - 5/2*w - 35/4], [81, 3, -3], [89, 89, 7/4*w^3 - 11/2*w^2 - 21/2*w + 119/4], [89, 89, 1/4*w^3 + 3/2*w^2 - 3/2*w - 23/4], [109, 109, -1/2*w^3 + 3*w^2 + 2*w - 41/2], [109, 109, -5/4*w^3 + 7/2*w^2 + 9/2*w - 45/4], [121, 11, -3/4*w^3 + 3/2*w^2 + 9/2*w - 23/4], [121, 11, -1/4*w^3 + 1/2*w^2 + 3/2*w - 21/4], [131, 131, -1/2*w^3 + w^2 + 5*w - 3/2], [131, 131, 3/4*w^3 - 1/2*w^2 - 7/2*w - 1/4], [139, 139, -5/4*w^3 + 11/2*w^2 + 9/2*w - 73/4], [139, 139, -1/4*w^3 + 7/2*w^2 - 3/2*w - 113/4], [149, 149, -1/4*w^3 + 3/2*w^2 + 1/2*w - 49/4], [149, 149, 1/4*w^3 - 3/2*w^2 - 1/2*w + 13/4], [151, 151, 1/2*w^3 - 3*w^2 - w + 23/2], [151, 151, -1/4*w^3 + 3/2*w^2 + 3/2*w - 45/4], [151, 151, 3/4*w^3 - 3/2*w^2 - 7/2*w + 11/4], [151, 151, -2*w^3 + 6*w^2 + 7*w - 20], [169, 13, -1/4*w^3 + 1/2*w^2 + 1/2*w - 25/4], [169, 13, 1/2*w^3 - w^2 - 4*w + 17/2], [179, 179, -1/4*w^3 + 1/2*w^2 + 7/2*w - 21/4], [179, 179, 1/4*w^3 - 1/2*w^2 - 7/2*w - 3/4], [181, 181, 1/2*w^3 - 3*w - 7/2], [181, 181, -5/4*w^3 + 7/2*w^2 + 15/2*w - 57/4], [191, 191, -3/4*w^3 + 5/2*w^2 + 9/2*w - 39/4], [191, 191, w^2 - 8], [199, 199, -3/4*w^3 + 7/2*w^2 + 5/2*w - 43/4], [199, 199, -1/4*w^3 + 5/2*w^2 - 1/2*w - 81/4], [199, 199, -3/4*w^3 + 7/2*w^2 + 5/2*w - 75/4], [199, 199, -1/4*w^3 + 5/2*w^2 - 1/2*w - 49/4], [229, 229, -1/2*w^3 + 2*w^2 + w - 21/2], [229, 229, -3/4*w^3 + 5/2*w^2 + 7/2*w - 55/4], [239, 239, 3/2*w^3 - 4*w^2 - 9*w + 35/2], [239, 239, 3/2*w^3 - 5*w^2 - 7*w + 45/2], [241, 241, -5/4*w^3 + 9/2*w^2 + 13/2*w - 93/4], [241, 241, -1/4*w^3 - 1/2*w^2 - 1/2*w + 3/4], [251, 251, -w^3 + 3*w^2 + 8*w - 13], [251, 251, -7/4*w^3 + 13/2*w^2 + 13/2*w - 99/4], [269, 269, -9/4*w^3 + 13/2*w^2 + 15/2*w - 85/4], [269, 269, -3/4*w^3 + 5/2*w^2 + 3/2*w - 47/4], [271, 271, 5/4*w^3 - 5/2*w^2 - 21/2*w + 53/4], [271, 271, 5/4*w^3 - 1/2*w^2 - 23/2*w - 23/4], [271, 271, 7/4*w^3 - 11/2*w^2 - 13/2*w + 83/4], [271, 271, -1/2*w^3 + w^2 - 13/2], [281, 281, 3/4*w^3 - 1/2*w^2 - 13/2*w - 1/4], [281, 281, -w^3 + 3*w^2 + 4*w - 13], [311, 311, -1/4*w^3 + 1/2*w^2 - 1/2*w - 5/4], [311, 311, 7/4*w^3 - 13/2*w^2 - 19/2*w + 155/4], [311, 311, -1/4*w^3 + 7/2*w^2 + 1/2*w - 49/4], [311, 311, -3/4*w^3 + 3/2*w^2 + 13/2*w - 23/4], [331, 331, -3/4*w^3 + 1/2*w^2 + 15/2*w + 5/4], [331, 331, -3/4*w^3 + 5/2*w^2 + 3/2*w - 39/4], [349, 349, -1/4*w^3 + 1/2*w^2 + 7/2*w - 17/4], [349, 349, 1/4*w^3 - 1/2*w^2 - 7/2*w + 1/4], [359, 359, -5/4*w^3 - 1/2*w^2 + 25/2*w + 55/4], [359, 359, w^2 - 2*w - 2], [359, 359, -9/4*w^3 + 15/2*w^2 + 17/2*w - 113/4], [359, 359, -1/4*w^3 + 3/2*w^2 - 1/2*w - 45/4], [361, 19, -w^3 + 2*w^2 + 6*w - 8], [379, 379, -5/4*w^3 + 5/2*w^2 + 17/2*w - 37/4], [379, 379, -5/4*w^3 + 7/2*w^2 + 9/2*w - 41/4], [379, 379, 5/4*w^3 - 3/2*w^2 - 21/2*w - 3/4], [379, 379, -w^3 + 2*w^2 + 5*w - 7], [389, 389, 9/4*w^3 - 17/2*w^2 - 15/2*w + 117/4], [389, 389, -3/4*w^3 - 5/2*w^2 + 21/2*w + 113/4], [401, 401, -3/4*w^3 - 7/2*w^2 + 23/2*w + 149/4], [401, 401, 11/4*w^3 - 21/2*w^2 - 19/2*w + 143/4], [409, 409, -3/4*w^3 + 5/2*w^2 + 9/2*w - 31/4], [409, 409, w^2 - 10], [421, 421, 5/4*w^3 - 5/2*w^2 - 19/2*w + 49/4], [421, 421, -1/4*w^3 + 5/2*w^2 - 1/2*w - 41/4], [421, 421, 5/4*w^3 - 5/2*w^2 - 7/2*w + 41/4], [421, 421, 9/4*w^3 - 9/2*w^2 - 35/2*w + 77/4], [431, 431, 7/4*w^3 - 9/2*w^2 - 11/2*w + 59/4], [431, 431, 5/4*w^3 - 9/2*w^2 - 13/2*w + 85/4], [431, 431, 5/4*w^3 - 5/2*w^2 - 11/2*w + 33/4], [431, 431, -2*w^2 + w + 12], [439, 439, -1/2*w^3 + w^2 + 5*w - 7/2], [439, 439, 2*w - 1], [461, 461, -7/4*w^3 + 13/2*w^2 + 21/2*w - 163/4], [461, 461, 3/2*w^3 - 5*w^2 - 4*w + 31/2], [491, 491, 5/4*w^3 - 3/2*w^2 - 17/2*w + 21/4], [491, 491, -7/4*w^3 + 9/2*w^2 + 19/2*w - 83/4], [509, 509, -3/2*w^3 + 4*w^2 + 7*w - 27/2], [509, 509, -3/2*w^3 + 3*w^2 + 10*w - 27/2], [521, 521, -5/4*w^3 + 3/2*w^2 + 11/2*w - 17/4], [521, 521, -5/2*w^3 + 6*w^2 + 17*w - 53/2], [521, 521, -1/2*w^3 + 3*w^2 + w - 31/2], [541, 541, 9/4*w^3 - 7/2*w^2 - 35/2*w + 37/4], [541, 541, w^3 - 2*w^2 - 8*w + 12], [541, 541, -1/4*w^3 - 3/2*w^2 + 7/2*w + 35/4], [541, 541, -3/4*w^3 + 9/2*w^2 + 5/2*w - 119/4], [569, 569, -w^3 + 3*w^2 + 6*w - 7], [569, 569, -7/4*w^3 + 9/2*w^2 + 23/2*w - 75/4], [599, 599, 1/4*w^3 + 5/2*w^2 - 9/2*w - 95/4], [599, 599, -7/4*w^3 + 13/2*w^2 + 15/2*w - 91/4], [619, 619, 9/4*w^3 - 13/2*w^2 - 19/2*w + 97/4], [619, 619, -7/4*w^3 + 3/2*w^2 + 29/2*w + 9/4], [619, 619, -1/4*w^3 + 3/2*w^2 - 3/2*w - 45/4], [619, 619, -1/4*w^3 - 1/2*w^2 + 9/2*w - 1/4], [631, 631, -9/4*w^3 + 13/2*w^2 + 15/2*w - 93/4], [631, 631, -13/4*w^3 + 21/2*w^2 + 39/2*w - 233/4], [631, 631, 7/4*w^3 - 9/2*w^2 - 15/2*w + 67/4], [631, 631, 1/4*w^3 + 7/2*w^2 - 3/2*w - 51/4], [641, 641, -1/4*w^3 + 5/2*w^2 + 1/2*w - 57/4], [641, 641, -w^3 + 4*w^2 + 5*w - 19], [661, 661, -2*w^3 + 20*w + 19], [661, 661, 5/4*w^3 - 11/2*w^2 - 13/2*w + 149/4], [691, 691, -5/4*w^3 + 3/2*w^2 + 13/2*w - 9/4], [691, 691, -7/4*w^3 + 11/2*w^2 + 23/2*w - 135/4], [739, 739, -2*w^3 + 7*w^2 + 10*w - 34], [739, 739, 1/4*w^3 + 5/2*w^2 - 7/2*w - 59/4], [751, 751, -1/2*w^3 + 3*w^2 + 2*w - 33/2], [751, 751, -3/4*w^3 + 7/2*w^2 + 7/2*w - 67/4], [761, 761, -w^3 + 4*w^2 + 5*w - 27], [761, 761, 7/4*w^3 - 9/2*w^2 - 15/2*w + 63/4], [769, 769, -5/4*w^3 + 1/2*w^2 + 25/2*w + 19/4], [769, 769, 3/2*w^3 - 5*w^2 - 4*w + 39/2], [769, 769, -3*w^3 + 10*w^2 + 11*w - 37], [769, 769, -7/4*w^3 + 5/2*w^2 + 27/2*w - 31/4], [811, 811, -w^3 + 3*w^2 + 6*w - 9], [811, 811, -3/4*w^3 + 7/2*w^2 + 5/2*w - 39/4], [811, 811, -1/4*w^3 + 5/2*w^2 - 1/2*w - 85/4], [811, 811, -5/4*w^3 + 7/2*w^2 + 17/2*w - 57/4], [821, 821, -3/4*w^3 + 3/2*w^2 + 3/2*w - 35/4], [821, 821, -5/4*w^3 + 11/2*w^2 + 9/2*w - 121/4], [829, 829, 7/4*w^3 - 13/2*w^2 - 19/2*w + 115/4], [829, 829, 1/4*w^3 - 7/2*w^2 - 1/2*w + 89/4], [839, 839, -3/4*w^3 + 5/2*w^2 + 3/2*w - 51/4], [839, 839, -3/4*w^3 + 1/2*w^2 + 15/2*w - 7/4], [911, 911, -3/4*w^3 + 1/2*w^2 + 11/2*w - 15/4], [911, 911, -3/4*w^3 + 7/2*w^2 + 3/2*w - 75/4], [929, 929, 9/4*w^3 - 9/2*w^2 - 35/2*w + 73/4], [929, 929, -3/4*w^3 + 7/2*w^2 + 9/2*w - 51/4], [929, 929, -3/4*w^3 + 7/2*w^2 + 9/2*w - 91/4], [929, 929, 5/2*w^3 - 4*w^2 - 19*w + 25/2], [961, 31, -5/4*w^3 + 5/2*w^2 + 15/2*w - 37/4], [961, 31, -1/2*w^3 + w^2 + 3*w - 19/2]]; primes := [ideal : I in primesArray]; heckePol := x^11 + 3*x^10 - 33*x^9 - 99*x^8 + 363*x^7 + 1115*x^6 - 1437*x^5 - 4903*x^4 + 957*x^3 + 6367*x^2 + 1161*x - 423; K := NumberField(heckePol); heckeEigenvaluesArray := [-11063/4603675*e^10 - 10098/4603675*e^9 + 618099/7365880*e^8 + 1764531/36829400*e^7 - 36974269/36829400*e^6 - 3555651/18414700*e^5 + 86293631/18414700*e^4 - 18850411/18414700*e^3 - 248954249/36829400*e^2 + 7132463/1473176*e + 66680731/36829400, e, -599641/55244100*e^10 - 162887/18414700*e^9 + 352253/920735*e^8 + 4350283/18414700*e^7 - 86409217/18414700*e^6 - 52204879/27622050*e^5 + 427940491/18414700*e^4 + 239257987/55244100*e^3 - 350915841/9207350*e^2 + 4323791/2209764*e + 96607383/18414700, -645899/110488200*e^10 - 295143/36829400*e^9 + 1511143/7365880*e^8 + 1172614/4603675*e^7 - 45577619/18414700*e^6 - 150729131/55244100*e^5 + 426062399/36829400*e^4 + 1226877143/110488200*e^3 - 549949473/36829400*e^2 - 25219693/2209764*e - 7019011/4603675, -1, -714893/55244100*e^10 - 522827/36829400*e^9 + 3424037/7365880*e^8 + 15049593/36829400*e^7 - 106922591/18414700*e^6 - 207851059/55244100*e^5 + 133554342/4603675*e^4 + 1317289927/110488200*e^3 - 1700445647/36829400*e^2 - 19269625/4419528*e + 33441217/9207350, -45569/22097640*e^10 - 2711/920735*e^9 + 61935/736588*e^8 + 732217/7365880*e^7 - 2302967/1841470*e^6 - 2911709/2762205*e^5 + 57859839/7365880*e^4 + 18013157/5524410*e^3 - 62904519/3682940*e^2 + 8097941/4419528*e + 5582889/920735, -45127/5524410*e^10 - 28871/7365880*e^9 + 55010/184147*e^8 + 344697/3682940*e^7 - 28102951/7365880*e^6 - 1464478/2762205*e^5 + 18063021/920735*e^4 - 22295639/22097640*e^3 - 29743382/920735*e^2 + 24136355/2209764*e + 32062939/7365880, 600497/110488200*e^10 + 645029/36829400*e^9 - 1270019/7365880*e^8 - 10522693/18414700*e^7 + 16790541/9207350*e^6 + 351569693/55244100*e^5 - 261411897/36829400*e^4 - 3057371129/110488200*e^3 + 220560769/36829400*e^2 + 18919255/552441*e + 24989757/18414700, 44699/55244100*e^10 - 7339/36829400*e^9 - 87443/3682940*e^8 - 27853/4603675*e^7 + 8899051/36829400*e^6 + 4613878/13811025*e^5 - 23633899/18414700*e^4 - 414948811/110488200*e^3 + 80526923/18414700*e^2 + 12079399/1104882*e - 51669799/36829400, 485777/110488200*e^10 + 222289/36829400*e^9 - 547877/3682940*e^8 - 6906401/36829400*e^7 + 64051399/36829400*e^6 + 56534869/27622050*e^5 - 297311027/36829400*e^4 - 1072781639/110488200*e^3 + 205991627/18414700*e^2 + 79124243/4419528*e + 38752949/36829400, -189659/9207350*e^10 - 1223031/36829400*e^9 + 659867/920735*e^8 + 19046277/18414700*e^7 - 315048421/36829400*e^6 - 49399946/4603675*e^5 + 182518776/4603675*e^4 + 1523789327/36829400*e^3 - 240709277/4603675*e^2 - 26210623/736588*e - 31279571/36829400, 3436231/110488200*e^10 + 719667/36829400*e^9 - 8114387/7365880*e^8 - 10530039/18414700*e^7 + 248060761/18414700*e^6 + 297707089/55244100*e^5 - 2398535431/36829400*e^4 - 1867488667/110488200*e^3 + 3646732737/36829400*e^2 - 993508/552441*e - 130991407/9207350, 1111859/110488200*e^10 + 254869/18414700*e^9 - 2530043/7365880*e^8 - 7205671/18414700*e^7 + 150323033/36829400*e^6 + 196695371/55244100*e^5 - 736505359/36829400*e^4 - 166032436/13811025*e^3 + 1329398093/36829400*e^2 + 12949943/1104882*e - 391598067/36829400, 908493/36829400*e^10 + 1123453/36829400*e^9 - 3225859/3682940*e^8 - 34511177/36829400*e^7 + 398117473/36829400*e^6 + 88674271/9207350*e^5 - 1966275129/36829400*e^4 - 1399541901/36829400*e^3 + 1565763129/18414700*e^2 + 61359253/1473176*e - 520675577/36829400, -278203/7365880*e^10 - 318603/7365880*e^9 + 1968257/1473176*e^8 + 2394863/1841470*e^7 - 59994439/3682940*e^6 - 47669407/3682940*e^5 + 575142849/7365880*e^4 + 353141021/7365880*e^3 - 841518903/7365880*e^2 - 30067881/736588*e + 5674913/1841470, 2375483/110488200*e^10 + 693453/18414700*e^9 - 5357761/7365880*e^8 - 20958327/18414700*e^7 + 307398871/36829400*e^6 + 631075427/55244100*e^5 - 1340447083/36829400*e^4 - 596117932/13811025*e^3 + 1571575691/36829400*e^2 + 22601707/552441*e - 22596929/36829400, -148599/18414700*e^10 - 368233/36829400*e^9 + 513097/1841470*e^8 + 1158684/4603675*e^7 - 123245903/36829400*e^6 - 15715431/9207350*e^5 + 305890797/18414700*e^4 + 47458461/36829400*e^3 - 145796136/4603675*e^2 + 4060155/368294*e + 533861147/36829400, -9269/516300*e^10 - 3983/172100*e^9 + 10629/17210*e^8 + 111747/172100*e^7 - 1264653/172100*e^6 - 732943/129075*e^5 + 6016769/172100*e^4 + 8577983/516300*e^3 - 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361081/36829400*e^9 - 365857/3682940*e^8 + 1942338/4603675*e^7 + 47387379/36829400*e^6 - 58390267/9207350*e^5 - 32098624/4603675*e^4 + 1416936077/36829400*e^3 + 234798667/18414700*e^2 - 13876784/184147*e - 193369471/36829400, -111698/13811025*e^10 - 219863/36829400*e^9 + 2127243/7365880*e^8 + 11133967/36829400*e^7 - 31911577/9207350*e^6 - 247937221/55244100*e^5 + 282022117/18414700*e^4 + 2554072063/110488200*e^3 - 586167393/36829400*e^2 - 166469299/4419528*e + 2858871/18414700, 207419/9207350*e^10 + 147512/4603675*e^9 - 1385259/1841470*e^8 - 13996007/18414700*e^7 + 82359409/9207350*e^6 + 40563897/9207350*e^5 - 212937991/4603675*e^4 + 10258917/9207350*e^3 + 435562032/4603675*e^2 - 18781535/736588*e - 127054633/4603675, -987083/27622050*e^10 - 419637/18414700*e^9 + 1155323/920735*e^8 + 2793477/4603675*e^7 - 279345817/18414700*e^6 - 69011177/13811025*e^5 + 330278829/4603675*e^4 + 726999337/55244100*e^3 - 453309133/4603675*e^2 + 2243371/1104882*e - 482950617/18414700, -2995391/110488200*e^10 - 1014987/36829400*e^9 + 6957717/7365880*e^8 + 3418826/4603675*e^7 - 211491371/18414700*e^6 - 366355079/55244100*e^5 + 2118075591/36829400*e^4 + 2934231587/110488200*e^3 - 3828141607/36829400*e^2 - 118678891/2209764*e + 153689651/4603675, 3412621/55244100*e^10 + 313468/4603675*e^9 - 3918511/1841470*e^8 - 8982612/4603675*e^7 + 458812027/18414700*e^6 + 496848949/27622050*e^5 - 2038156971/18414700*e^4 - 1572123611/27622050*e^3 + 581925323/4603675*e^2 + 5100142/552441*e + 622443127/18414700, -1386629/110488200*e^10 + 196897/36829400*e^9 + 3690713/7365880*e^8 - 4185249/18414700*e^7 - 32600781/4603675*e^6 + 213705799/55244100*e^5 + 1518943229/36829400*e^4 - 3350723797/110488200*e^3 - 3111124383/36829400*e^2 + 96251587/1104882*e + 570917751/18414700, 6610391/110488200*e^10 + 4146437/36829400*e^9 - 1839409/920735*e^8 - 126156583/36829400*e^7 + 827166467/36829400*e^6 + 477733601/13811025*e^5 - 3495544241/36829400*e^4 - 14422475687/110488200*e^3 + 492380804/4603675*e^2 + 542842121/4419528*e - 314907083/36829400, 3429349/55244100*e^10 + 4098811/36829400*e^9 - 15206711/7365880*e^8 - 127058099/36829400*e^7 + 425465213/18414700*e^6 + 1989012587/55244100*e^5 - 447081156/4603675*e^4 - 16032375611/110488200*e^3 + 3920115721/36829400*e^2 + 706881191/4419528*e + 15427119/9207350, -194641/110488200*e^10 - 423337/36829400*e^9 + 4431/3682940*e^8 + 8085083/36829400*e^7 + 36982433/36829400*e^6 - 7383676/13811025*e^5 - 427630309/36829400*e^4 - 527457713/110488200*e^3 + 611972809/18414700*e^2 + 49768919/4419528*e - 341313317/36829400, 1068999/36829400*e^10 + 2162979/36829400*e^9 - 7768979/7365880*e^8 - 32287493/18414700*e^7 + 251287457/18414700*e^6 + 313419581/18414700*e^5 - 2736273197/36829400*e^4 - 2190016893/36829400*e^3 + 5367798469/36829400*e^2 + 8414161/184147*e - 204193242/4603675, 206439/18414700*e^10 - 228703/9207350*e^9 - 1376459/3682940*e^8 + 4303351/4603675*e^7 + 19201701/4603675*e^6 - 114917709/9207350*e^5 - 321994967/18414700*e^4 + 318816438/4603675*e^3 + 391875809/18414700*e^2 - 23724866/184147*e - 15343873/9207350, 10403921/110488200*e^10 + 835361/18414700*e^9 - 12414941/3682940*e^8 - 45592523/36829400*e^7 + 770993851/18414700*e^6 + 278873587/27622050*e^5 - 7633345271/36829400*e^4 - 1195302961/55244100*e^3 + 5983700471/18414700*e^2 - 118492939/4419528*e - 460805149/18414700, -198443/55244100*e^10 - 151794/4603675*e^9 + 34499/920735*e^8 + 9640767/9207350*e^7 + 9309167/9207350*e^6 - 322137017/27622050*e^5 - 271522857/18414700*e^4 + 1518238663/27622050*e^3 + 423204207/9207350*e^2 - 49722437/552441*e + 616071/4603675, 938887/55244100*e^10 + 1094209/18414700*e^9 - 2232779/3682940*e^8 - 18687203/9207350*e^7 + 135983119/18414700*e^6 + 653363503/27622050*e^5 - 624935237/18414700*e^4 - 5900925409/55244100*e^3 + 660614249/18414700*e^2 + 158990345/1104882*e + 289328519/18414700, 6387271/55244100*e^10 + 1472711/9207350*e^9 - 3593283/920735*e^8 - 22080837/4603675*e^7 + 828997777/18414700*e^6 + 653841212/13811025*e^5 - 3738037921/18414700*e^4 - 4757266061/27622050*e^3 + 1304126823/4603675*e^2 + 77914825/552441*e - 864992573/18414700, 2196773/18414700*e^10 + 5516291/36829400*e^9 - 3781552/920735*e^8 - 39181761/9207350*e^7 + 1792282431/36829400*e^6 + 176537856/4603675*e^5 - 4190052919/18414700*e^4 - 4406537847/36829400*e^3 + 1503614572/4603675*e^2 + 12883762/184147*e + 366191581/36829400, 3760021/36829400*e^10 + 4134891/36829400*e^9 - 13127383/3682940*e^8 - 121076069/36829400*e^7 + 1580161081/36829400*e^6 + 145401431/4603675*e^5 - 7542231313/36829400*e^4 - 4224290747/36829400*e^3 + 5763815963/18414700*e^2 + 157793001/1473176*e - 1568667169/36829400, 254253/7365880*e^10 + 10221/920735*e^9 - 1798293/1473176*e^8 - 316643/1841470*e^7 + 108297983/7365880*e^6 - 1970583/3682940*e^5 - 501422939/7365880*e^4 + 48224867/3682940*e^3 + 667656943/7365880*e^2 - 27231111/736588*e + 62329323/7365880, 120991/1104882*e^10 + 109329/736588*e^9 - 2848583/736588*e^8 - 3281005/736588*e^7 + 34817575/736588*e^6 + 24025873/552441*e^5 - 84772389/368294*e^4 - 335356103/2209764*e^3 + 265979595/736588*e^2 + 248706431/2209764*e - 38152139/736588, 399416/13811025*e^10 + 2399121/36829400*e^9 - 3608343/3682940*e^8 - 9031658/4603675*e^7 + 422584861/36829400*e^6 + 265362058/13811025*e^5 - 494844857/9207350*e^4 - 7624866871/110488200*e^3 + 1523873103/18414700*e^2 + 86585905/1104882*e - 823196289/36829400]; 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;