/* This code can be loaded, or copied and paste using cpaste, into Sage. It will load the data associated to the HMF, including the field, level, and Hecke and Atkin-Lehner eigenvalue data. */ P. = PolynomialRing(QQ) g = P([3, 2, -5, -2, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([19, 19, -w^2 + w + 4]) primes_array = [ [3, 3, w],\ [4, 2, -w^3 + 3*w^2 + w - 2],\ [11, 11, -w^3 + 3*w^2 + 2*w - 5],\ [13, 13, w^3 - 2*w^2 - 4*w + 2],\ [13, 13, -w + 2],\ [17, 17, w^3 - 3*w^2 - 2*w + 2],\ [19, 19, -w^2 + w + 4],\ [19, 19, w^3 - 2*w^2 - 3*w + 2],\ [23, 23, w^2 - 2*w - 1],\ [27, 3, w^3 - 2*w^2 - 5*w - 1],\ [29, 29, -w^3 + 2*w^2 + 5*w - 1],\ [37, 37, -w^3 + 2*w^2 + 5*w - 4],\ [41, 41, 2*w^3 - 5*w^2 - 6*w + 4],\ [53, 53, w^3 - 2*w^2 - 3*w - 2],\ [59, 59, w - 4],\ [67, 67, 2*w^2 - 3*w - 8],\ [73, 73, 2*w^3 - 5*w^2 - 6*w + 5],\ [89, 89, -2*w^3 + 6*w^2 + 5*w - 7],\ [97, 97, -2*w^3 + 6*w^2 + 3*w - 7],\ [97, 97, -w^3 + 3*w^2 + 3*w - 1],\ [103, 103, w^2 - 4*w - 4],\ [113, 113, 2*w^3 - 5*w^2 - 4*w + 1],\ [113, 113, -w^3 + 3*w^2 + w - 5],\ [139, 139, 2*w^2 - 3*w - 4],\ [139, 139, w^3 - w^2 - 6*w - 1],\ [149, 149, 2*w^3 - 5*w^2 - 5*w + 2],\ [149, 149, w^3 - 4*w^2 + 10],\ [151, 151, -3*w - 1],\ [157, 157, w^2 - w - 8],\ [157, 157, w^3 - 7*w - 8],\ [163, 163, -3*w^3 + 8*w^2 + 8*w - 10],\ [163, 163, w^3 - 3*w^2 - 4*w + 7],\ [163, 163, -w^3 + 4*w^2 + w - 8],\ [163, 163, -2*w^3 + 5*w^2 + 7*w - 4],\ [167, 167, -2*w^3 + 5*w^2 + 8*w - 4],\ [169, 13, -2*w^3 + 4*w^2 + 7*w - 5],\ [173, 173, 2*w^3 - 4*w^2 - 9*w + 4],\ [173, 173, -w^3 + 5*w^2 - 2*w - 5],\ [181, 181, 3*w^3 - 7*w^2 - 12*w + 7],\ [191, 191, w^3 - 3*w^2 - 3*w - 1],\ [199, 199, 2*w^3 - 3*w^2 - 10*w - 1],\ [199, 199, 2*w^3 - 6*w^2 - 6*w + 13],\ [223, 223, -2*w^3 + 6*w^2 + 4*w - 11],\ [223, 223, 3*w^3 - 7*w^2 - 12*w + 8],\ [223, 223, -2*w^3 + 5*w^2 + 6*w - 1],\ [229, 229, -w^3 + 3*w^2 + 2*w + 1],\ [229, 229, 3*w^3 - 4*w^2 - 15*w - 5],\ [233, 233, -3*w^3 + 7*w^2 + 11*w - 5],\ [233, 233, 2*w^3 - 5*w^2 - 4*w + 5],\ [241, 241, w^3 - 3*w^2 - 5*w + 7],\ [257, 257, w^3 - 3*w^2 - 4*w + 8],\ [263, 263, w^2 - 4*w - 5],\ [271, 271, -2*w^3 + 6*w^2 + 5*w - 13],\ [277, 277, -2*w^3 + 6*w^2 + 3*w - 8],\ [277, 277, w^3 - 2*w^2 - 2*w - 2],\ [281, 281, 2*w^3 - 5*w^2 - 4*w + 2],\ [283, 283, -3*w^2 + 7*w + 10],\ [283, 283, -w^3 + 4*w^2 - w - 7],\ [293, 293, 2*w^3 - 5*w^2 - 3*w + 4],\ [293, 293, w^3 - 8*w - 4],\ [307, 307, -w^3 + 2*w^2 + 6*w - 2],\ [311, 311, w^2 - 5],\ [317, 317, -w^3 + 5*w^2 - 2*w - 11],\ [331, 331, -w^3 + 3*w^2 - 5],\ [349, 349, -2*w^2 + 5*w + 2],\ [353, 353, -2*w^3 + 5*w^2 + 4*w - 4],\ [353, 353, 3*w^3 - 7*w^2 - 10*w + 2],\ [353, 353, -2*w^3 + 7*w^2 + 2*w - 10],\ [353, 353, w^3 + w^2 - 9*w - 13],\ [361, 19, -w^3 + w^2 + 5*w - 1],\ [373, 373, -w^3 + 3*w^2 + 4*w - 11],\ [373, 373, 2*w^3 - 5*w^2 - 5*w - 2],\ [383, 383, w^3 - 5*w^2 + w + 13],\ [389, 389, 3*w^3 - 10*w^2 - 4*w + 16],\ [397, 397, -w^3 + 2*w^2 + 3*w + 5],\ [397, 397, -3*w^3 + 7*w^2 + 10*w - 7],\ [419, 419, -w^3 + 2*w^2 + 4*w + 4],\ [419, 419, 3*w^3 - 7*w^2 - 12*w + 5],\ [421, 421, -w^2 + 5*w - 2],\ [421, 421, 3*w^3 - 7*w^2 - 10*w + 5],\ [431, 431, w^2 - 10],\ [439, 439, w^3 - w^2 - 7*w - 7],\ [461, 461, -2*w^3 + 7*w^2 + 5*w - 10],\ [467, 467, -3*w^3 + 6*w^2 + 14*w - 4],\ [479, 479, 3*w^2 - 6*w - 5],\ [479, 479, -4*w^3 + 11*w^2 + 10*w - 10],\ [491, 491, -w^3 + w^2 + 6*w - 1],\ [499, 499, -4*w^3 + 9*w^2 + 15*w - 10],\ [499, 499, 2*w^3 - 5*w^2 - 7*w + 2],\ [503, 503, -2*w^3 + 7*w^2 - 8],\ [521, 521, 2*w^2 - w - 7],\ [521, 521, -3*w^3 + 7*w^2 + 11*w - 11],\ [569, 569, 2*w^2 - 5*w - 1],\ [569, 569, -4*w - 1],\ [571, 571, -w^3 + 3*w^2 - 7],\ [577, 577, -w^3 + w^2 + 8*w - 4],\ [587, 587, -3*w^3 + 8*w^2 + 7*w - 8],\ [593, 593, 3*w^2 - 4*w - 11],\ [593, 593, -3*w^3 + 9*w^2 + 6*w - 14],\ [601, 601, -3*w^3 + 9*w^2 + 4*w - 10],\ [601, 601, 5*w^3 - 9*w^2 - 24*w - 2],\ [607, 607, 2*w^2 - 7*w - 5],\ [613, 613, w^3 - 6*w - 2],\ [613, 613, -w^2 + 3*w + 8],\ [617, 617, -w^2 + 4*w - 5],\ [619, 619, 3*w^3 - 6*w^2 - 13*w + 5],\ [625, 5, -5],\ [631, 631, -2*w^3 + 4*w^2 + 8*w - 7],\ [647, 647, -2*w^3 + 4*w^2 + 5*w - 1],\ [647, 647, w^3 - 3*w^2 - 2*w - 2],\ [653, 653, w^3 - 4*w^2 - w + 13],\ [653, 653, w^3 - 2*w^2 - 7*w + 5],\ [659, 659, -4*w^3 + 10*w^2 + 15*w - 14],\ [659, 659, 5*w^3 - 12*w^2 - 17*w + 14],\ [661, 661, w^2 - 5*w - 2],\ [661, 661, -w^3 + 5*w^2 - 2*w - 7],\ [673, 673, -w^3 + w^2 + 6*w - 2],\ [683, 683, 2*w^3 - 8*w^2 + w + 8],\ [683, 683, 5*w^3 - 12*w^2 - 19*w + 14],\ [701, 701, -2*w^3 + 6*w^2 + 5*w - 4],\ [709, 709, -w^2 + 3*w - 4],\ [709, 709, -2*w^3 + 5*w^2 + 9*w - 8],\ [719, 719, w^2 - 7],\ [719, 719, -2*w^3 + 3*w^2 + 8*w + 5],\ [733, 733, -2*w^3 + 7*w^2 + 3*w - 10],\ [739, 739, 4*w^3 - 11*w^2 - 13*w + 14],\ [739, 739, -5*w^3 + 10*w^2 + 21*w - 4],\ [739, 739, 3*w^3 - 8*w^2 - 6*w + 8],\ [739, 739, -w^3 + w^2 + 8*w - 5],\ [743, 743, -w - 5],\ [743, 743, w^3 - w^2 - 8*w - 8],\ [751, 751, -2*w^3 + 5*w^2 + 8*w - 2],\ [751, 751, -2*w^3 + 6*w^2 - 1],\ [757, 757, -2*w^3 + 6*w^2 + 3*w - 10],\ [757, 757, 5*w^2 - 9*w - 22],\ [761, 761, -2*w^3 + 4*w^2 + 9*w - 7],\ [761, 761, 2*w^3 - 3*w^2 - 12*w - 1],\ [769, 769, 5*w^3 - 13*w^2 - 14*w + 10],\ [773, 773, -3*w^3 + 5*w^2 + 16*w + 1],\ [773, 773, w^3 - 11*w + 2],\ [787, 787, -w^3 + 4*w^2 - w - 11],\ [797, 797, -5*w^3 + 12*w^2 + 17*w - 11],\ [809, 809, -2*w^3 + 7*w^2 + 4*w - 11],\ [809, 809, 4*w^3 - 9*w^2 - 14*w + 4],\ [821, 821, -w^3 + 5*w^2 - 2*w - 17],\ [827, 827, -3*w^3 + 8*w^2 + 9*w - 7],\ [829, 829, 2*w^3 - 6*w^2 - 5*w + 2],\ [829, 829, -w^3 + 4*w^2 - w - 10],\ [839, 839, w^2 - 8],\ [853, 853, -5*w^3 + 12*w^2 + 18*w - 16],\ [853, 853, w^3 - 6*w - 8],\ [863, 863, 5*w^3 - 11*w^2 - 19*w + 7],\ [881, 881, -3*w^3 + 9*w^2 + 9*w - 11],\ [907, 907, -w^3 + 2*w^2 + 7*w - 4],\ [907, 907, -2*w^3 + 3*w^2 + 7*w + 4],\ [929, 929, -w^3 + 5*w^2 - 3*w - 11],\ [937, 937, 2*w^3 - 5*w^2 - 6*w + 11],\ [937, 937, -2*w^3 + 6*w^2 + 3*w - 11],\ [941, 941, -3*w^3 + 8*w^2 + 11*w - 8],\ [953, 953, -4*w^3 + 10*w^2 + 10*w - 13],\ [953, 953, w^3 - 5*w^2 - w + 13],\ [967, 967, 3*w^3 - 7*w^2 - 9*w + 7],\ [967, 967, -2*w^3 + 5*w^2 + 8*w + 1],\ [967, 967, 4*w^3 - 13*w^2 - 6*w + 20],\ [967, 967, -2*w^2 + 3*w + 13],\ [971, 971, -6*w^3 + 13*w^2 + 25*w - 8],\ [977, 977, w^3 - w^2 - 4*w - 10],\ [977, 977, 2*w^3 - 6*w^2 - 7*w + 13],\ [983, 983, -4*w^3 + 10*w^2 + 11*w - 13],\ [983, 983, -w^3 + 5*w^2 - w - 7],\ [983, 983, 3*w^3 - 9*w^2 - 7*w + 19],\ [983, 983, -2*w^3 + 3*w^2 + 10*w - 1],\ [997, 997, 5*w^3 - 12*w^2 - 17*w + 13],\ [997, 997, -2*w^3 + 3*w^2 + 13*w - 4],\ [997, 997, w^3 - 7*w - 2],\ [997, 997, w^3 - 9*w - 2]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^20 - 43*x^18 + 762*x^16 - 7243*x^14 + 40390*x^12 - 136052*x^10 + 274255*x^8 - 313426*x^6 + 177576*x^4 - 35924*x^2 + 2304 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, -282311889/11109605344*e^19 + 11229092235/11109605344*e^17 - 89251608885/5554802672*e^15 + 1452712185083/11109605344*e^13 - 3226046574827/5554802672*e^11 + 3846311520701/2777401336*e^9 - 17560008848095/11109605344*e^7 + 2776201288913/5554802672*e^5 + 385522799999/1388700668*e^3 - 149357119803/2777401336*e, -24194657/347175167*e^18 + 1950290689/694350334*e^16 - 15811071802/347175167*e^14 + 132758641625/347175167*e^12 - 1243101944407/694350334*e^10 + 3268480922915/694350334*e^8 - 4622656996337/694350334*e^6 + 1554818196974/347175167*e^4 - 367857704769/347175167*e^2 + 28563790338/347175167, 329700391/16664408016*e^19 - 13248653653/16664408016*e^17 + 35654327117/2777401336*e^15 - 1785713165389/16664408016*e^13 + 4144683034793/8332204008*e^11 - 5379261739559/4166102004*e^9 + 29784655736905/16664408016*e^7 - 9540387484931/8332204008*e^5 + 153578737231/694350334*e^3 - 33264546083/4166102004*e, 54587871/694350334*e^18 - 2193715957/694350334*e^16 + 17710223026/347175167*e^14 - 295558519525/694350334*e^12 + 685197604125/347175167*e^10 - 1773312429017/347175167*e^8 + 4877811161789/694350334*e^6 - 1546536216302/347175167*e^4 + 303158953205/347175167*e^2 - 16503358648/347175167, 493129375/5554802672*e^19 - 19674243533/5554802672*e^17 + 157137250619/2777401336*e^15 - 2579047470501/5554802672*e^13 + 5817538396993/2777401336*e^11 - 7171010858033/1388700668*e^9 + 35757248436265/5554802672*e^7 - 8649853454079/2777401336*e^5 - 49444868243/694350334*e^3 + 91905374337/1388700668*e, -1, 1467771307/33328816032*e^19 - 59172684265/33328816032*e^17 + 160016328789/5554802672*e^15 - 8075298841849/33328816032*e^13 + 18981194495273/16664408016*e^11 - 25201999019939/8332204008*e^9 + 146103916346389/33328816032*e^7 - 52374198857987/16664408016*e^5 + 1241454420673/1388700668*e^3 - 626146522751/8332204008*e, 66930575/694350334*e^19 - 1356691005/347175167*e^17 + 22186164906/347175167*e^15 - 377456323897/694350334*e^13 + 1804177994907/694350334*e^11 - 4910744173141/694350334*e^9 + 3692849526480/347175167*e^7 - 2804377100998/347175167*e^5 + 875389194339/347175167*e^3 - 77968788302/347175167*e, 779949265/16664408016*e^19 - 31558750867/16664408016*e^17 + 85794264527/2777401336*e^15 - 4363971383611/16664408016*e^13 + 10383573248207/8332204008*e^11 - 14058110805047/4166102004*e^9 + 84182440148983/16664408016*e^7 - 31958112752945/8332204008*e^5 + 840494346599/694350334*e^3 - 407914118033/4166102004*e, 2446911259/16664408016*e^19 - 98954304121/16664408016*e^17 + 268759198881/2777401336*e^15 - 13646966001817/16664408016*e^13 + 32362832575073/8332204008*e^11 - 43517635610921/4166102004*e^9 + 256853749906045/16664408016*e^7 - 94486604357879/8332204008*e^5 + 2329396208491/694350334*e^3 - 1229531770415/4166102004*e, -2013876161/33328816032*e^19 + 81543532235/33328816032*e^17 - 221877153687/5554802672*e^15 + 11298305825003/33328816032*e^13 - 26916925440211/16664408016*e^11 + 36484452053041/8332204008*e^9 - 218580685891679/33328816032*e^7 + 83002106586961/16664408016*e^5 - 2217174857979/1388700668*e^3 + 1363694585605/8332204008*e, -1665654449/16664408016*e^19 + 66593790539/16664408016*e^17 - 177795618271/2777401336*e^15 + 8787828492731/16664408016*e^13 - 19922967461467/8332204008*e^11 + 24708592839421/4166102004*e^9 - 123849481792895/16664408016*e^7 + 29733313052689/8332204008*e^5 + 106922524065/694350334*e^3 - 351707084615/4166102004*e, 175084515/1388700668*e^18 - 7027501045/1388700668*e^16 + 56632138333/694350334*e^14 - 942495883913/1388700668*e^12 + 2175180895905/694350334*e^10 - 2792493001238/347175167*e^8 + 15118926001473/1388700668*e^6 - 4595684486233/694350334*e^4 + 370698746937/347175167*e^2 - 11852918221/347175167, 71952147/1388700668*e^18 - 2904647325/1388700668*e^16 + 23621482815/694350334*e^14 - 399174030549/1388700668*e^12 + 946799726161/694350334*e^10 - 1280760350605/347175167*e^8 + 7722137603785/1388700668*e^6 - 3008525937725/694350334*e^4 + 508376178470/347175167*e^2 - 48281590415/347175167, -49591407/1388700668*e^18 + 1995334567/1388700668*e^16 - 16134149843/694350334*e^14 + 269816011093/1388700668*e^12 - 313585997955/347175167*e^10 + 1627864417013/694350334*e^8 - 4483474465091/1388700668*e^6 + 1406611722403/694350334*e^4 - 122897092319/347175167*e^2 + 3714896253/347175167, -129040279/11109605344*e^19 + 5463803805/11109605344*e^17 - 47389445851/5554802672*e^15 + 874191660029/11109605344*e^13 - 2334715854853/5554802672*e^11 + 3694082725419/2777401336*e^9 - 27147468906841/11109605344*e^7 + 13405446387599/5554802672*e^5 - 1449755802879/1388700668*e^3 + 245263104219/2777401336*e, -1773349555/33328816032*e^19 + 69582797137/33328816032*e^17 - 180458616861/5554802672*e^15 + 8498871104977/33328816032*e^13 - 17630025900329/16664408016*e^11 + 18008186581415/8332204008*e^9 - 46541221795837/33328816032*e^7 - 25524497979973/16664408016*e^5 + 2889897149859/1388700668*e^3 - 2517914687737/8332204008*e, 267977581/694350334*e^18 - 10794132803/694350334*e^16 + 87442619872/347175167*e^14 - 1467168038739/694350334*e^12 + 3431967536198/347175167*e^10 - 9025516928030/347175167*e^8 + 25622860640567/694350334*e^6 - 8744908732908/347175167*e^4 + 2179856202885/347175167*e^2 - 170457752184/347175167, 216371799/5554802672*e^19 - 8756345645/5554802672*e^17 + 71427472011/2777401336*e^15 - 1211328106509/5554802672*e^13 + 2882775718445/2777401336*e^11 - 3902946370839/1388700668*e^9 + 23360977272569/5554802672*e^7 - 8875626723175/2777401336*e^5 + 722840827439/694350334*e^3 - 173370389127/1388700668*e, 265732945/1388700668*e^18 - 10670412859/1388700668*e^16 + 86037092397/694350334*e^14 - 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103778414240963/1388700668*e^6 + 33459978435261/694350334*e^4 - 3539839507222/347175167*e^2 + 244776046471/347175167, -617728173/694350334*e^18 + 12438552036/347175167*e^16 - 201437023196/347175167*e^14 + 3376341491907/694350334*e^12 - 15757284244697/694350334*e^10 + 41194046857767/694350334*e^8 - 28797895998832/347175167*e^6 + 18842868461722/347175167*e^4 - 4058370572020/347175167*e^2 + 275171182994/347175167, 3403795679/8332204008*e^19 - 137376665453/8332204008*e^17 + 372057070013/1388700668*e^15 - 18814099852181/8332204008*e^13 + 44340763071601/4166102004*e^11 - 59062331595895/2083051002*e^9 + 343461814409345/8332204008*e^7 - 123260741166403/4166102004*e^5 + 2919840631526/347175167*e^3 - 1714873027321/2083051002*e, 666337683/1388700668*e^18 - 26866589913/1388700668*e^16 + 217921343621/694350334*e^14 - 3662203280433/1388700668*e^12 + 8581585039635/694350334*e^10 - 11298209566599/347175167*e^8 + 64046392894461/1388700668*e^6 - 21584847098733/694350334*e^4 + 2529629690824/347175167*e^2 - 176951350565/347175167, -9007407/33870748*e^18 + 359811977/33870748*e^16 - 2878553301/16935374*e^14 + 47351886869/33870748*e^12 - 107147636337/16935374*e^10 + 132646556528/8467687*e^8 - 665088295333/33870748*e^6 + 161776418707/16935374*e^4 + 2073068929/8467687*e^2 - 1781816307/8467687, 105022129/2083051002*e^19 - 4099104403/2083051002*e^17 + 10553722755/347175167*e^15 - 492061834747/2083051002*e^13 + 1006871844698/1041525501*e^11 - 2022848292814/1041525501*e^9 + 2612595660619/2083051002*e^7 + 1256653243744/1041525501*e^5 - 550716670359/347175167*e^3 + 185529126476/1041525501*e, -876070791/1388700668*e^18 + 35179781681/1388700668*e^16 - 283712146829/694350334*e^14 + 4727784990825/1388700668*e^12 - 10938303633563/694350334*e^10 + 14116647238575/347175167*e^8 - 77415889598241/1388700668*e^6 + 24477457780115/694350334*e^4 - 2417756025656/347175167*e^2 + 145965897769/347175167, -1048542479/1388700668*e^18 + 42132509089/1388700668*e^16 - 340021844181/694350334*e^14 + 5669559975073/1388700668*e^12 - 13116891726147/694350334*e^10 + 16889749479111/347175167*e^8 - 91745869817825/1388700668*e^6 + 28010121350515/694350334*e^4 - 2315153721828/347175167*e^2 + 104017917337/347175167, -775983155/33328816032*e^19 + 33049879121/33328816032*e^17 - 96124603965/5554802672*e^15 + 5346192845393/33328816032*e^13 - 14303961788137/16664408016*e^11 + 22533643784959/8332204008*e^9 - 163754112631613/33328816032*e^7 + 80611372102651/16664408016*e^5 - 3159713470221/1388700668*e^3 + 3187047951679/8332204008*e, -2220015503/11109605344*e^19 + 87829198997/11109605344*e^17 - 692429368987/5554802672*e^15 + 11123492470949/11109605344*e^13 - 24140194480437/5554802672*e^11 + 27481008166923/2777401336*e^9 - 110755297240769/11109605344*e^7 + 4430180331295/5554802672*e^5 + 5385918995193/1388700668*e^3 - 1628701400901/2777401336*e, 607758215/694350334*e^19 - 12338366284/347175167*e^17 + 404447020199/694350334*e^15 - 3452045106663/694350334*e^13 + 16586665996765/694350334*e^11 - 22764494729217/347175167*e^9 + 69459040088033/694350334*e^7 - 54155005262245/694350334*e^5 + 8922121111680/347175167*e^3 - 839492808142/347175167*e, -1196338517/16664408016*e^19 + 51468624407/16664408016*e^17 - 152066108023/2777401336*e^15 + 8670607467767/16664408016*e^13 - 24129187340623/8332204008*e^11 + 40381799588767/4166102004*e^9 - 319969163934899/16664408016*e^7 + 174651696394513/8332204008*e^5 - 7244776454655/694350334*e^3 + 4980882861001/4166102004*e, -2574014557/5554802672*e^19 + 104252753687/5554802672*e^17 - 851280406301/2777401336*e^15 + 14454220568399/5554802672*e^13 - 34441483465555/2777401336*e^11 + 46661988778837/1388700668*e^9 - 278862939533939/5554802672*e^7 + 104980454020785/2777401336*e^5 - 8177921630611/694350334*e^3 + 1578247896609/1388700668*e] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([19, 19, -w^2 + w + 4])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]