/* 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([-55, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([3, 3, w + 1]) primes_array = [ [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 = [ZF.ideal(I) for I in primes_array] heckePol = x^16 + 60*x^14 + 1346*x^12 + 14040*x^10 + 67921*x^8 + 129884*x^6 + 74064*x^4 + 13504*x^2 + 256 K. = NumberField(heckePol) hecke_eigenvalues_array = [309706439/197457236608*e^15 + 4608179083/49364309152*e^13 + 203959158615/98728618304*e^11 + 518614665811/24682154576*e^9 + 18983190523319/197457236608*e^7 + 7648327613915/49364309152*e^5 + 385740600025/12341077288*e^3 - 13299755859/1542634661*e, -11523/1313824*e^15 - 686685/1313824*e^13 - 951877/82114*e^11 - 9723097/82114*e^9 - 719100509/1313824*e^7 - 1202061251/1313824*e^5 - 11145560/41057*e^3 + 59629/41057*e, -309706439/197457236608*e^15 - 4608179083/49364309152*e^13 - 203959158615/98728618304*e^11 - 518614665811/24682154576*e^9 - 18983190523319/197457236608*e^7 - 7648327613915/49364309152*e^5 - 385740600025/12341077288*e^3 + 14842390520/1542634661*e, -613193579/98728618304*e^14 - 9144636301/24682154576*e^12 - 406310234195/49364309152*e^10 - 1040689053985/12341077288*e^8 - 38733324226299/98728618304*e^6 - 16508446158437/24682154576*e^4 - 345272360250/1542634661*e^2 - 9640594767/1542634661, 739098001/493643091520*e^14 + 11048649721/123410772880*e^12 + 492864465101/246821545760*e^10 + 158939488012/7713173305*e^8 + 48055823468073/493643091520*e^6 + 21418363384359/123410772880*e^4 + 1129134042769/15426346610*e^2 + 38024452499/7713173305, 37559336411/987286183040*e^15 + 69925647997/30852693220*e^13 + 24802398249611/493643091520*e^11 + 3955102905131/7713173305*e^9 + 2335562296423083/987286183040*e^7 + 121250306349703/30852693220*e^5 + 68713781579563/61705386440*e^3 - 179638820343/7713173305*e, 19900248969/987286183040*e^15 + 296626654249/246821545760*e^13 + 13168913801609/493643091520*e^11 + 33681272516329/123410772880*e^9 + 1249771919509177/987286183040*e^7 + 527925246430761/246821545760*e^5 + 42094866345327/61705386440*e^3 + 116306484003/7713173305*e, -19900248969/987286183040*e^15 - 296626654249/246821545760*e^13 - 13168913801609/493643091520*e^11 - 33681272516329/123410772880*e^9 - 1249771919509177/987286183040*e^7 - 527925246430761/246821545760*e^5 - 42094866345327/61705386440*e^3 - 116306484003/7713173305*e, 30194153467/493643091520*e^15 + 899990912019/246821545760*e^13 + 19972822495387/246821545760*e^11 + 102120577526919/123410772880*e^9 + 1892814668116091/493643091520*e^7 + 1594446324255451/246821545760*e^5 + 15597431127649/7713173305*e^3 + 215905589138/7713173305*e, -1348130443/246821545760*e^14 - 20065153503/61705386440*e^12 - 888635749153/123410772880*e^10 - 4525560906431/61705386440*e^8 - 83160808349639/246821545760*e^6 - 4261344772794/7713173305*e^4 - 2192535373039/15426346610*e^2 + 20971276491/7713173305, -2291962289/123410772880*e^14 - 34171642499/30852693220*e^12 - 758827409637/30852693220*e^10 - 15536822670301/61705386440*e^8 - 144333491499947/123410772880*e^6 - 122508809413287/61705386440*e^4 - 10121470910839/15426346610*e^2 - 161712757149/7713173305, 47077655/1542634661*e^15 + 89779139831/49364309152*e^13 + 497839792641/12341077288*e^11 + 10171772451181/24682154576*e^9 + 23514015560645/12341077288*e^7 + 157278744005731/49364309152*e^5 + 2907286492725/3085269322*e^3 - 14345137561/1542634661*e, -290131281/49364309152*e^15 - 8629861611/24682154576*e^13 - 190847892775/24682154576*e^11 - 484751531589/6170538644*e^9 - 17712352789869/49364309152*e^7 - 14243936079161/24682154576*e^5 - 389031902189/3085269322*e^3 + 10801580202/1542634661*e, -14393896651/197457236608*e^15 - 214634309891/49364309152*e^13 - 9535175561123/98728618304*e^11 - 24418403404905/24682154576*e^9 - 908686079737995/197457236608*e^7 - 387464679928047/49364309152*e^5 - 32891696948741/12341077288*e^3 - 164339698904/1542634661*e, 14327183695/197457236608*e^15 + 106749886195/24682154576*e^13 + 9474305202215/98728618304*e^11 + 1513285416205/1542634661*e^9 + 896875588432655/197457236608*e^7 + 188418086654301/24682154576*e^5 + 29016858604309/12341077288*e^3 + 27540971699/1542634661*e, -3379136249/98728618304*e^14 - 50369926127/24682154576*e^12 - 2236317229233/49364309152*e^10 - 5720179933851/12341077288*e^8 - 212287321917097/98728618304*e^6 - 89710309134455/24682154576*e^4 - 1790169374014/1542634661*e^2 - 33844657779/1542634661, 2252603969/197457236608*e^15 + 16784664869/24682154576*e^13 + 1489807213169/98728618304*e^11 + 1903971504037/12341077288*e^9 + 141090951049745/197457236608*e^7 + 29650160812085/24682154576*e^5 + 4543051356795/12341077288*e^3 + 4978133793/1542634661*e, 4881627333/197457236608*e^15 + 18220344015/12341077288*e^13 + 3244443943861/98728618304*e^11 + 521271089224/1542634661*e^9 + 312975583928149/197457236608*e^7 + 34219369370425/12341077288*e^5 + 13346681658047/12341077288*e^3 + 131912604271/1542634661*e, 43753465191/987286183040*e^15 + 162892191409/61705386440*e^13 + 28881581421911/493643091520*e^11 + 18413484949579/30852693220*e^9 + 2715226106889463/987286183040*e^7 + 280742250768981/61705386440*e^5 + 76428593580063/61705386440*e^3 - 445633937523/7713173305*e, -18549205529/246821545760*e^15 - 34569471773/7713173305*e^13 - 12282966685829/123410772880*e^11 - 31440029252809/30852693220*e^9 - 1168676076489697/246821545760*e^7 - 248228937148913/30852693220*e^5 - 41041406739677/15426346610*e^3 - 587067966252/7713173305*e, -6075793829/123410772880*e^14 - 181159134963/61705386440*e^12 - 4022554108069/61705386440*e^10 - 10294138958659/15426346610*e^8 - 382451815277037/123410772880*e^6 - 324286452391297/61705386440*e^4 - 26360710739809/15426346610*e^2 - 327665811259/7713173305, -1482664013/246821545760*e^14 - 22096018483/61705386440*e^12 - 980650603873/123410772880*e^10 - 1253258178599/15426346610*e^8 - 92848144172469/246821545760*e^6 - 38954917402297/61705386440*e^4 - 1473244054422/7713173305*e^2 - 81648824564/7713173305, -2154119509/98728618304*e^14 - 32129887219/24682154576*e^12 - 1428062958573/49364309152*e^10 - 3660520345999/12341077288*e^8 - 136515517676229/98728618304*e^6 - 58607904352219/24682154576*e^4 - 1288593408454/1542634661*e^2 - 51435448223/1542634661, 365401507/12341077288*e^14 + 5446659873/3085269322*e^12 + 241821640379/6170538644*e^10 + 618589683986/1542634661*e^8 + 22963971273683/12341077288*e^6 + 9717893239309/3085269322*e^4 + 1571285177616/1542634661*e^2 + 47801972261/1542634661, 11320007613/197457236608*e^15 + 168629430279/49364309152*e^13 + 7478733383877/98728618304*e^11 + 19091488329321/24682154576*e^9 + 705399893761725/197457236608*e^7 + 293935267144683/49364309152*e^5 + 21258283842315/12341077288*e^3 - 57969814062/1542634661*e, 341172539/12341077288*e^15 + 10143867365/6170538644*e^13 + 448332347387/12341077288*e^11 + 4548520630719/12341077288*e^9 + 2588007758403/1542634661*e^7 + 32892726459021/12341077288*e^5 + 769150666958/1542634661*e^3 - 146697812348/1542634661*e, -20273586989/246821545760*e^14 - 302143331109/61705386440*e^12 - 13410417178039/123410772880*e^10 - 68567588963163/61705386440*e^8 - 1270969542427857/246821545760*e^6 - 267717550061363/30852693220*e^4 - 41961725022307/15426346610*e^2 - 321044678097/7713173305, 5797521451/246821545760*e^14 + 86377573521/61705386440*e^12 + 3831930532981/123410772880*e^10 + 19574911768767/61705386440*e^8 + 362140455526703/246821545760*e^6 + 75867704187847/30852693220*e^4 + 11560053548073/15426346610*e^2 + 38121379603/7713173305, 328903713/30852693220*e^14 + 19630173377/30852693220*e^12 + 872912139507/61705386440*e^10 + 8955701763643/61705386440*e^8 + 41784421751253/61705386440*e^6 + 71863847066521/61705386440*e^4 + 6424620188067/15426346610*e^2 + 209255229697/7713173305, -46548731387/493643091520*e^14 - 693856005447/123410772880*e^12 - 30805347379207/246821545760*e^10 - 39397403315451/30852693220*e^8 - 2924270282333811/493643091520*e^6 - 1236016257762853/123410772880*e^4 - 49515479253143/15426346610*e^2 - 568324360293/7713173305, 31253730469/493643091520*e^14 + 465858890759/123410772880*e^12 + 20682242314069/246821545760*e^10 + 52898830792649/61705386440*e^8 + 1962973756703717/493643091520*e^6 + 829335433456091/123410772880*e^4 + 33094779522341/15426346610*e^2 + 375305210911/7713173305, 3335576511/123410772880*e^14 + 49728544111/30852693220*e^12 + 2208404926401/61705386440*e^10 + 5651353846231/15426346610*e^8 + 209941807408803/123410772880*e^6 + 89008965489999/30852693220*e^4 + 7286137184708/7713173305*e^2 + 216790385656/7713173305, -2252603969/98728618304*e^15 - 16784664869/12341077288*e^13 - 1489807213169/49364309152*e^11 - 1903971504037/6170538644*e^9 - 141090951049745/98728618304*e^7 - 29650160812085/12341077288*e^5 - 4543051356795/6170538644*e^3 - 9956267586/1542634661*e, 7747474233/49364309152*e^15 + 231032685735/24682154576*e^13 + 5131101287585/24682154576*e^11 + 26273242087509/12341077288*e^9 + 488563341334425/49364309152*e^7 + 415843890247555/24682154576*e^5 + 17431317551437/3085269322*e^3 + 332314966026/1542634661*e, 13324296949/987286183040*e^15 + 12408407229/15426346610*e^13 + 8809165100069/493643091520*e^11 + 1406910627804/7713173305*e^9 + 833914345791717/987286183040*e^7 + 21932406092211/15426346610*e^5 + 27639990570837/61705386440*e^3 + 37592232103/7713173305*e, 1654548301/987286183040*e^15 + 24381343341/246821545760*e^13 + 1061881577901/493643091520*e^11 + 2621368826901/123410772880*e^9 + 89854057976573/987286183040*e^7 + 28954887820029/246821545760*e^5 - 1276359848277/61705386440*e^3 + 89820867407/7713173305*e, -298654807/98728618304*e^14 - 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8352471231/3085269322*e^13 - 5974780640409/98728618304*e^11 - 3870255374819/6170538644*e^9 - 591086240395049/197457236608*e^7 - 34032710547465/6170538644*e^5 - 33873981659691/12341077288*e^3 - 575581193135/1542634661*e, -438212671/1542634661*e^15 - 835956584749/49364309152*e^13 - 2319008244983/6170538644*e^11 - 94859487221813/24682154576*e^9 - 109897118876817/6170538644*e^7 - 1481423162052253/49364309152*e^5 - 28992069937535/3085269322*e^3 - 203825950025/1542634661*e, -30690094707/98728618304*e^15 - 914778623127/49364309152*e^13 - 20301258195491/49364309152*e^11 - 103803094954775/24682154576*e^9 - 1924172991273027/98728618304*e^7 - 1621494511551991/49364309152*e^5 - 63713636592227/6170538644*e^3 - 261364934005/1542634661*e, -4759747111/246821545760*e^14 - 70933925021/61705386440*e^12 - 3148621629531/123410772880*e^10 - 8052476853381/30852693220*e^8 - 298837568477743/246821545760*e^6 - 126209156787069/61705386440*e^4 - 4877480492649/7713173305*e^2 + 124230831562/7713173305, -24745321247/493643091520*e^14 - 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401804415129/49364309152*e^13 - 8935869528313/49364309152*e^11 - 45867091492821/24682154576*e^9 - 857423460317337/98728618304*e^7 - 741407553592753/49364309152*e^5 - 8479387862821/1542634661*e^3 - 483291890470/1542634661*e, -789265127/12341077288*e^14 - 11755353045/3085269322*e^12 - 521218254607/6170538644*e^10 - 1329994894936/1542634661*e^8 - 49101540654967/12341077288*e^6 - 20420834755049/3085269322*e^4 - 2954828581968/1542634661*e^2 - 21843769793/1542634661, -51007376223/197457236608*e^15 - 760022389245/49364309152*e^13 - 33721179320359/98728618304*e^11 - 86150574060235/24682154576*e^9 - 3188903913307711/197457236608*e^7 - 1336872916965625/49364309152*e^5 - 101380059598937/12341077288*e^3 - 5237818822/1542634661*e, 7887213849/49364309152*e^15 + 470867789637/49364309152*e^13 + 5237603752323/24682154576*e^11 + 53803475219663/24682154576*e^9 + 503669961985901/49364309152*e^7 + 875456944553049/49364309152*e^5 + 10320624164361/1542634661*e^3 + 712250710817/1542634661*e, 7585421507/123410772880*e^15 + 906895381981/246821545760*e^13 + 5054840463667/61705386440*e^11 + 104264563887361/123410772880*e^9 + 492225430546111/123410772880*e^7 + 1755521078728069/246821545760*e^5 + 94362702069379/30852693220*e^3 + 2531987815432/7713173305*e, 249234286077/493643091520*e^15 + 7429896656119/246821545760*e^13 + 164923953723197/246821545760*e^11 + 843608298272299/123410772880*e^9 + 15650888812237341/493643091520*e^7 + 13222133224080191/246821545760*e^5 + 263650717808233/15426346610*e^3 + 2643672741438/7713173305*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([3, 3, w + 1])] = 11523/1313824*e^15 + 686685/1313824*e^13 + 951877/82114*e^11 + 9723097/82114*e^9 + 719100509/1313824*e^7 + 1202061251/1313824*e^5 + 11145560/41057*e^3 - 59629/41057*e # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]