/* 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([-11, -12, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([27, 3, 3]) primes_array = [ [8, 2, 2],\ [11, 11, w],\ [11, 11, -w^2 + 3*w + 7],\ [11, 11, w^2 - 2*w - 8],\ [23, 23, w^2 - 2*w - 7],\ [23, 23, w^2 - 3*w - 8],\ [23, 23, w - 1],\ [27, 3, 3],\ [29, 29, w^2 - 2*w - 5],\ [29, 29, w^2 - 3*w - 10],\ [29, 29, w - 3],\ [31, 31, w^2 - 2*w - 6],\ [31, 31, -w^2 + 3*w + 9],\ [31, 31, w - 2],\ [37, 37, -w^2 + 4*w + 7],\ [43, 43, 2*w^2 - 6*w - 13],\ [43, 43, 2*w^2 - 4*w - 17],\ [43, 43, w^2 - 2*w - 12],\ [47, 47, w^2 - 4*w - 4],\ [47, 47, w^2 - w - 5],\ [47, 47, 2*w^2 - 5*w - 18],\ [73, 73, -2*w^2 + 5*w + 14],\ [73, 73, w^2 - w - 9],\ [73, 73, -w^2 + 4*w + 8],\ [97, 97, -2*w^2 + 5*w + 10],\ [97, 97, 2*w^2 - 3*w - 15],\ [97, 97, w^2 - 4*w - 12],\ [101, 101, -w - 5],\ [101, 101, w^2 - 3*w - 2],\ [101, 101, w^2 - 2*w - 13],\ [103, 103, w^2 - w - 10],\ [103, 103, w^2 - 4*w - 9],\ [103, 103, -2*w^2 + 5*w + 13],\ [125, 5, -5],\ [137, 137, 2*w - 1],\ [137, 137, 2*w^2 - 6*w - 15],\ [137, 137, -2*w^2 + 4*w + 15],\ [149, 149, 3*w^2 - 7*w - 24],\ [149, 149, -2*w^2 + 3*w + 18],\ [149, 149, 3*w^2 - 8*w - 20],\ [179, 179, 2*w^2 - 6*w - 21],\ [179, 179, -w^2 + 4*w + 15],\ [179, 179, -2*w^2 + 5*w + 7],\ [191, 191, -w - 6],\ [191, 191, w^2 - 3*w - 1],\ [191, 191, w^2 - 2*w - 14],\ [193, 193, -3*w^2 + 9*w + 20],\ [193, 193, 2*w^2 - 4*w - 23],\ [193, 193, -3*w^2 + 6*w + 25],\ [199, 199, w^2 - 3*w - 14],\ [199, 199, w^2 - 2*w - 1],\ [199, 199, w - 7],\ [211, 211, 2*w^2 - 3*w - 19],\ [211, 211, w^2 - 5*w - 10],\ [211, 211, 3*w^2 - 8*w - 19],\ [223, 223, 2*w^2 - 6*w - 17],\ [223, 223, 2*w^2 - 4*w - 13],\ [223, 223, 2*w - 3],\ [233, 233, -3*w^2 + 7*w + 23],\ [233, 233, w^2 - 8],\ [233, 233, -2*w^2 + 7*w + 14],\ [251, 251, w^2 - 5*w - 4],\ [251, 251, 3*w^2 - 8*w - 25],\ [251, 251, 2*w^2 - 3*w - 13],\ [269, 269, w^2 - 5*w - 12],\ [269, 269, 2*w^2 - 3*w - 21],\ [269, 269, 3*w^2 - 8*w - 17],\ [307, 307, w^2 - 5*w - 3],\ [307, 307, 2*w^2 - 3*w - 12],\ [307, 307, -3*w^2 + 8*w + 26],\ [343, 7, -7],\ [347, 347, 2*w^2 - 3*w - 10],\ [347, 347, 3*w^2 - 8*w - 28],\ [347, 347, w^2 - 5*w - 1],\ [359, 359, 3*w^2 - 5*w - 23],\ [359, 359, 2*w^2 - 7*w - 23],\ [359, 359, -3*w^2 + 7*w + 14],\ [397, 397, 2*w^2 - 8*w - 7],\ [397, 397, 4*w^2 - 10*w - 37],\ [397, 397, 2*w^2 - 2*w - 9],\ [401, 401, 2*w^2 - 7*w - 16],\ [401, 401, 3*w^2 - 7*w - 21],\ [401, 401, w^2 - 10],\ [421, 421, 3*w^2 - 5*w - 29],\ [421, 421, w^2 - 6*w - 12],\ [421, 421, -4*w^2 + 11*w + 24],\ [433, 433, w^2 - 6*w - 15],\ [433, 433, 3*w^2 - 5*w - 32],\ [433, 433, -4*w^2 + 11*w + 21],\ [443, 443, 5*w^2 - 13*w - 36],\ [443, 443, -2*w^2 + 9*w + 15],\ [443, 443, 5*w^2 - 12*w - 40],\ [467, 467, w^2 - 6*w - 13],\ [467, 467, 3*w^2 - 5*w - 30],\ [467, 467, -4*w^2 + 11*w + 23],\ [487, 487, -3*w^2 + 10*w + 20],\ [487, 487, -4*w^2 + 9*w + 32],\ [487, 487, w^2 + w - 7],\ [491, 491, -w - 8],\ [491, 491, -w^2 + 3*w - 1],\ [491, 491, w^2 - 2*w - 16],\ [541, 541, 3*w^2 - 7*w - 16],\ [541, 541, 2*w^2 - 7*w - 21],\ [541, 541, w^2 - 15],\ [547, 547, 2*w^2 - 4*w - 25],\ [547, 547, 2*w^2 - 6*w - 5],\ [547, 547, -2*w - 9],\ [563, 563, 3*w^2 - 7*w - 18],\ [563, 563, 2*w^2 - 7*w - 19],\ [563, 563, w^2 - 13],\ [569, 569, w^2 - 14],\ [569, 569, 3*w^2 - 7*w - 17],\ [569, 569, 2*w^2 - 7*w - 20],\ [593, 593, w^2 - w - 18],\ [593, 593, w^2 - 4*w - 17],\ [593, 593, 2*w^2 - 5*w - 5],\ [619, 619, -5*w^2 + 14*w + 30],\ [619, 619, 4*w^2 - 7*w - 38],\ [619, 619, -5*w^2 + 11*w + 42],\ [643, 643, w^2 + w - 8],\ [643, 643, -3*w^2 + 10*w + 21],\ [643, 643, -4*w^2 + 9*w + 31],\ [677, 677, -2*w^2 + 7*w + 27],\ [677, 677, -3*w^2 + 7*w + 10],\ [677, 677, 3*w^2 - 9*w - 31],\ [709, 709, w^2 - 7*w - 14],\ [709, 709, 4*w^2 - 7*w - 39],\ [709, 709, 5*w^2 - 14*w - 29],\ [739, 739, -5*w^2 + 14*w + 27],\ [739, 739, w^2 - 7*w - 16],\ [739, 739, 4*w^2 - 7*w - 41],\ [751, 751, 4*w^2 - 9*w - 43],\ [751, 751, 4*w^2 - 7*w - 40],\ [751, 751, -3*w^2 + 10*w + 9],\ [769, 769, w^2 - 3*w - 17],\ [769, 769, -w^2 + 2*w - 2],\ [769, 769, w - 10],\ [787, 787, 3*w^2 - 6*w - 19],\ [787, 787, 3*w - 5],\ [787, 787, 3*w^2 - 9*w - 26],\ [857, 857, 3*w^2 - 9*w - 28],\ [857, 857, 3*w^2 - 6*w - 17],\ [857, 857, 3*w - 7],\ [859, 859, 4*w^2 - 11*w - 40],\ [859, 859, 3*w^2 - 4*w - 19],\ [859, 859, 5*w^2 - 13*w - 42],\ [877, 877, 6*w^2 - 14*w - 49],\ [877, 877, -3*w^2 + 13*w + 21],\ [877, 877, -4*w^2 + 14*w + 25],\ [887, 887, 2*w^2 - 5*w - 4],\ [887, 887, w^2 - w - 19],\ [887, 887, w^2 - 4*w - 18],\ [911, 911, 2*w^2 - 2*w - 21],\ [911, 911, 4*w^2 - 10*w - 25],\ [911, 911, 2*w^2 - 8*w - 19],\ [919, 919, 4*w^2 - 11*w - 34],\ [919, 919, 3*w^2 - 5*w - 19],\ [919, 919, w^2 - 6*w - 2],\ [991, 991, -w - 10],\ [991, 991, -w^2 + 3*w - 3],\ [991, 991, w^2 - 2*w - 18]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^12 + 11*x^11 + 10*x^10 - 249*x^9 - 854*x^8 + 107*x^7 + 4717*x^6 + 8436*x^5 + 5004*x^4 - 1184*x^3 - 2740*x^2 - 1136*x - 152 K. = NumberField(heckePol) hecke_eigenvalues_array = [77192061/357733226*e^11 + 932339133/357733226*e^10 + 1466204189/357733226*e^9 - 10223694147/178866613*e^8 - 42653289602/178866613*e^7 - 4706911272/178866613*e^6 + 470915801833/357733226*e^5 + 838747271819/357733226*e^4 + 368798894273/357733226*e^3 - 126757523947/178866613*e^2 - 125330711839/178866613*e - 26004354115/178866613, e, -47267661267/30765057436*e^11 - 464746544673/30765057436*e^10 + 383280206/178866613*e^9 + 11653733965969/30765057436*e^8 + 13389077310487/15382528718*e^7 - 35319960419671/30765057436*e^6 - 179762795585607/30765057436*e^5 - 96334188640623/15382528718*e^4 - 12900751838863/15382528718*e^3 + 18703213719086/7691264359*e^2 + 11132087374731/7691264359*e + 1859190266088/7691264359, 8418523335/30765057436*e^11 + 89656954713/30765057436*e^10 + 317373105/178866613*e^9 - 2100663479581/30765057436*e^8 - 3240562299573/15382528718*e^7 + 2767214202571/30765057436*e^6 + 38392937928535/30765057436*e^5 + 29819273237287/15382528718*e^4 + 12430415345207/15382528718*e^3 - 4510303790578/7691264359*e^2 - 4577816682858/7691264359*e - 973245229152/7691264359, -933651515/30765057436*e^11 - 24470588261/30765057436*e^10 - 800214247/178866613*e^9 + 353543499319/30765057436*e^8 + 2097361112779/15382528718*e^7 + 5261715576019/30765057436*e^6 - 19081872800115/30765057436*e^5 - 12562616103597/7691264359*e^4 - 15035339845115/15382528718*e^3 + 3242839671684/7691264359*e^2 + 4255254178864/7691264359*e + 920991632732/7691264359, -23003224009/30765057436*e^11 - 251261747345/30765057436*e^10 - 1142194473/178866613*e^9 + 5869817193415/30765057436*e^8 + 9529691348983/15382528718*e^7 - 7304818853025/30765057436*e^6 - 112974988533849/30765057436*e^5 - 42608728685669/7691264359*e^4 - 30738412009321/15382528718*e^3 + 13674754278213/7691264359*e^2 + 11946975539197/7691264359*e + 2407995771748/7691264359, 5018976553/7691264359*e^11 + 99983395359/15382528718*e^10 - 33305100/178866613*e^9 - 2490312489069/15382528718*e^8 - 2994979263955/7691264359*e^7 + 7136701120195/15382528718*e^6 + 19840030133061/7691264359*e^5 + 22077123153624/7691264359*e^4 + 3165001004616/7691264359*e^3 - 8665105906957/7691264359*e^2 - 5001589497041/7691264359*e - 796098558284/7691264359, -1, -16654376149/7691264359*e^11 - 167443933032/7691264359*e^10 - 261040081/178866613*e^9 + 4128974501256/7691264359*e^8 + 10336519526055/7691264359*e^7 - 10812970326395/7691264359*e^6 - 66985333497174/7691264359*e^5 - 80127757444277/7691264359*e^4 - 17643347894116/7691264359*e^3 + 29140768440660/7691264359*e^2 + 20217745656885/7691264359*e + 3682222504136/7691264359, 31419648005/30765057436*e^11 + 321785575919/30765057436*e^10 + 454034228/178866613*e^9 - 7807583437619/30765057436*e^8 - 10476204218729/15382528718*e^7 + 17297604473833/30765057436*e^6 + 131365203046033/30765057436*e^5 + 86547726005531/15382528718*e^4 + 26685930132953/15382528718*e^3 - 14332733017856/7691264359*e^2 - 12071480762798/7691264359*e - 2442124217920/7691264359, 43157182219/30765057436*e^11 + 441877343833/30765057436*e^10 + 571699110/178866613*e^9 - 10790773253085/30765057436*e^8 - 14325856563175/15382528718*e^7 + 25654344925775/30765057436*e^6 + 182397812353223/30765057436*e^5 + 113979693704463/15382528718*e^4 + 27056824393383/15382528718*e^3 - 20524062068952/7691264359*e^2 - 14348598288415/7691264359*e - 2618405918712/7691264359, -27064259193/30765057436*e^11 - 269256624391/30765057436*e^10 + 27837035/178866613*e^9 + 6662063052303/30765057436*e^8 + 8074201210371/15382528718*e^7 - 18007400272901/30765057436*e^6 - 105142985872909/30765057436*e^5 - 62456768969349/15382528718*e^4 - 15212248195283/15382528718*e^3 + 10840082518902/7691264359*e^2 + 8296791380585/7691264359*e + 1669328009841/7691264359, -4066772379/15382528718*e^11 - 44397994063/15382528718*e^10 - 416043360/178866613*e^9 + 1027873623571/15382528718*e^8 + 1693934189573/7691264359*e^7 - 1029757529181/15382528718*e^6 - 19840091214203/15382528718*e^5 - 15779420201140/7691264359*e^4 - 6303481584674/7691264359*e^3 + 5040376515664/7691264359*e^2 + 4647933002991/7691264359*e + 920825938537/7691264359, 83463007063/30765057436*e^11 + 817738988465/30765057436*e^10 - 821956617/178866613*e^9 - 20545095857805/30765057436*e^8 - 23309856478123/15382528718*e^7 + 63230783034179/30765057436*e^6 + 314493803071943/30765057436*e^5 + 166474346880115/15382528718*e^4 + 20562502920561/15382528718*e^3 - 32722027979576/7691264359*e^2 - 19084487336942/7691264359*e - 3125745742495/7691264359, 280611646/178866613*e^11 + 2672595209/178866613*e^10 - 1210162942/178866613*e^9 - 68763288130/178866613*e^8 - 137764963821/178866613*e^7 + 250952719203/178866613*e^6 + 986457178899/178866613*e^5 + 842543459401/178866613*e^4 - 84380330745/178866613*e^3 - 391664394070/178866613*e^2 - 142785184962/178866613*e - 14065422636/178866613, 10317947419/7691264359*e^11 + 99787760695/7691264359*e^10 - 1376300135/357733226*e^9 - 2531096016235/7691264359*e^8 - 10887113213545/15382528718*e^7 + 8375657124241/7691264359*e^6 + 75037848501947/15382528718*e^5 + 36891295728316/7691264359*e^4 + 2510517786233/7691264359*e^3 - 14906771709937/7691264359*e^2 - 7991418024663/7691264359*e - 1262568899141/7691264359, 73736320561/30765057436*e^11 + 710192290479/30765057436*e^10 - 1422049551/178866613*e^9 - 18120440024141/30765057436*e^8 - 19059544243341/15382528718*e^7 + 62522985090551/30765057436*e^6 + 266926335481261/30765057436*e^5 + 61770764058620/7691264359*e^4 - 1871217760395/15382528718*e^3 - 26981510750387/7691264359*e^2 - 11705600442291/7691264359*e - 1470111604633/7691264359, -60002934045/30765057436*e^11 - 604123270203/30765057436*e^10 - 495603123/357733226*e^9 + 14929848974845/30765057436*e^8 + 9342632562147/7691264359*e^7 - 39912065161563/30765057436*e^6 - 243560612199843/30765057436*e^5 - 71185188124392/7691264359*e^4 - 26198804794511/15382528718*e^3 + 27052268679252/7691264359*e^2 + 16929336769778/7691264359*e + 2769898997383/7691264359, -42929236177/30765057436*e^11 - 405327208351/30765057436*e^10 + 1256881351/178866613*e^9 + 10481886596025/30765057436*e^8 + 10121387064631/15382528718*e^7 - 39516778893299/30765057436*e^6 - 147059520800301/30765057436*e^5 - 29859989800716/7691264359*e^4 + 8634607370065/15382528718*e^3 + 14204336869478/7691264359*e^2 + 4772527551361/7691264359*e + 378856796940/7691264359, -2703751216/7691264359*e^11 - 55899820841/15382528718*e^10 - 266619515/178866613*e^9 + 654608984409/7691264359*e^8 + 1903445011816/7691264359*e^7 - 853022751813/7691264359*e^6 - 11162311282172/7691264359*e^5 - 36990792295349/15382528718*e^4 - 9569241204257/7691264359*e^3 + 4846373439771/7691264359*e^2 + 6409975892895/7691264359*e + 1483026898068/7691264359, 40229819545/30765057436*e^11 + 414408421309/30765057436*e^10 + 730268358/178866613*e^9 - 9986384490605/30765057436*e^8 - 13737077414629/15382528718*e^7 + 20387381357323/30765057436*e^6 + 170283048206593/30765057436*e^5 + 116683488241475/15382528718*e^4 + 39218611083275/15382528718*e^3 - 18938512689539/7691264359*e^2 - 16603009273814/7691264359*e - 3389891944604/7691264359, 34852917183/30765057436*e^11 + 313761820521/30765057436*e^10 - 1830445623/178866613*e^9 - 8388645835853/30765057436*e^8 - 6382424775461/15382528718*e^7 + 38109093297627/30765057436*e^6 + 103994504210467/30765057436*e^5 + 25102012024653/15382528718*e^4 - 22332848369675/15382528718*e^3 - 8783376814919/7691264359*e^2 + 262176755190/7691264359*e + 660848158835/7691264359, 55313524137/15382528718*e^11 + 272299896593/7691264359*e^10 - 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6668797810/178866613*e^9 - 2008674060221/15382528718*e^8 + 4512377111983/7691264359*e^7 + 31788042982833/15382528718*e^6 - 23891050849181/15382528718*e^5 - 146695569625521/15382528718*e^4 - 57316462118903/7691264359*e^3 + 14472005381080/7691264359*e^2 + 27520207978992/7691264359*e + 6782935306962/7691264359, 37116059665/7691264359*e^11 + 665590642343/15382528718*e^10 - 16162330137/357733226*e^9 - 17847958705919/15382528718*e^8 - 26561942279871/15382528718*e^7 + 82273399981621/15382528718*e^6 + 219297520036801/15382528718*e^5 + 49419927804053/7691264359*e^4 - 51720602330469/7691264359*e^3 - 37738788828107/7691264359*e^2 + 3430215638250/7691264359*e + 3540428327522/7691264359] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([27, 3, 3])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]