/* 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![-40, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, w + 6], [2, 2, -w + 7], [5, 5, -6*w - 35], [5, 5, -6*w + 41], [7, 7, 32*w - 219], [9, 3, 3], [17, 17, 2*w - 13], [17, 17, 2*w + 11], [19, 19, 4*w + 23], [19, 19, 4*w - 27], [23, 23, 58*w - 397], [29, 29, 20*w - 137], [29, 29, -20*w - 117], [61, 61, 2*w - 11], [61, 61, -2*w - 9], [71, 71, -10*w - 59], [71, 71, 10*w - 69], [83, 83, -44*w + 301], [83, 83, 44*w + 257], [89, 89, -70*w - 409], [89, 89, 70*w - 479], [97, 97, 2*w - 9], [97, 97, -2*w - 7], [103, 103, -16*w - 93], [103, 103, 16*w - 109], [121, 11, -11], [127, 127, 34*w + 199], [127, 127, 34*w - 233], [151, 151, 6*w - 43], [151, 151, -6*w - 37], [157, 157, 2*w - 3], [157, 157, -2*w - 1], [163, 163, 2*w - 19], [163, 163, -2*w - 17], [169, 13, -13], [179, 179, -110*w - 643], [179, 179, 110*w - 753], [181, 181, 10*w + 57], [181, 181, 10*w - 67], [193, 193, 72*w - 493], [193, 193, -72*w - 421], [197, 197, 4*w + 27], [197, 197, 4*w - 31], [199, 199, 56*w - 383], [199, 199, -56*w - 327], [211, 211, -14*w - 83], [211, 211, 14*w - 97], [227, 227, -28*w - 163], [227, 227, -28*w + 191], [229, 229, -134*w - 783], [229, 229, 134*w - 917], [233, 233, 8*w - 57], [233, 233, -8*w - 49], [239, 239, 2*w - 21], [239, 239, -2*w - 19], [241, 241, -82*w + 561], [241, 241, 82*w + 479], [251, 251, 108*w + 631], [251, 251, 108*w - 739], [277, 277, -36*w + 247], [277, 277, -36*w - 211], [283, 283, -4*w - 17], [283, 283, 4*w - 21], [293, 293, 6*w + 31], [293, 293, -6*w + 37], [313, 313, -26*w - 151], [313, 313, 26*w - 177], [317, 317, 4*w - 33], [317, 317, -4*w - 29], [331, 331, 10*w - 71], [331, 331, -10*w - 61], [347, 347, -26*w - 153], [347, 347, 26*w - 179], [367, 367, 8*w + 43], [367, 367, -8*w + 51], [383, 383, 24*w - 163], [383, 383, -24*w - 139], [419, 419, -4*w - 13], [419, 419, 4*w - 17], [433, 433, 94*w + 549], [433, 433, -94*w + 643], [443, 443, 38*w - 261], [443, 443, 38*w + 223], [449, 449, 8*w - 59], [449, 449, -8*w - 51], [463, 463, -162*w - 947], [463, 463, -162*w + 1109], [467, 467, -12*w + 79], [467, 467, 12*w + 67], [479, 479, -120*w + 821], [479, 479, 120*w + 701], [487, 487, 6*w + 41], [487, 487, 6*w - 47], [491, 491, 74*w + 433], [491, 491, 74*w - 507], [499, 499, -50*w + 343], [499, 499, -50*w - 293], [503, 503, 16*w + 91], [503, 503, 16*w - 107], [521, 521, 38*w - 259], [521, 521, -38*w - 221], [523, 523, -4*w - 9], [523, 523, 4*w - 13], [541, 541, -20*w - 119], [541, 541, 20*w - 139], [547, 547, 226*w - 1547], [547, 547, -354*w + 2423], [563, 563, 4*w - 11], [563, 563, -4*w - 7], [599, 599, -10*w - 63], [599, 599, 10*w - 73], [619, 619, 4*w - 7], [619, 619, -4*w - 3], [643, 643, 4*w - 3], [643, 643, 4*w - 1], [653, 653, -188*w - 1099], [653, 653, -188*w + 1287], [661, 661, -10*w + 63], [661, 661, 10*w + 53], [673, 673, -8*w - 53], [673, 673, 8*w - 61], [677, 677, -378*w + 2587], [677, 677, -262*w + 1793], [683, 683, -22*w - 131], [683, 683, 22*w - 153], [727, 727, 8*w - 47], [727, 727, 8*w + 39], [733, 733, 34*w - 231], [733, 733, -34*w - 197], [739, 739, 2*w - 31], [739, 739, -2*w - 29], [769, 769, -50*w - 291], [769, 769, 50*w - 341], [773, 773, 6*w - 29], [773, 773, -6*w - 23], [787, 787, -92*w + 629], [787, 787, 92*w + 537], [797, 797, -158*w + 1081], [797, 797, 158*w + 923], [809, 809, -32*w - 189], [809, 809, 32*w - 221], [821, 821, -316*w - 1847], [821, 821, 316*w - 2163], [823, 823, -42*w - 247], [823, 823, 42*w - 289], [839, 839, 64*w + 373], [839, 839, 64*w - 437], [863, 863, 2*w - 33], [863, 863, -2*w - 31], [877, 877, 4*w - 41], [877, 877, -4*w - 37], [881, 881, 30*w - 203], [881, 881, -30*w - 173], [883, 883, 18*w - 127], [883, 883, -18*w - 109], [937, 937, -118*w - 689], [937, 937, 118*w - 807], [941, 941, 18*w + 101], [941, 941, -18*w + 119], [947, 947, 14*w - 101], [947, 947, -14*w - 87], [961, 31, -31], [967, 967, 54*w - 371], [967, 967, 54*w + 317], [971, 971, -20*w + 133], [971, 971, 20*w + 113], [983, 983, 24*w - 161], [983, 983, 24*w + 137], [991, 991, -126*w + 863], [991, 991, -126*w - 737]]; primes := [ideal : I in primesArray]; heckePol := x^12 - 44*x^10 + 696*x^8 - 5024*x^6 + 17152*x^4 - 25344*x^2 + 10816; K := NumberField(heckePol); heckeEigenvaluesArray := [0, -37/6848*e^10 + 765/3424*e^8 - 2711/856*e^6 + 15989/856*e^4 - 18125/428*e^2 + 2565/107, e, 205/22256*e^11 - 1043/2782*e^9 + 7208/1391*e^7 - 163945/5564*e^5 + 178337/2782*e^3 - 50264/1391*e, -193/89024*e^11 + 3921/44512*e^9 - 6725/5564*e^7 + 74443/11128*e^5 - 73971/5564*e^3 + 9062/1391*e, 19/3424*e^10 - 777/3424*e^8 + 5369/1712*e^6 - 3725/214*e^4 + 14465/428*e^2 - 2397/214, -905/89024*e^11 + 9175/22256*e^9 - 126231/22256*e^7 + 357519/11128*e^5 - 195271/2782*e^3 + 118175/2782*e, -1121/89024*e^11 + 5717/11128*e^9 - 158225/22256*e^7 + 447147/11128*e^5 - 117687/1391*e^3 + 119129/2782*e, -129/6848*e^11 + 1309/1712*e^9 - 17987/1712*e^7 + 50407/856*e^5 - 26187/214*e^3 + 11885/214*e, 857/89024*e^11 - 8673/22256*e^9 + 118503/22256*e^7 - 327401/11128*e^5 + 81047/1391*e^3 - 51195/2782*e, 37/3424*e^10 - 765/1712*e^8 + 2711/428*e^6 - 15989/428*e^4 + 18125/214*e^2 - 4916/107, -23/3424*e^10 + 873/3424*e^8 - 5373/1712*e^6 + 1576/107*e^4 - 9581/428*e^2 + 1077/214, -19/3424*e^10 + 777/3424*e^8 - 5369/1712*e^6 + 3725/214*e^4 - 14465/428*e^2 + 2397/214, 1077/44512*e^11 - 44359/44512*e^9 + 312019/22256*e^7 - 227041/2782*e^5 + 1014587/5564*e^3 - 293973/2782*e, -647/44512*e^11 + 25907/44512*e^9 - 173559/22256*e^7 + 231453/5564*e^5 - 431525/5564*e^3 + 67151/2782*e, 1/32*e^10 - 41/32*e^8 + 285/16*e^6 - 201/2*e^4 + 825/4*e^2 - 185/2, 3/1712*e^10 - 251/3424*e^8 + 1825/1712*e^6 - 2775/428*e^4 + 5835/428*e^2 + 375/214, 1617/89024*e^11 - 32779/44512*e^9 + 112781/11128*e^7 - 640595/11128*e^5 + 707113/5564*e^3 - 109113/1391*e, -627/89024*e^11 + 12767/44512*e^9 - 22107/5564*e^7 + 253447/11128*e^5 - 282703/5564*e^3 + 39811/1391*e, 25/5564*e^11 - 7903/44512*e^9 + 51881/22256*e^7 - 33303/2782*e^5 + 103657/5564*e^3 + 23735/2782*e, 11/2782*e^11 - 6899/44512*e^9 + 44153/22256*e^7 - 26469/2782*e^5 + 73469/5564*e^3 - 3873/2782*e, -71/22256*e^11 + 1601/11128*e^9 - 6411/2782*e^7 + 89215/5564*e^5 - 129877/2782*e^3 + 63454/1391*e, 285/5564*e^11 - 46299/22256*e^9 + 318395/11128*e^7 - 893679/5564*e^5 + 466585/1391*e^3 - 220111/1391*e, 173/44512*e^11 - 7469/44512*e^9 + 57097/22256*e^7 - 96437/5564*e^5 + 281823/5564*e^3 - 131123/2782*e, 425/22256*e^11 - 34631/44512*e^9 + 238925/22256*e^7 - 167619/2782*e^5 + 687097/5564*e^3 - 131771/2782*e, 7/214*e^10 - 2307/1712*e^8 + 16213/856*e^6 - 23439/214*e^4 + 50715/214*e^2 - 12871/107, 55/1712*e^10 - 4459/3424*e^8 + 30605/1712*e^6 - 42957/428*e^4 + 90711/428*e^2 - 22657/214, -93/3424*e^10 + 3837/3424*e^8 - 27057/1712*e^6 + 9857/107*e^4 - 86537/428*e^2 + 21633/214, -35/428*e^10 + 11321/3424*e^8 - 77427/1712*e^6 + 108207/428*e^4 - 226825/428*e^2 + 55795/214, -151/3424*e^10 + 6085/3424*e^8 - 41453/1712*e^6 + 14450/107*e^4 - 122077/428*e^2 + 31169/214, -643/44512*e^11 + 26667/44512*e^9 - 188963/22256*e^7 + 137539/2782*e^5 - 591559/5564*e^3 + 132029/2782*e, -945/44512*e^11 + 38837/44512*e^9 - 272297/22256*e^7 + 394209/5564*e^5 - 868027/5564*e^3 + 223257/2782*e, -135/3424*e^10 + 5487/3424*e^8 - 37799/1712*e^6 + 26591/214*e^4 - 110583/428*e^2 + 26819/214, 37/1712*e^10 - 2953/3424*e^8 + 19655/1712*e^6 - 25879/428*e^4 + 47141/428*e^2 - 6075/214, -23/3424*e^10 + 873/3424*e^8 - 5373/1712*e^6 + 1576/107*e^4 - 9153/428*e^2 - 2347/214, -277/3424*e^10 + 11249/3424*e^8 - 77317/1712*e^6 + 54239/214*e^4 - 228241/428*e^2 + 55929/214, 35/856*e^10 - 5607/3424*e^8 + 37697/1712*e^6 - 51161/428*e^4 + 103087/428*e^2 - 26881/214, -95/6848*e^11 + 499/856*e^9 - 14439/1712*e^7 + 43563/856*e^5 - 25491/214*e^3 + 15997/214*e, 1121/89024*e^11 - 5717/11128*e^9 + 158225/22256*e^7 - 447147/11128*e^5 + 117687/1391*e^3 - 119129/2782*e, -161/3424*e^10 + 6539/3424*e^8 - 44887/1712*e^6 + 15633/107*e^4 - 128699/428*e^2 + 31935/214, 347/3424*e^10 - 14213/3424*e^8 + 99001/1712*e^6 - 35347/107*e^4 + 301773/428*e^2 - 74345/214, -223/3424*e^10 + 9097/3424*e^8 - 62925/1712*e^6 + 22240/107*e^4 - 188673/428*e^2 + 48925/214, -39/3424*e^10 + 1685/3424*e^8 - 12665/1712*e^6 + 9955/214*e^4 - 46113/428*e^2 + 10349/214, 1013/89024*e^11 - 20609/44512*e^9 + 35557/5564*e^7 - 402333/11128*e^5 + 430645/5564*e^3 - 55153/1391*e, 437/89024*e^11 - 8561/44512*e^9 + 3441/1391*e^7 - 135505/11128*e^5 + 107337/5564*e^3 - 9369/1391*e, -75/1712*e^10 + 771/428*e^8 - 10791/428*e^6 + 30889/214*e^4 - 32376/107*e^2 + 14198/107, -9/107*e^10 + 2905/856*e^8 - 4940/107*e^6 + 54509/214*e^4 - 55468/107*e^2 + 25026/107, 1297/89024*e^11 - 27013/44512*e^9 + 48379/5564*e^7 - 580763/11128*e^5 + 690399/5564*e^3 - 122780/1391*e, -599/89024*e^11 + 12645/44512*e^9 - 46133/11128*e^7 + 279231/11128*e^5 - 298941/5564*e^3 + 10770/1391*e, -143/6848*e^11 + 2893/3424*e^9 - 9853/856*e^7 + 53993/856*e^5 - 52815/428*e^3 + 4520/107*e, 4529/89024*e^11 - 92761/44512*e^9 + 161775/5564*e^7 - 1862205/11128*e^5 + 2050741/5564*e^3 - 287564/1391*e, 247/3424*e^10 - 10101/3424*e^8 + 70225/1712*e^6 - 50137/214*e^4 + 216293/428*e^2 - 57269/214, 331/3424*e^10 - 13401/3424*e^8 + 91709/1712*e^6 - 63891/214*e^4 + 264385/428*e^2 - 60793/214, 335/3424*e^10 - 13711/3424*e^8 + 95351/1712*e^6 - 67919/214*e^4 + 289247/428*e^2 - 71243/214, 3/1712*e^10 - 251/3424*e^8 + 1825/1712*e^6 - 2775/428*e^4 + 4979/428*e^2 + 2087/214, 235/89024*e^11 - 513/5564*e^9 + 21143/22256*e^7 - 30203/11128*e^5 - 8577/2782*e^3 + 31071/2782*e, -1149/89024*e^11 + 11495/22256*e^9 - 154387/22256*e^7 + 418581/11128*e^5 - 105977/1391*e^3 + 107661/2782*e, 115/11128*e^11 - 19707/44512*e^9 + 146729/22256*e^7 - 229977/5564*e^5 + 556811/5564*e^3 - 163101/2782*e, 31/3424*e^11 - 1279/3424*e^9 + 9019/1712*e^7 - 13107/428*e^5 + 28061/428*e^3 - 5285/214*e, 137/3424*e^10 - 5535/3424*e^8 + 37587/1712*e^6 - 25609/214*e^4 + 100223/428*e^2 - 21023/214, 239/3424*e^10 - 9695/3424*e^8 + 66579/1712*e^6 - 23341/107*e^4 + 195031/428*e^2 - 45143/214, 621/89024*e^11 - 13337/44512*e^9 + 25343/5564*e^7 - 340445/11128*e^5 + 496621/5564*e^3 - 121519/1391*e, -795/89024*e^11 + 16281/44512*e^9 - 56347/11128*e^7 + 312957/11128*e^5 - 302119/5564*e^3 + 30445/1391*e, 4171/89024*e^11 - 85041/44512*e^9 + 294731/11128*e^7 - 1684753/11128*e^5 + 1861523/5564*e^3 - 285962/1391*e, -817/89024*e^11 + 16973/44512*e^9 - 15225/2782*e^7 + 371389/11128*e^5 - 458069/5564*e^3 + 77208/1391*e, -931/89024*e^11 + 20053/44512*e^9 - 75641/11128*e^7 + 490561/11128*e^5 - 655651/5564*e^3 + 138316/1391*e, -4415/89024*e^11 + 89681/44512*e^9 - 308809/11128*e^7 + 1743033/11128*e^5 - 1858723/5564*e^3 + 240366/1391*e, 115/3424*e^10 - 2343/1712*e^8 + 4067/214*e^6 - 46821/428*e^4 + 51719/214*e^2 - 12162/107, -137/3424*e^10 + 2821/1712*e^8 - 9905/428*e^6 + 57103/428*e^4 - 60865/214*e^2 + 14524/107, 93/856*e^10 - 15027/3424*e^8 + 102557/1712*e^6 - 142625/428*e^4 + 295323/428*e^2 - 68021/214, 35/3424*e^10 - 1375/3424*e^8 + 9023/1712*e^6 - 5927/214*e^4 + 20823/428*e^2 + 957/214, 149/3424*e^10 - 192/107*e^8 + 21635/856*e^6 - 63081/428*e^4 + 35597/107*e^2 - 20783/107, 265/3424*e^10 - 665/214*e^8 + 36031/856*e^6 - 99611/428*e^4 + 51976/107*e^2 - 25825/107, -4239/89024*e^11 + 86927/44512*e^9 - 152189/5564*e^7 + 1770773/11128*e^5 - 1999341/5564*e^3 + 294690/1391*e, 2125/89024*e^11 - 42941/44512*e^9 + 18340/1391*e^7 - 821403/11128*e^5 + 881475/5564*e^3 - 122522/1391*e, 1407/89024*e^11 - 14541/22256*e^9 + 207053/22256*e^7 - 622543/11128*e^5 + 185664/1391*e^3 - 213821/2782*e, 641/89024*e^11 - 3207/11128*e^9 + 86509/22256*e^7 - 237773/11128*e^5 + 121991/2782*e^3 - 66933/2782*e, 3473/44512*e^11 - 70673/22256*e^9 + 122053/2782*e^7 - 346153/1391*e^5 + 742683/1391*e^3 - 388243/1391*e, 519/44512*e^11 - 2627/5564*e^9 + 72431/11128*e^7 - 208633/5564*e^5 + 121300/1391*e^3 - 87491/1391*e, -6881/89024*e^11 + 70283/22256*e^9 - 977087/22256*e^7 + 2798279/11128*e^5 - 1525029/2782*e^3 + 837287/2782*e, 3987/89024*e^11 - 10207/5564*e^9 + 569575/22256*e^7 - 1635605/11128*e^5 + 442826/1391*e^3 - 465859/2782*e, -45/3424*e^10 + 1829/3424*e^8 - 12457/1712*e^6 + 8293/214*e^4 - 26589/428*e^2 - 1475/214, -253/1712*e^10 + 20597/3424*e^8 - 142067/1712*e^6 + 199999/428*e^4 - 418897/428*e^2 + 101055/214, 291/1712*e^10 - 11799/1712*e^8 + 80969/856*e^6 - 113455/214*e^4 + 237683/214*e^2 - 58157/107, -225/1712*e^10 + 9145/1712*e^8 - 62927/856*e^6 + 88173/214*e^4 - 182165/214*e^2 + 41203/107, -133/3424*e^10 + 1333/856*e^8 - 17989/856*e^6 + 49475/428*e^4 - 26143/107*e^2 + 15291/107, 93/3424*e^10 - 493/428*e^8 + 14545/856*e^6 - 45527/428*e^4 + 27974/107*e^2 - 18039/107, -2439/44512*e^11 + 49277/22256*e^9 - 41956/1391*e^7 + 923721/5564*e^5 - 917367/2782*e^3 + 175258/1391*e, -2083/44512*e^11 + 21379/11128*e^9 - 299197/11128*e^7 + 216063/1391*e^5 - 957299/2782*e^3 + 281720/1391*e, 37/856*e^11 - 6013/3424*e^9 + 41343/1712*e^7 - 14491/107*e^5 + 121995/428*e^3 - 32801/214*e, -27/44512*e^11 + 1825/44512*e^9 - 21213/22256*e^7 + 50847/5564*e^5 - 187667/5564*e^3 + 88567/2782*e, -5/1712*e^10 + 347/3424*e^8 - 1829/1712*e^6 + 1629/428*e^4 - 95/428*e^2 - 5119/214, -431/3424*e^10 + 17299/3424*e^8 - 116847/1712*e^6 + 39977/107*e^4 - 322687/428*e^2 + 72519/214, 163/856*e^10 - 26455/3424*e^8 + 181589/1712*e^6 - 253935/428*e^4 + 527819/428*e^2 - 124993/214, -583/3424*e^10 + 23729/3424*e^8 - 163865/1712*e^6 + 57980/107*e^4 - 492977/428*e^2 + 123581/214, -29/856*e^10 + 4817/3424*e^8 - 34463/1712*e^6 + 51831/428*e^4 - 121049/428*e^2 + 35871/214, -101/3424*e^10 + 4029/3424*e^8 - 27065/1712*e^6 + 18675/214*e^4 - 81477/428*e^2 + 25413/214, -1997/89024*e^11 + 4975/5564*e^9 - 265877/22256*e^7 + 717905/11128*e^5 - 364089/2782*e^3 + 191157/2782*e, 10513/89024*e^11 - 106413/22256*e^9 + 1456123/22256*e^7 - 4051577/11128*e^5 + 1038447/1391*e^3 - 905259/2782*e, -2757/44512*e^11 + 111857/44512*e^9 - 766737/22256*e^7 + 532937/2782*e^5 - 2162393/5564*e^3 + 439567/2782*e, -3409/44512*e^11 + 138205/44512*e^9 - 948861/22256*e^7 + 667127/2782*e^5 - 2848037/5564*e^3 + 772675/2782*e, -31/5564*e^11 + 9911/44512*e^9 - 67337/22256*e^7 + 95333/5564*e^5 - 200199/5564*e^3 + 4509/2782*e, 583/44512*e^11 - 24157/44512*e^9 + 172425/22256*e^7 - 258991/5564*e^5 + 607895/5564*e^3 - 159319/2782*e, 27/1712*e^10 - 2045/3424*e^8 + 12359/1712*e^6 - 13419/428*e^4 + 13781/428*e^2 + 5729/214, -15/856*e^10 + 2403/3424*e^8 - 16217/1712*e^6 + 22507/428*e^4 - 49683/428*e^2 + 11337/214, 53/1712*e^10 - 4149/3424*e^8 + 26963/1712*e^6 - 34901/428*e^4 + 65421/428*e^2 - 14347/214, 369/3424*e^10 - 15169/3424*e^8 + 106085/1712*e^6 - 37971/107*e^4 + 322205/428*e^2 - 73933/214, -2181/89024*e^11 + 1393/1391*e^9 - 312193/22256*e^7 + 922845/11128*e^5 - 558731/2782*e^3 + 404329/2782*e, 349/89024*e^11 - 449/2782*e^9 + 50625/22256*e^7 - 149375/11128*e^5 + 45107/1391*e^3 - 88363/2782*e, 39/1712*e^10 - 789/856*e^8 + 5423/428*e^6 - 7708/107*e^4 + 17118/107*e^2 - 10028/107, 13/107*e^10 - 526/107*e^8 + 7195/107*e^6 - 40075/107*e^4 + 82308/107*e^2 - 36220/107, 7/6848*e^11 - 21/856*e^9 - 157/1712*e^7 + 4627/856*e^5 - 3944/107*e^3 + 12711/214*e, -3127/89024*e^11 + 15801/11128*e^9 - 431115/22256*e^7 + 1190347/11128*e^5 - 594121/2782*e^3 + 236255/2782*e, 1227/44512*e^11 - 49243/44512*e^9 + 332499/22256*e^7 - 456047/5564*e^5 + 954273/5564*e^3 - 302253/2782*e, -1021/22256*e^11 + 83569/44512*e^9 - 581019/22256*e^7 + 826297/5564*e^5 - 1743169/5564*e^3 + 383407/2782*e, -1/856*e^10 - 11/3424*e^8 + 2029/1712*e^6 - 7245/428*e^4 + 31099/428*e^2 - 17905/214, -137/1712*e^10 + 11177/3424*e^8 - 77635/1712*e^6 + 111745/428*e^4 - 250201/428*e^2 + 72327/214, -41/832*e^11 + 837/416*e^9 - 363/13*e^7 + 16531/104*e^5 - 17471/52*e^3 + 1897/13*e, 693/89024*e^11 - 14843/44512*e^9 + 55091/11128*e^7 - 342501/11128*e^5 + 413931/5564*e^3 - 67429/1391*e, -55/1712*e^10 + 4245/3424*e^8 - 26539/1712*e^6 + 30759/428*e^4 - 39565/428*e^2 - 3237/214, -169/1712*e^10 + 13997/3424*e^8 - 99099/1712*e^6 + 144983/428*e^4 - 322285/428*e^2 + 87587/214, -8201/89024*e^11 + 166785/44512*e^9 - 287317/5564*e^7 + 3235653/11128*e^5 - 3417025/5564*e^3 + 438946/1391*e, 503/6848*e^11 - 10209/3424*e^9 + 34981/856*e^7 - 193957/856*e^5 + 195123/428*e^3 - 20188/107*e, 301/1712*e^10 - 24613/3424*e^8 + 170839/1712*e^6 - 242473/428*e^4 + 512685/428*e^2 - 121591/214, 231/3424*e^10 - 9289/3424*e^8 + 62933/1712*e^6 - 21560/107*e^4 + 171201/428*e^2 - 38153/214, -1679/44512*e^11 + 69079/44512*e^9 - 483607/22256*e^7 + 173149/1391*e^5 - 1459755/5564*e^3 + 293263/2782*e, 283/22256*e^11 - 21823/44512*e^9 + 136349/22256*e^7 - 158199/5564*e^5 + 198191/5564*e^3 + 58059/2782*e, -1507/89024*e^11 + 15355/22256*e^9 - 212025/22256*e^7 + 593251/11128*e^5 - 145631/1391*e^3 + 58007/2782*e, -1995/89024*e^11 + 19995/22256*e^9 - 268337/22256*e^7 + 718157/11128*e^5 - 339929/2782*e^3 + 117657/2782*e, 655/3424*e^10 - 26741/3424*e^8 + 185337/1712*e^6 - 65770/107*e^4 + 562141/428*e^2 - 145189/214, 141/1712*e^10 - 11369/3424*e^8 + 77215/1712*e^6 - 106029/428*e^4 + 211329/428*e^2 - 41439/214, -65/3424*e^11 + 1315/1712*e^9 - 4537/428*e^7 + 26371/428*e^5 - 31847/214*e^3 + 13243/107*e, 3167/44512*e^11 - 8121/2782*e^9 + 454247/11128*e^7 - 1310497/5564*e^5 + 721225/1391*e^3 - 408452/1391*e, 1379/89024*e^11 - 27569/44512*e^9 + 92231/11128*e^7 - 492535/11128*e^5 + 465393/5564*e^3 - 50745/1391*e, -5701/89024*e^11 + 114957/44512*e^9 - 97519/2782*e^7 + 2129051/11128*e^5 - 2053847/5564*e^3 + 155457/1391*e, 399/11128*e^11 - 3999/2782*e^9 + 107613/5564*e^7 - 582037/5564*e^5 + 569759/2782*e^3 - 118329/1391*e, 1949/22256*e^11 - 78983/22256*e^9 + 542727/11128*e^7 - 383537/1391*e^5 + 1664443/2782*e^3 - 477869/1391*e, -4419/89024*e^11 + 89301/44512*e^9 - 303567/11128*e^7 + 1653505/11128*e^5 - 1565883/5564*e^3 + 94088/1391*e, 85/89024*e^11 - 3053/44512*e^9 + 9333/5564*e^7 - 190985/11128*e^5 + 394137/5564*e^3 - 119408/1391*e, 365/3424*e^10 - 7483/1712*e^8 + 26119/428*e^6 - 150141/428*e^4 + 163919/214*e^2 - 43030/107, -43/3424*e^10 + 837/1712*e^8 - 2659/428*e^6 + 12451/428*e^4 - 6009/214*e^2 - 3778/107, -91/3424*e^10 + 1841/1712*e^8 - 6309/428*e^6 + 35293/428*e^4 - 34913/214*e^2 + 3496/107, -447/3424*e^10 + 9109/1712*e^8 - 7859/107*e^6 + 176617/428*e^4 - 182873/214*e^2 + 43838/107, -159/1712*e^10 + 13089/3424*e^8 - 91803/1712*e^6 + 132309/428*e^4 - 285929/428*e^2 + 71931/214, 381/3424*e^10 - 15671/3424*e^8 + 109735/1712*e^6 - 78717/214*e^4 + 334731/428*e^2 - 77463/214, 1123/22256*e^11 - 92103/44512*e^9 + 643925/22256*e^7 - 232489/1391*e^5 + 2047061/5564*e^3 - 521527/2782*e, 1137/44512*e^11 - 46869/44512*e^9 + 334121/22256*e^7 - 511899/5564*e^5 + 1320963/5564*e^3 - 517063/2782*e, 217/3424*e^10 - 8953/3424*e^8 + 63133/1712*e^6 - 22964/107*e^4 + 199637/428*e^2 - 44485/214, 19/428*e^10 - 6109/3424*e^8 + 41347/1712*e^6 - 56711/428*e^4 + 113045/428*e^2 - 16715/214, -187/3424*e^10 + 1871/856*e^8 - 25185/856*e^6 + 68993/428*e^4 - 36142/107*e^2 + 20077/107, 75/3424*e^10 - 83/107*e^8 + 7153/856*e^6 - 12699/428*e^4 + 2278/107*e^2 - 1535/107, -1731/22256*e^11 + 70327/22256*e^9 - 483659/11128*e^7 + 338849/1391*e^5 - 1399045/2782*e^3 + 285840/1391*e, -2897/22256*e^11 + 58625/11128*e^9 - 100260/1391*e^7 + 558892/1391*e^5 - 1160667/1391*e^3 + 553011/1391*e, -29/1712*e^10 + 1231/1712*e^8 - 8803/856*e^6 + 6091/107*e^4 - 20231/214*e^2 + 167/107, 5/428*e^10 - 801/1712*e^8 + 5477/856*e^6 - 8073/214*e^4 + 20413/214*e^2 - 4421/107, -841/44512*e^11 + 17475/22256*e^9 - 15708/1391*e^7 + 96612/1391*e^5 - 245623/1391*e^3 + 182342/1391*e, 4089/44512*e^11 - 41547/11128*e^9 + 572087/11128*e^7 - 401863/1391*e^5 + 1672011/2782*e^3 - 383545/1391*e, -4821/44512*e^11 + 98405/22256*e^9 - 85373/1391*e^7 + 1950603/5564*e^5 - 2117077/2782*e^3 + 568999/1391*e, 99/44512*e^11 - 1723/22256*e^9 + 1123/1391*e^7 - 7673/2782*e^5 + 3833/1391*e^3 - 4958/1391*e, -415/3424*e^10 + 16915/3424*e^8 - 116831/1712*e^6 + 82353/214*e^4 - 346075/428*e^2 + 80795/214, 223/1712*e^10 - 18301/3424*e^8 + 127883/1712*e^6 - 184019/428*e^4 + 401849/428*e^2 - 105447/214, 383/3424*e^10 - 15505/3424*e^8 + 105885/1712*e^6 - 36567/107*e^4 + 293341/428*e^2 - 62893/214, -101/1712*e^10 + 8165/3424*e^8 - 55735/1712*e^6 + 77589/428*e^4 - 163061/428*e^2 + 48151/214, 203/3424*e^10 - 7975/3424*e^8 + 51991/1712*e^6 - 33649/214*e^4 + 124711/428*e^2 - 26635/214, 6027/89024*e^11 - 15193/5564*e^9 + 825683/22256*e^7 - 2266023/11128*e^5 + 1126539/2782*e^3 - 449831/2782*e, -1763/89024*e^11 + 1156/1391*e^9 - 267151/22256*e^7 + 803029/11128*e^5 - 229930/1391*e^3 + 238113/2782*e, 3451/89024*e^11 - 34295/22256*e^9 + 452677/22256*e^7 - 1163433/11128*e^5 + 471359/2782*e^3 - 8171/2782*e, 6137/89024*e^11 - 31251/11128*e^9 + 865649/22256*e^7 - 2474723/11128*e^5 + 684484/1391*e^3 - 803399/2782*e, 95/1712*e^10 - 3885/1712*e^8 + 27059/856*e^6 - 39069/214*e^4 + 88375/214*e^2 - 25681/107, -193/1712*e^10 + 7735/1712*e^8 - 52195/856*e^6 + 35777/107*e^4 - 146551/214*e^2 + 38067/107]; 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;