/*
  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.<x> = PolynomialRing(QQ)
g = P([1, -2, -6, 0, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([13, 13, -w^3 + w^2 + 4*w + 1])

primes_array = [
[2, 2, w + 1],\
[3, 3, -w^3 + w^2 + 5*w - 2],\
[11, 11, -w^3 + 5*w + 1],\
[11, 11, -w + 2],\
[13, 13, -w^3 + w^2 + 4*w + 1],\
[17, 17, -w^3 + w^2 + 6*w - 1],\
[17, 17, -w^2 - w + 3],\
[23, 23, w^3 - 6*w],\
[27, 3, w^3 + w^2 - 5*w - 4],\
[41, 41, -w^3 + w^2 + 5*w],\
[41, 41, 2*w^3 - 11*w - 4],\
[53, 53, 2*w^3 - 2*w^2 - 11*w + 6],\
[59, 59, 2*w^3 - 11*w - 2],\
[67, 67, w^3 - 7*w - 1],\
[71, 71, 3*w^3 - 2*w^2 - 15*w + 1],\
[79, 79, -3*w^3 + w^2 + 16*w + 3],\
[89, 89, -4*w^3 + 2*w^2 + 22*w - 3],\
[101, 101, -2*w^3 + 13*w + 6],\
[101, 101, w^3 + w^2 - 6*w - 3],\
[101, 101, -3*w^3 + 2*w^2 + 17*w - 3],\
[101, 101, w^3 - w^2 - 8*w - 3],\
[107, 107, 2*w^3 - 3*w^2 - 6*w],\
[109, 109, -w^3 + w^2 + 4*w - 3],\
[113, 113, w^3 - 8*w - 4],\
[113, 113, 7*w^3 - 4*w^2 - 39*w + 7],\
[121, 11, w^3 - w^2 - 6*w - 1],\
[127, 127, -2*w^3 + w^2 + 9*w + 3],\
[127, 127, 2*w^3 - 13*w],\
[131, 131, 6*w^3 - 3*w^2 - 34*w + 6],\
[139, 139, w^2 - 2*w - 4],\
[149, 149, 3*w^3 - w^2 - 18*w - 3],\
[151, 151, -2*w^3 + 2*w^2 + 9*w - 4],\
[163, 163, 3*w^3 - 2*w^2 - 16*w + 2],\
[163, 163, 2*w^3 - 10*w + 1],\
[167, 167, 3*w^3 - 18*w - 4],\
[173, 173, -3*w^3 + w^2 + 18*w - 3],\
[193, 193, 4*w^3 - 3*w^2 - 23*w + 9],\
[197, 197, 3*w^3 - w^2 - 17*w + 2],\
[197, 197, -3*w^3 - 2*w^2 + 15*w + 9],\
[199, 199, 2*w^3 - w^2 - 8*w - 6],\
[223, 223, -6*w^3 + 3*w^2 + 35*w - 7],\
[227, 227, 2*w^3 - 12*w - 1],\
[227, 227, -w^3 + w^2 + 7*w - 8],\
[227, 227, 3*w^3 - w^2 - 18*w - 1],\
[227, 227, -4*w^3 - 2*w^2 + 21*w + 14],\
[229, 229, -w^3 + 2*w^2 + 3*w - 5],\
[233, 233, w^3 - 3*w - 3],\
[233, 233, -3*w^3 + 3*w^2 + 15*w - 8],\
[241, 241, 4*w^2 - 7*w - 8],\
[241, 241, w^3 + w^2 - 4*w - 5],\
[251, 251, 2*w^3 - w^2 - 13*w - 1],\
[251, 251, 2*w^3 - 9*w - 6],\
[257, 257, w^3 - 8*w],\
[263, 263, -w^3 + 2*w^2 + 5*w - 7],\
[269, 269, 3*w^3 - 3*w^2 - 12*w - 1],\
[271, 271, 2*w^3 - w^2 - 12*w - 2],\
[271, 271, 9*w^3 - 5*w^2 - 50*w + 9],\
[277, 277, w^3 - 2*w^2 - 6*w + 6],\
[277, 277, w^3 + w^2 - 7*w - 2],\
[281, 281, 4*w^3 - 2*w^2 - 24*w + 5],\
[281, 281, -7*w^3 + 4*w^2 + 40*w - 12],\
[289, 17, -5*w^3 + 5*w^2 + 26*w - 15],\
[293, 293, -w^3 + 2*w^2 + 4*w - 4],\
[293, 293, -9*w^3 + 5*w^2 + 51*w - 12],\
[307, 307, -2*w^3 + 2*w^2 + 12*w - 7],\
[311, 311, -4*w^3 + 24*w + 11],\
[313, 313, 2*w^3 - 9*w - 4],\
[317, 317, 2*w^3 - 4*w^2 - 2*w - 3],\
[317, 317, 2*w^3 - 14*w - 7],\
[331, 331, -3*w^3 + 19*w + 3],\
[349, 349, -w^3 - 2*w^2 + 8*w + 8],\
[349, 349, w^3 + w^2 - 5*w - 8],\
[361, 19, -w^3 + w^2 + 7*w - 6],\
[361, 19, w^3 + w^2 - 3*w - 4],\
[373, 373, 2*w^3 - w^2 - 11*w - 3],\
[379, 379, -4*w^3 + 25*w + 14],\
[383, 383, -3*w^3 + w^2 + 15*w - 2],\
[383, 383, 3*w^3 - 4*w^2 - 10*w],\
[401, 401, 4*w^3 - w^2 - 24*w - 4],\
[409, 409, w^2 + 5*w + 5],\
[419, 419, w^2 - 2*w - 6],\
[419, 419, -6*w^3 + 3*w^2 + 34*w - 8],\
[433, 433, 3*w^3 - 16*w - 6],\
[433, 433, 3*w^3 - w^2 - 16*w - 5],\
[443, 443, -w^3 + 2*w^2 + w + 3],\
[449, 449, 2*w^2 - 2*w - 5],\
[457, 457, -4*w^3 + 4*w^2 + 20*w - 9],\
[457, 457, w^2 + w - 7],\
[463, 463, w^3 - 7*w - 7],\
[463, 463, w^3 + 2*w^2 - 5*w - 9],\
[487, 487, w^3 + 2*w^2 - 5*w - 5],\
[487, 487, -w^3 + w^2 + 6*w - 7],\
[509, 509, -2*w^3 - 2*w^2 + 13*w + 10],\
[509, 509, 3*w^3 - 17*w - 3],\
[509, 509, -5*w^3 + 3*w^2 + 30*w - 11],\
[509, 509, -w^3 - 2*w^2 + 4*w + 6],\
[521, 521, -2*w^3 - w^2 + 9*w + 7],\
[541, 541, -2*w^3 - 2*w^2 + 12*w + 9],\
[541, 541, -2*w^3 + 12*w + 9],\
[547, 547, 2*w^3 + w^2 - 7*w - 3],\
[547, 547, 5*w^3 - 3*w^2 - 30*w + 3],\
[557, 557, w^3 + 2*w^2 - 5*w - 11],\
[557, 557, -w^3 + w^2 + 3*w - 4],\
[563, 563, 3*w^3 - w^2 - 15*w - 4],\
[569, 569, 2*w^2 - w - 8],\
[577, 577, 4*w^3 - 22*w - 7],\
[593, 593, -3*w^3 + w^2 + 17*w + 4],\
[607, 607, -w^3 - 3*w^2 + w + 4],\
[613, 613, 3*w^3 - 2*w^2 - 14*w],\
[613, 613, w^2 - 3*w - 5],\
[619, 619, -w^3 + 2*w^2 + 4*w - 10],\
[625, 5, -5],\
[641, 641, -3*w^3 - 3*w^2 + 12*w + 7],\
[643, 643, 2*w^3 + 2*w^2 - 14*w - 13],\
[643, 643, 3*w^3 - 19*w - 7],\
[647, 647, w^3 - 5*w - 7],\
[653, 653, -3*w^3 + 2*w^2 + 14*w - 2],\
[659, 659, -6*w^3 + 2*w^2 + 34*w - 1],\
[659, 659, -2*w^3 + w^2 + 17*w + 9],\
[683, 683, w^3 + 2*w^2 - 7*w - 7],\
[691, 691, 3*w^3 - 15*w - 7],\
[701, 701, w^3 + 2*w^2 - 6*w - 10],\
[701, 701, -w^3 + 3*w^2 + 6*w - 13],\
[701, 701, 5*w^3 - 2*w^2 - 30*w + 4],\
[701, 701, 4*w^3 - 2*w^2 - 21*w + 2],\
[727, 727, -2*w^3 - 3*w^2 + 9*w + 7],\
[743, 743, 3*w - 4],\
[751, 751, w^3 - 9*w - 3],\
[757, 757, w^3 - 5*w^2 + 4*w + 7],\
[757, 757, 4*w^3 - 5*w^2 - 11*w - 5],\
[761, 761, 4*w^3 - 5*w^2 - 20*w + 16],\
[769, 769, -3*w^3 + 16*w + 12],\
[773, 773, 2*w^3 + 2*w^2 - 11*w - 10],\
[773, 773, w^3 - 3*w - 5],\
[773, 773, -w^3 + 2*w^2 + 9*w + 3],\
[773, 773, 5*w^3 - w^2 - 27*w],\
[787, 787, -4*w^3 + 3*w^2 + 22*w - 4],\
[787, 787, -2*w^3 + w^2 + 14*w - 6],\
[797, 797, 2*w^3 - w^2 - 14*w + 2],\
[797, 797, -3*w^3 + 2*w^2 + 19*w - 7],\
[811, 811, 2*w^3 + w^2 - 14*w - 12],\
[811, 811, 2*w^3 + 2*w^2 - 8*w - 3],\
[829, 829, -2*w^3 + 14*w + 11],\
[857, 857, -3*w^3 + 3*w^2 + 14*w - 7],\
[859, 859, 3*w^3 - w^2 - 15*w],\
[859, 859, -11*w^3 + 7*w^2 + 60*w - 15],\
[863, 863, 2*w^2 - 2*w - 13],\
[863, 863, 2*w^3 + w^2 - 12*w - 4],\
[877, 877, -6*w^3 - 2*w^2 + 32*w + 19],\
[887, 887, 3*w^3 + w^2 - 14*w - 7],\
[887, 887, 2*w^3 - w^2 - 15*w - 5],\
[911, 911, 3*w^3 - 18*w - 2],\
[919, 919, w^3 - w^2 - 6*w - 5],\
[919, 919, -4*w^3 + w^2 + 24*w - 2],\
[941, 941, 3*w^3 - 3*w^2 - 13*w],\
[947, 947, 2*w^3 - 2*w^2 - 10*w - 1],\
[947, 947, -3*w^3 + 2*w^2 + 14*w + 6],\
[953, 953, -4*w^3 + 3*w^2 + 21*w - 3],\
[953, 953, 4*w^3 - w^2 - 23*w + 1],\
[961, 31, -2*w^3 + 3*w^2 + 12*w - 8],\
[961, 31, 3*w^2 - 4*w - 6],\
[967, 967, -5*w^3 - 3*w^2 + 23*w + 14],\
[971, 971, 5*w^3 - w^2 - 27*w - 8],\
[971, 971, -9*w^3 + 6*w^2 + 50*w - 12],\
[983, 983, -w^3 + w^2 + 10*w + 3],\
[997, 997, 2*w^3 + w^2 - 11*w - 3],\
[997, 997, w^3 + 2*w^2 - 6*w - 8],\
[997, 997, -5*w^3 + 4*w^2 + 29*w - 15]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^14 - 3*x^13 - 16*x^12 + 52*x^11 + 88*x^10 - 332*x^9 - 189*x^8 + 990*x^7 + 65*x^6 - 1421*x^5 + 281*x^4 + 867*x^3 - 285*x^2 - 129*x + 34
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, 162/10427*e^13 - 951/10427*e^12 + 717/10427*e^11 + 12738/10427*e^10 - 37368/10427*e^9 - 50794/10427*e^8 + 245324/10427*e^7 + 52093/10427*e^6 - 595274/10427*e^5 + 46094/10427*e^4 + 542697/10427*e^3 - 79735/10427*e^2 - 127987/10427*e + 23042/10427, -2463/10427*e^13 + 4611/10427*e^12 + 47606/10427*e^11 - 83602/10427*e^10 - 349444/10427*e^9 + 561786/10427*e^8 + 1228012/10427*e^7 - 1752256/10427*e^6 - 2103431/10427*e^5 + 2559762/10427*e^4 + 1532418/10427*e^3 - 1493153/10427*e^2 - 271792/10427*e + 162144/10427, -129/10427*e^13 + 178/10427*e^12 + 3484/10427*e^11 - 5509/10427*e^10 - 32806/10427*e^9 + 52805/10427*e^8 + 133293/10427*e^7 - 196921/10427*e^6 - 227684/10427*e^5 + 258341/10427*e^4 + 117587/10427*e^3 - 35950/10427*e^2 + 15217/10427*e - 51174/10427, -1, -612/10427*e^13 + 117/10427*e^12 + 11194/10427*e^11 + 538/10427*e^10 - 77799/10427*e^9 - 25920/10427*e^8 + 263057/10427*e^7 + 163515/10427*e^6 - 439036/10427*e^5 - 384990/10427*e^4 + 278508/10427*e^3 + 351039/10427*e^2 - 5404/10427*e - 75462/10427, -4884/10427*e^13 + 10134/10427*e^12 + 87288/10427*e^11 - 173170/10427*e^10 - 583452/10427*e^9 + 1078736/10427*e^8 + 1855388/10427*e^7 - 3079333/10427*e^6 - 2877855/10427*e^5 + 4086071/10427*e^4 + 1866245/10427*e^3 - 2158143/10427*e^2 - 253915/10427*e + 201276/10427, 2130/10427*e^13 - 4394/10427*e^12 - 36915/10427*e^11 + 71321/10427*e^10 + 238570/10427*e^9 - 413351/10427*e^8 - 739407/10427*e^7 + 1078063/10427*e^6 + 1142014/10427*e^5 - 1303635/10427*e^4 - 748510/10427*e^3 + 625359/10427*e^2 + 73192/10427*e - 29932/10427, -4497/10427*e^13 + 9600/10427*e^12 + 76836/10427*e^11 - 156643/10427*e^10 - 485034/10427*e^9 + 909894/10427*e^8 + 1455509/10427*e^7 - 2342592/10427*e^6 - 2194803/10427*e^5 + 2664574/10427*e^4 + 1523911/10427*e^3 - 1059728/10427*e^2 - 351701/10427*e + 21134/10427, 6312/10427*e^13 - 10407/10427*e^12 - 116883/10427*e^11 + 178866/10427*e^10 + 817118/10427*e^9 - 1122526/10427*e^8 - 2736814/10427*e^7 + 3229575/10427*e^6 + 4500087/10427*e^5 - 4313877/10427*e^4 - 3172998/10427*e^3 + 2298336/10427*e^2 + 561956/10427*e - 192030/10427, -1239/10427*e^13 + 4377/10427*e^12 + 14791/10427*e^11 - 63824/10427*e^10 - 37441/10427*e^9 + 300816/10427*e^8 - 90554/10427*e^7 - 494382/10427*e^6 + 422107/10427*e^5 + 76518/10427*e^4 - 390535/10427*e^3 + 265541/10427*e^2 + 10118/10427*e + 258/10427, -349/10427*e^13 - 1620/10427*e^12 + 10234/10427*e^11 + 26238/10427*e^10 - 100232/10427*e^9 - 155239/10427*e^8 + 420994/10427*e^7 + 445362/10427*e^6 - 802860/10427*e^5 - 693769/10427*e^4 + 717178/10427*e^3 + 519792/10427*e^2 - 306964/10427*e - 44362/10427, -10256/10427*e^13 + 21588/10427*e^12 + 176278/10427*e^11 - 354975/10427*e^10 - 1113425/10427*e^9 + 2081602/10427*e^8 + 3280466/10427*e^7 - 5415133/10427*e^6 - 4636947/10427*e^5 + 6257354/10427*e^4 + 2729498/10427*e^3 - 2701921/10427*e^2 - 392876/10427*e + 199264/10427, -5296/10427*e^13 + 7146/10427*e^12 + 98981/10427*e^11 - 117129/10427*e^10 - 700433/10427*e^9 + 677941/10427*e^8 + 2384884/10427*e^7 - 1692049/10427*e^6 - 4021546/10427*e^5 + 1736179/10427*e^4 + 2968451/10427*e^3 - 518880/10427*e^2 - 583516/10427*e - 33940/10427, -4269/10427*e^13 + 7103/10427*e^12 + 75528/10427*e^11 - 119020/10427*e^10 - 495918/10427*e^9 + 717144/10427*e^8 + 1540491/10427*e^7 - 1933681/10427*e^6 - 2381874/10427*e^5 + 2339400/10427*e^4 + 1750137/10427*e^3 - 1110158/10427*e^2 - 496688/10427*e + 154744/10427, -72/10427*e^13 - 3053/10427*e^12 + 3157/10427*e^11 + 42998/10427*e^10 - 25100/10427*e^9 - 177855/10427*e^8 + 13774/10427*e^7 + 121667/10427*e^6 + 322494/10427*e^5 + 547206/10427*e^4 - 706938/10427*e^3 - 731526/10427*e^2 + 336095/10427*e + 112566/10427, 744/10427*e^13 - 3209/10427*e^12 - 4817/10427*e^11 + 38805/10427*e^10 - 36065/10427*e^9 - 101587/10427*e^8 + 323408/10427*e^7 - 231904/10427*e^6 - 673553/10427*e^5 + 1143942/10427*e^4 + 298082/10427*e^3 - 1126589/10427*e^2 + 127809/10427*e + 192328/10427, -14964/10427*e^13 + 20648/10427*e^12 + 279020/10427*e^11 - 347088/10427*e^10 - 1970344/10427*e^9 + 2110985/10427*e^8 + 6703308/10427*e^7 - 5825972/10427*e^6 - 11333902/10427*e^5 + 7403528/10427*e^4 + 8499581/10427*e^3 - 3742693/10427*e^2 - 1863424/10427*e + 340870/10427, 13034/10427*e^13 - 29786/10427*e^12 - 220752/10427*e^11 + 487675/10427*e^10 + 1369355/10427*e^9 - 2844107/10427*e^8 - 3966580/10427*e^7 + 7358469/10427*e^6 + 5577108/10427*e^5 - 8492302/10427*e^4 - 3371766/10427*e^3 + 3759084/10427*e^2 + 548459/10427*e - 387276/10427, 592/10427*e^13 + 5407/10427*e^12 - 24799/10427*e^11 - 80120/10427*e^10 + 294428/10427*e^9 + 381431/10427*e^8 - 1450226/10427*e^7 - 591403/10427*e^6 + 3215308/10427*e^5 - 144239/10427*e^4 - 2994738/10427*e^3 + 710902/10427*e^2 + 818806/10427*e - 98334/10427, 4040/10427*e^13 - 10586/10427*e^12 - 67080/10427*e^11 + 174389/10427*e^10 + 409714/10427*e^9 - 1034746/10427*e^8 - 1201540/10427*e^7 + 2808836/10427*e^6 + 1833976/10427*e^5 - 3684504/10427*e^4 - 1324936/10427*e^3 + 2192259/10427*e^2 + 251710/10427*e - 324154/10427, -3073/10427*e^13 + 6261/10427*e^12 + 54947/10427*e^11 - 112643/10427*e^10 - 368618/10427*e^9 + 753691/10427*e^8 + 1195834/10427*e^7 - 2353002/10427*e^6 - 2000532/10427*e^5 + 3429928/10427*e^4 + 1623761/10427*e^3 - 1933362/10427*e^2 - 466977/10427*e + 241698/10427, -221/10427*e^13 + 2649/10427*e^12 - 6964/10427*e^11 - 31666/10427*e^10 + 148296/10427*e^9 + 74056/10427*e^8 - 894124/10427*e^7 + 268456/10427*e^6 + 2214181/10427*e^5 - 1141754/10427*e^4 - 2221173/10427*e^3 + 1003501/10427*e^2 + 622510/10427*e - 49842/10427, 14580/10427*e^13 - 23028/10427*e^12 - 269134/10427*e^11 + 395676/10427*e^10 + 1871234/10427*e^9 - 2486060/10427*e^8 - 6209291/10427*e^7 + 7190850/10427*e^6 + 10040467/10427*e^5 - 9761158/10427*e^4 - 6806169/10427*e^3 + 5409239/10427*e^2 + 972716/10427*e - 574678/10427, 5153/10427*e^13 - 7272/10427*e^12 - 99807/10427*e^11 + 130987/10427*e^10 + 736894/10427*e^9 - 877817/10427*e^8 - 2624864/10427*e^7 + 2747144/10427*e^6 + 4619478/10427*e^5 - 4018157/10427*e^4 - 3521839/10427*e^3 + 2215811/10427*e^2 + 634252/10427*e - 59646/10427, -5484/10427*e^13 + 9022/10427*e^12 + 96218/10427*e^11 - 151993/10427*e^10 - 622311/10427*e^9 + 931267/10427*e^8 + 1879804/10427*e^7 - 2621548/10427*e^6 - 2765874/10427*e^5 + 3484756/10427*e^4 + 1908058/10427*e^3 - 1914577/10427*e^2 - 553418/10427*e + 263458/10427, -871/10427*e^13 + 4920/10427*e^12 + 4448/10427*e^11 - 73893/10427*e^10 + 72311/10427*e^9 + 361790/10427*e^8 - 631328/10427*e^7 - 604154/10427*e^6 + 1634278/10427*e^5 - 47525/10427*e^4 - 1399933/10427*e^3 + 643482/10427*e^2 + 187696/10427*e - 25924/10427, 19967/10427*e^13 - 38625/10427*e^12 - 350527/10427*e^11 + 642381/10427*e^10 + 2293477/10427*e^9 - 3852009/10427*e^8 - 7163679/10427*e^7 + 10473186/10427*e^6 + 11135427/10427*e^5 - 13196019/10427*e^4 - 7639447/10427*e^3 + 6691937/10427*e^2 + 1413988/10427*e - 598724/10427, 824/10427*e^13 - 4451/10427*e^12 - 2532/10427*e^11 + 54750/10427*e^10 - 89275/10427*e^9 - 147267/10427*e^8 + 682317/10427*e^7 - 345077/10427*e^6 - 1737440/10427*e^5 + 1780224/10427*e^4 + 1664005/10427*e^3 - 1797892/10427*e^2 - 310509/10427*e + 282746/10427, 2768/10427*e^13 - 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30968/10427*e^12 - 172890/10427*e^11 + 498014/10427*e^10 + 931355/10427*e^9 - 2826882/10427*e^8 - 2256554/10427*e^7 + 7017944/10427*e^6 + 2956894/10427*e^5 - 7513653/10427*e^4 - 2724449/10427*e^3 + 2631754/10427*e^2 + 1431064/10427*e - 59642/10427, -2816/10427*e^13 - 75/10427*e^12 + 65546/10427*e^11 - 8633/10427*e^10 - 566926/10427*e^9 + 116875/10427*e^8 + 2249903/10427*e^7 - 414419/10427*e^6 - 4115285/10427*e^5 + 283684/10427*e^4 + 3093141/10427*e^3 + 181628/10427*e^2 - 762252/10427*e + 99706/10427, 17070/10427*e^13 - 33011/10427*e^12 - 300980/10427*e^11 + 533536/10427*e^10 + 1995483/10427*e^9 - 3042409/10427*e^8 - 6423229/10427*e^7 + 7545881/10427*e^6 + 10550149/10427*e^5 - 7972130/10427*e^4 - 7930114/10427*e^3 + 2883397/10427*e^2 + 1794924/10427*e - 229010/10427, -5109/10427*e^13 + 9717/10427*e^12 + 77603/10427*e^11 - 135251/10427*e^10 - 406428/10427*e^9 + 581591/10427*e^8 + 926114/10427*e^7 - 740151/10427*e^6 - 986373/10427*e^5 - 306312/10427*e^4 + 426055/10427*e^3 + 625967/10427*e^2 + 7840/10427*e + 91650/10427, 49070/10427*e^13 - 81450/10427*e^12 - 898895/10427*e^11 + 1395653/10427*e^10 + 6195071/10427*e^9 - 8729020/10427*e^8 - 20401172/10427*e^7 + 25057648/10427*e^6 + 32960123/10427*e^5 - 33549404/10427*e^4 - 22794189/10427*e^3 + 17997451/10427*e^2 + 3820567/10427*e - 1682426/10427, -11588/10427*e^13 + 12029/10427*e^12 + 239896/10427*e^11 - 237267/10427*e^10 - 1901012/10427*e^9 + 1757766/10427*e^8 + 7257724/10427*e^7 - 6068213/10427*e^6 - 13727396/10427*e^5 + 9749093/10427*e^4 + 11704250/10427*e^3 - 6235659/10427*e^2 - 3053709/10427*e + 707258/10427, -24462/10427*e^13 + 39331/10427*e^12 + 454791/10427*e^11 - 682625/10427*e^10 - 3199528/10427*e^9 + 4364535/10427*e^8 + 10816006/10427*e^7 - 12996127/10427*e^6 - 17991368/10427*e^5 + 18398270/10427*e^4 + 12823756/10427*e^3 - 10576178/10427*e^2 - 2309988/10427*e + 962560/10427, -45586/10427*e^13 + 93051/10427*e^12 + 787260/10427*e^11 - 1527588/10427*e^10 - 5034962/10427*e^9 + 8971034/10427*e^8 + 15273066/10427*e^7 - 23600316/10427*e^6 - 22907878/10427*e^5 + 28275756/10427*e^4 + 14953518/10427*e^3 - 13460165/10427*e^2 - 2433816/10427*e + 1369042/10427, -32694/10427*e^13 + 69119/10427*e^12 + 558542/10427*e^11 - 1108620/10427*e^10 - 3542485/10427*e^9 + 6260144/10427*e^8 + 10819832/10427*e^7 - 15407264/10427*e^6 - 17040569/10427*e^5 + 16471161/10427*e^4 + 12785574/10427*e^3 - 6418644/10427*e^2 - 3154684/10427*e + 402630/10427, 11825/10427*e^13 - 23268/10427*e^12 - 201194/10427*e^11 + 386819/10427*e^10 + 1241578/10427*e^9 - 2320600/10427*e^8 - 3470272/10427*e^7 + 6327668/10427*e^6 + 4326860/10427*e^5 - 8033807/10427*e^4 - 1710794/10427*e^3 + 4105247/10427*e^2 - 98468/10427*e - 355718/10427, -3446/10427*e^13 + 7099/10427*e^12 + 38428/10427*e^11 - 117256/10427*e^10 - 25380/10427*e^9 + 726853/10427*e^8 - 950574/10427*e^7 - 2245932/10427*e^6 + 3529025/10427*e^5 + 3669305/10427*e^4 - 3921513/10427*e^3 - 2533280/10427*e^2 + 1107205/10427*e + 172296/10427, 12974/10427*e^13 - 34068/10427*e^12 - 230286/10427*e^11 + 582593/10427*e^10 + 1539600/10427*e^9 - 3623153/10427*e^8 - 4994326/10427*e^7 + 10276886/10427*e^6 + 8202355/10427*e^5 - 13447910/10427*e^4 - 5921157/10427*e^3 + 6861491/10427*e^2 + 924119/10427*e - 408168/10427, -5617/10427*e^13 + 16561/10427*e^12 + 107408/10427*e^11 - 306884/10427*e^10 - 782794/10427*e^9 + 2122899/10427*e^8 + 2739118/10427*e^7 - 6857965/10427*e^6 - 4609205/10427*e^5 + 10356410/10427*e^4 + 3015168/10427*e^3 - 6165443/10427*e^2 - 64796/10427*e + 798974/10427, 16888/10427*e^13 - 47390/10427*e^12 - 273594/10427*e^11 + 764453/10427*e^10 + 1592579/10427*e^9 - 4335705/10427*e^8 - 4247980/10427*e^7 + 10631934/10427*e^6 + 5402836/10427*e^5 - 11029079/10427*e^4 - 2670182/10427*e^3 + 3952084/10427*e^2 + 87855/10427*e - 425848/10427, 35200/10427*e^13 - 77265/10427*e^12 - 600358/10427*e^11 + 1260096/10427*e^10 + 3781216/10427*e^9 - 7305271/10427*e^8 - 11341531/10427*e^7 + 18719697/10427*e^6 + 17308278/10427*e^5 - 21178107/10427*e^4 - 12758381/10427*e^3 + 8750989/10427*e^2 + 3470063/10427*e - 547716/10427, -21788/10427*e^13 + 34833/10427*e^12 + 402133/10427*e^11 - 596721/10427*e^10 - 2801436/10427*e^9 + 3723976/10427*e^8 + 9372397/10427*e^7 - 10610582/10427*e^6 - 15535731/10427*e^5 + 13957706/10427*e^4 + 11299382/10427*e^3 - 7360672/10427*e^2 - 2191443/10427*e + 867630/10427, 13086/10427*e^13 - 10782/10427*e^12 - 258368/10427*e^11 + 198263/10427*e^10 + 1913467/10427*e^9 - 1341030/10427*e^8 - 6596022/10427*e^7 + 4097894/10427*e^6 + 10533366/10427*e^5 - 5512634/10427*e^4 - 6569736/10427*e^3 + 2695553/10427*e^2 + 869361/10427*e - 173142/10427, 40983/10427*e^13 - 91711/10427*e^12 - 704328/10427*e^11 + 1530987/10427*e^10 + 4477060/10427*e^9 - 9220573/10427*e^8 - 13497792/10427*e^7 + 25154360/10427*e^6 + 20153847/10427*e^5 - 31584861/10427*e^4 - 13019286/10427*e^3 + 15589849/10427*e^2 + 1953329/10427*e - 1263864/10427, 462/10427*e^13 - 395/10427*e^12 + 6679/10427*e^11 - 13491/10427*e^10 - 179557/10427*e^9 + 236694/10427*e^8 + 1098557/10427*e^7 - 1078735/10427*e^6 - 2637608/10427*e^5 + 1561497/10427*e^4 + 2476853/10427*e^3 - 284934/10427*e^2 - 442237/10427*e - 310432/10427, 18578/10427*e^13 - 44953/10427*e^12 - 297009/10427*e^11 + 722009/10427*e^10 + 1685870/10427*e^9 - 4112017/10427*e^8 - 4307715/10427*e^7 + 10439948/10427*e^6 + 5196385/10427*e^5 - 12255804/10427*e^4 - 2511466/10427*e^3 + 6033018/10427*e^2 - 66235/10427*e - 754966/10427, -12987/10427*e^13 + 18890/10427*e^12 + 250117/10427*e^11 - 343408/10427*e^10 - 1821606/10427*e^9 + 2306347/10427*e^8 + 6334655/10427*e^7 - 7107847/10427*e^6 - 10864705/10427*e^5 + 9939838/10427*e^4 + 8508880/10427*e^3 - 5148862/10427*e^2 - 2333787/10427*e + 464118/10427, 25866/10427*e^13 - 26719/10427*e^12 - 531993/10427*e^11 + 501065/10427*e^10 + 4189474/10427*e^9 - 3487472/10427*e^8 - 15870592/10427*e^7 + 11160611/10427*e^6 + 29470343/10427*e^5 - 16337420/10427*e^4 - 23813017/10427*e^3 + 9377694/10427*e^2 + 5552302/10427*e - 853230/10427, 4739/10427*e^13 - 1366/10427*e^12 - 84261/10427*e^11 - 2360/10427*e^10 + 561288/10427*e^9 + 244872/10427*e^8 - 1810560/10427*e^7 - 1586905/10427*e^6 + 3058976/10427*e^5 + 3523258/10427*e^4 - 2701683/10427*e^3 - 2475319/10427*e^2 + 1047063/10427*e + 154888/10427, 25243/10427*e^13 - 40247/10427*e^12 - 452686/10427*e^11 + 659074/10427*e^10 + 3043707/10427*e^9 - 3861629/10427*e^8 - 9786875/10427*e^7 + 10142016/10427*e^6 + 15524608/10427*e^5 - 12103933/10427*e^4 - 10613108/10427*e^3 + 5613873/10427*e^2 + 1809914/10427*e - 321500/10427, -15965/10427*e^13 + 19766/10427*e^12 + 325373/10427*e^11 - 375206/10427*e^10 - 2538850/10427*e^9 + 2672129/10427*e^8 + 9548766/10427*e^7 - 8909015/10427*e^6 - 17700502/10427*e^5 + 13889440/10427*e^4 + 14416818/10427*e^3 - 8411825/10427*e^2 - 3364278/10427*e + 770176/10427, 30516/10427*e^13 - 80663/10427*e^12 - 512571/10427*e^11 + 1340542/10427*e^10 + 3148155/10427*e^9 - 7998628/10427*e^8 - 9032018/10427*e^7 + 21431159/10427*e^6 + 12589225/10427*e^5 - 26116017/10427*e^4 - 7399126/10427*e^3 + 12510658/10427*e^2 + 1108934/10427*e - 1194376/10427, -20463/10427*e^13 + 44240/10427*e^12 + 367641/10427*e^11 - 741240/10427*e^10 - 2495352/10427*e^9 + 4479316/10427*e^8 + 8216692/10427*e^7 - 12230200/10427*e^6 - 13612903/10427*e^5 + 15290389/10427*e^4 + 9929303/10427*e^3 - 7605722/10427*e^2 - 2041398/10427*e + 943196/10427, -14870/10427*e^13 + 19710/10427*e^12 + 316896/10427*e^11 - 402645/10427*e^10 - 2562036/10427*e^9 + 3089584/10427*e^8 + 9760711/10427*e^7 - 11017870/10427*e^6 - 17540183/10427*e^5 + 18181412/10427*e^4 + 12548890/10427*e^3 - 11631479/10427*e^2 - 1513528/10427*e + 1078466/10427, 26863/10427*e^13 - 49757/10427*e^12 - 466370/10427*e^11 + 807308/10427*e^10 + 3003691/10427*e^9 - 4651098/10427*e^8 - 9210495/10427*e^7 + 11851624/10427*e^6 + 14191029/10427*e^5 - 13446864/10427*e^4 - 10222379/10427*e^3 + 5900931/10427*e^2 + 2542455/10427*e - 695846/10427, 4707/10427*e^13 - 5040/10427*e^12 - 85175/10427*e^11 + 95532/10427*e^10 + 572145/10427*e^9 - 685713/10427*e^8 - 1820658/10427*e^7 + 2320525/10427*e^6 + 2969437/10427*e^5 - 3640185/10427*e^4 - 2564041/10427*e^3 + 2218421/10427*e^2 + 809481/10427*e - 385946/10427, 15110/10427*e^13 - 33863/10427*e^12 - 268333/10427*e^11 + 565177/10427*e^10 + 1808067/10427*e^9 - 3403883/10427*e^8 - 6025099/10427*e^7 + 9291560/10427*e^6 + 10573948/10427*e^5 - 11622124/10427*e^4 - 8857774/10427*e^3 + 5384208/10427*e^2 + 2304828/10427*e - 98176/10427, -12417/10427*e^13 + 28288/10427*e^12 + 236420/10427*e^11 - 515239/10427*e^10 - 1692411/10427*e^9 + 3461511/10427*e^8 + 5702523/10427*e^7 - 10647382/10427*e^6 - 9127072/10427*e^5 + 14799191/10427*e^4 + 6071469/10427*e^3 - 7422899/10427*e^2 - 970586/10427*e + 370636/10427, 5856/10427*e^13 - 15840/10427*e^12 - 93413/10427*e^11 + 249598/10427*e^10 + 536503/10427*e^9 - 1373073/10427*e^8 - 1436571/10427*e^7 + 3256340/10427*e^6 + 1881680/10427*e^5 - 3298584/10427*e^4 - 904003/10427*e^3 + 1241799/10427*e^2 + 132467/10427*e - 125586/10427, 36448/10427*e^13 - 59103/10427*e^12 - 679409/10427*e^11 + 1029196/10427*e^10 + 4786292/10427*e^9 - 6599807/10427*e^8 - 16150782/10427*e^7 + 19658962/10427*e^6 + 26681127/10427*e^5 - 27680889/10427*e^4 - 18689477/10427*e^3 + 15675069/10427*e^2 + 3007370/10427*e - 1393598/10427, 27007/10427*e^13 - 54078/10427*e^12 - 472684/10427*e^11 + 898571/10427*e^10 + 3074745/10427*e^9 - 5358942/10427*e^8 - 9498718/10427*e^7 + 14319310/10427*e^6 + 14484471/10427*e^5 - 17283577/10427*e^4 - 9673944/10427*e^3 + 8218997/10427*e^2 + 1901546/10427*e - 900124/10427, -7111/10427*e^13 + 18380/10427*e^12 + 107747/10427*e^11 - 285330/10427*e^10 - 573729/10427*e^9 + 1537047/10427*e^8 + 1445238/10427*e^7 - 3559170/10427*e^6 - 2124749/10427*e^5 + 3531460/10427*e^4 + 1593708/10427*e^3 - 1027598/10427*e^2 + 61243/10427*e - 259270/10427, -5448/10427*e^13 + 15762/10427*e^12 + 68572/10427*e^11 - 236054/10427*e^10 - 192681/10427*e^9 + 1192240/10427*e^8 - 473158/10427*e^7 - 2458201/10427*e^6 + 2818156/10427*e^5 + 2043329/10427*e^4 - 3838268/10427*e^3 - 641665/10427*e^2 + 1567261/10427*e - 11792/10427, 9745/10427*e^13 - 1403/10427*e^12 - 208469/10427*e^11 + 45238/10427*e^10 + 1686608/10427*e^9 - 455165/10427*e^8 - 6420582/10427*e^7 + 1846142/10427*e^6 + 11613564/10427*e^5 - 3014103/10427*e^4 - 8759082/10427*e^3 + 1633128/10427*e^2 + 1819657/10427*e - 183252/10427, 2902/10427*e^13 - 6995/10427*e^12 - 33112/10427*e^11 + 113100/10427*e^10 + 43117/10427*e^9 - 656050/10427*e^8 + 591222/10427*e^7 + 1730902/10427*e^6 - 2131628/10427*e^5 - 2127707/10427*e^4 + 1923795/10427*e^3 + 933698/10427*e^2 - 57723/10427*e - 3028/10427, 12449/10427*e^13 - 35041/10427*e^12 - 193798/10427*e^11 + 573752/10427*e^10 + 1055934/10427*e^9 - 3354659/10427*e^8 - 2564325/10427*e^7 + 8721082/10427*e^6 + 3095962/10427*e^5 - 9929688/10427*e^4 - 2111300/10427*e^3 + 3678016/10427*e^2 + 801515/10427*e + 24220/10427, -3347/10427*e^13 + 4780/10427*e^12 + 82312/10427*e^11 - 95569/10427*e^10 - 767679/10427*e^9 + 712032/10427*e^8 + 3365512/10427*e^7 - 2461449/10427*e^6 - 6926931/10427*e^5 + 4061260/10427*e^4 + 5639768/10427*e^3 - 3088875/10427*e^2 - 920279/10427*e + 859498/10427, 5518/10427*e^13 - 24669/10427*e^12 - 57449/10427*e^11 + 368613/10427*e^10 + 34032/10427*e^9 - 1836976/10427*e^8 + 1129991/10427*e^7 + 3559583/10427*e^6 - 3578315/10427*e^5 - 2125236/10427*e^4 + 3428996/10427*e^3 - 265052/10427*e^2 - 702156/10427*e - 109812/10427, -9357/10427*e^13 + 6849/10427*e^12 + 180450/10427*e^11 - 111276/10427*e^10 - 1313843/10427*e^9 + 650697/10427*e^8 + 4522789/10427*e^7 - 1757420/10427*e^6 - 7380253/10427*e^5 + 2428724/10427*e^4 + 4793626/10427*e^3 - 1847913/10427*e^2 - 578621/10427*e + 455990/10427, -21272/10427*e^13 + 34121/10427*e^12 + 377770/10427*e^11 - 564258/10427*e^10 - 2503380/10427*e^9 + 3387632/10427*e^8 + 7890368/10427*e^7 - 9426672/10427*e^6 - 12205931/10427*e^5 + 12767937/10427*e^4 + 8065879/10427*e^3 - 7560963/10427*e^2 - 1272173/10427*e + 1030618/10427, -19639/10427*e^13 + 29362/10427*e^12 + 365109/10427*e^11 - 498804/10427*e^10 - 2574200/10427*e^9 + 3070382/10427*e^8 + 8784312/10427*e^7 - 8540028/10427*e^6 - 15089668/10427*e^5 + 10730997/10427*e^4 + 12020815/10427*e^3 - 5211960/10427*e^2 - 3519731/10427*e + 412636/10427, 4810/10427*e^13 - 4293/10427*e^12 - 80278/10427*e^11 + 47634/10427*e^10 + 489300/10427*e^9 - 66771/10427*e^8 - 1400401/10427*e^7 - 695608/10427*e^6 + 1970232/10427*e^5 + 2338047/10427*e^4 - 1103497/10427*e^3 - 1786103/10427*e^2 + 18620/10427*e + 89938/10427, -15075/10427*e^13 + 24196/10427*e^12 + 289535/10427*e^11 - 431122/10427*e^10 - 2111572/10427*e^9 + 2841694/10427*e^8 + 7359721/10427*e^7 - 8699161/10427*e^6 - 12325178/10427*e^5 + 12483106/10427*e^4 + 8194350/10427*e^3 - 7125301/10427*e^2 - 918205/10427*e + 725556/10427]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([13, 13, -w^3 + w^2 + 4*w + 1])] = 1

# EXAMPLE:
# pp = ZF.ideal(2).factor()[0][0]
# hecke_eigenvalues[pp]