/* 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![-1, 1, 6, -2, -6, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [5, 5, -w^4 + w^3 + 4*w^2 - 2*w - 1], [7, 7, w^5 - w^4 - 5*w^3 + 3*w^2 + 4*w - 3], [19, 19, w^5 - w^4 - 5*w^3 + 2*w^2 + 4*w], [23, 23, -w^2 + w + 2], [25, 5, w^5 + w^4 - 6*w^3 - 7*w^2 + 4*w + 2], [41, 41, 2*w^5 - w^4 - 10*w^3 + 6*w - 1], [47, 47, w^3 - w^2 - 4*w], [53, 53, w^5 - 5*w^3 - 3*w^2 + w + 3], [59, 59, 2*w^5 - 11*w^3 - 4*w^2 + 8*w], [61, 61, w^2 - 2*w - 2], [64, 2, -2], [67, 67, -w^4 + 6*w^2 + w - 3], [67, 67, -2*w^4 + w^3 + 10*w^2 - 6], [73, 73, w^5 - 5*w^3 - 2*w^2 + 2*w - 2], [73, 73, -w^5 - w^4 + 7*w^3 + 6*w^2 - 8*w - 2], [79, 79, w^5 - w^4 - 5*w^3 + 3*w^2 + 5*w - 1], [83, 83, 2*w^5 - 11*w^3 - 4*w^2 + 6*w], [89, 89, -2*w^5 + 11*w^3 + 4*w^2 - 7*w - 2], [97, 97, w^5 - w^4 - 5*w^3 + 3*w^2 + 2*w - 2], [97, 97, w^5 - w^4 - 6*w^3 + 3*w^2 + 7*w - 1], [97, 97, -w^5 + 2*w^4 + 5*w^3 - 8*w^2 - 5*w + 4], [101, 101, w^5 - 6*w^3 - 3*w^2 + 6*w + 2], [103, 103, -w^5 - w^4 + 5*w^3 + 8*w^2 - 5], [107, 107, 2*w^4 - w^3 - 9*w^2 + w + 2], [109, 109, 2*w^5 - 10*w^3 - 5*w^2 + 4*w + 1], [113, 113, -w^4 + 6*w^2 + 2*w - 3], [125, 5, 2*w^5 - 10*w^3 - 6*w^2 + 5*w + 6], [127, 127, 2*w^5 - w^4 - 11*w^3 + 10*w], [131, 131, 2*w^5 - w^4 - 11*w^3 + 9*w + 1], [137, 137, -w^5 + 2*w^4 + 6*w^3 - 9*w^2 - 9*w + 5], [139, 139, -w^5 - 2*w^4 + 7*w^3 + 11*w^2 - 5*w - 5], [157, 157, 2*w^5 - 10*w^3 - 5*w^2 + 5*w + 1], [167, 167, -2*w^5 + 10*w^3 + 5*w^2 - 3*w + 1], [173, 173, 4*w^5 + 2*w^4 - 22*w^3 - 19*w^2 + 10*w + 5], [179, 179, 2*w^5 - w^4 - 11*w^3 + 10*w + 1], [179, 179, 2*w^5 - 11*w^3 - 5*w^2 + 8*w + 1], [181, 181, -4*w^5 - w^4 + 22*w^3 + 14*w^2 - 12*w - 4], [191, 191, w^4 + w^3 - 6*w^2 - 4*w + 2], [191, 191, -3*w^5 + 18*w^3 + 5*w^2 - 15*w - 1], [193, 193, -w^4 + w^3 + 6*w^2 - 2*w - 7], [197, 197, -3*w^5 + 2*w^4 + 16*w^3 - 3*w^2 - 12*w + 1], [197, 197, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 7*w - 5], [197, 197, -w^5 + w^4 + 6*w^3 - 3*w^2 - 9*w + 1], [211, 211, -2*w^5 + w^4 + 10*w^3 - w^2 - 4*w + 4], [223, 223, 2*w^5 - 10*w^3 - 6*w^2 + 5*w + 4], [227, 227, w^5 - 7*w^3 - w^2 + 9*w], [227, 227, -2*w^5 + w^4 + 9*w^3 - 2*w + 1], [229, 229, w^5 + w^4 - 5*w^3 - 6*w^2 - 2*w - 2], [233, 233, -2*w^5 - w^4 + 12*w^3 + 8*w^2 - 8*w - 4], [269, 269, 3*w^5 - 16*w^3 - 6*w^2 + 8*w], [269, 269, -2*w^5 + 12*w^3 + 3*w^2 - 12*w - 1], [277, 277, 3*w^5 - w^4 - 17*w^3 - w^2 + 16*w], [289, 17, -2*w^5 - w^4 + 12*w^3 + 8*w^2 - 10*w - 3], [307, 307, w^4 - 5*w^2 - 2*w - 1], [307, 307, 3*w^5 - w^4 - 16*w^3 - 3*w^2 + 12*w + 3], [307, 307, -w^4 - w^3 + 5*w^2 + 6*w - 2], [311, 311, -w^4 - w^3 + 7*w^2 + 4*w - 4], [311, 311, 3*w^5 - w^4 - 17*w^3 - 2*w^2 + 16*w + 2], [313, 313, -w^5 + w^4 + 6*w^3 - 3*w^2 - 7*w], [317, 317, 2*w^5 + w^4 - 12*w^3 - 8*w^2 + 7*w], [331, 331, 2*w^5 - 2*w^4 - 10*w^3 + 5*w^2 + 8*w - 5], [337, 337, w^5 - 6*w^3 - 3*w^2 + 6*w], [347, 347, -2*w^5 + 2*w^4 + 10*w^3 - 6*w^2 - 7*w + 2], [347, 347, 3*w^5 + w^4 - 17*w^3 - 11*w^2 + 9*w + 2], [349, 349, -2*w^5 + 11*w^3 + 5*w^2 - 7*w - 6], [367, 367, -w^5 - 2*w^4 + 5*w^3 + 13*w^2 + 2*w - 5], [367, 367, 2*w^5 + 2*w^4 - 10*w^3 - 16*w^2 - w + 6], [373, 373, 3*w^5 - w^4 - 17*w^3 - w^2 + 15*w + 1], [373, 373, 2*w^4 - 2*w^3 - 9*w^2 + 2*w + 4], [379, 379, 2*w^5 + w^4 - 11*w^3 - 10*w^2 + 5*w + 7], [389, 389, w^2 - 5], [389, 389, w^5 - 7*w^3 + 8*w], [397, 397, -2*w^5 - w^4 + 12*w^3 + 9*w^2 - 7*w - 5], [397, 397, -3*w^5 - w^4 + 18*w^3 + 10*w^2 - 14*w - 2], [397, 397, -2*w^5 - 2*w^4 + 12*w^3 + 13*w^2 - 6*w - 4], [419, 419, 2*w^5 + 2*w^4 - 11*w^3 - 14*w^2 + 2*w + 5], [419, 419, -w^5 + 4*w^3 + 3*w^2 + w - 2], [433, 433, -w^5 + 5*w^3 + 4*w^2 - w - 3], [433, 433, w^5 - 2*w^4 - 5*w^3 + 7*w^2 + 6*w - 4], [439, 439, 4*w^5 + 2*w^4 - 22*w^3 - 18*w^2 + 9*w + 4], [439, 439, w^5 + w^4 - 5*w^3 - 9*w^2 + 6], [439, 439, -3*w^5 + 2*w^4 + 16*w^3 - 3*w^2 - 13*w + 2], [443, 443, -3*w^5 + w^4 + 15*w^3 + 3*w^2 - 8*w - 2], [449, 449, -w^5 + w^4 + 5*w^3 - 4*w^2 - 2*w + 3], [449, 449, 3*w^5 - 16*w^3 - 8*w^2 + 9*w + 4], [449, 449, w^4 + w^3 - 6*w^2 - 6*w + 5], [449, 449, -w^5 + w^4 + 6*w^3 - 5*w^2 - 7*w + 4], [457, 457, -2*w^5 + 2*w^4 + 10*w^3 - 5*w^2 - 6*w + 4], [457, 457, w^5 - 3*w^4 - 4*w^3 + 12*w^2 + 5*w - 4], [461, 461, -2*w^4 + w^3 + 10*w^2 + w - 5], [463, 463, -w^5 + w^4 + 6*w^3 - 4*w^2 - 5*w + 4], [487, 487, -3*w^4 + 2*w^3 + 14*w^2 - w - 7], [487, 487, 3*w^5 + w^4 - 18*w^3 - 10*w^2 + 14*w + 3], [491, 491, -3*w^5 + w^4 + 16*w^3 + w^2 - 12*w + 2], [491, 491, 3*w^5 - w^4 - 16*w^3 - 2*w^2 + 12*w + 3], [499, 499, 4*w^5 - 22*w^3 - 9*w^2 + 15*w + 4], [499, 499, w^4 + w^3 - 7*w^2 - 6*w + 6], [523, 523, w^4 - 2*w^3 - 4*w^2 + 5*w + 2], [529, 23, -w^5 + 7*w^3 + w^2 - 10*w - 1], [541, 541, w^5 + 2*w^4 - 8*w^3 - 10*w^2 + 9*w + 3], [557, 557, w^5 - w^4 - 5*w^3 + 3*w^2 + 5*w - 5], [557, 557, 4*w^5 - 22*w^3 - 8*w^2 + 14*w + 1], [563, 563, 2*w^5 - w^4 - 12*w^3 + 2*w^2 + 13*w - 1], [563, 563, -w^5 - 2*w^4 + 6*w^3 + 13*w^2 - 2*w - 6], [563, 563, -w^5 + w^4 + 4*w^3 - 2*w^2 + w + 3], [569, 569, -3*w^5 + w^4 + 16*w^3 + w^2 - 9*w + 1], [599, 599, -w^4 - w^3 + 7*w^2 + 6*w - 5], [607, 607, -w^5 + 6*w^3 + 3*w^2 - 7*w - 1], [613, 613, -w^5 - 2*w^4 + 7*w^3 + 12*w^2 - 4*w - 7], [619, 619, 4*w^5 - w^4 - 22*w^3 - 3*w^2 + 16*w], [653, 653, 2*w^5 - 2*w^4 - 10*w^3 + 4*w^2 + 7*w], [653, 653, -2*w^5 + w^4 + 12*w^3 - w^2 - 14*w + 1], [659, 659, -2*w^5 - w^4 + 11*w^3 + 11*w^2 - 5*w - 7], [661, 661, 2*w^5 - 11*w^3 - 3*w^2 + 7*w], [661, 661, 2*w^5 - w^4 - 11*w^3 + 11*w + 2], [673, 673, -w^4 - w^3 + 5*w^2 + 7*w - 1], [677, 677, 3*w^5 - 16*w^3 - 7*w^2 + 10*w + 3], [677, 677, w^5 + 3*w^4 - 6*w^3 - 18*w^2 + 12], [691, 691, w^5 - w^4 - 4*w^3 + 2*w^2 - 4], [709, 709, -w^4 + w^3 + 3*w^2 - 2*w + 2], [709, 709, -2*w^5 - w^4 + 11*w^3 + 9*w^2 - 6*w - 1], [719, 719, 2*w^5 + 2*w^4 - 12*w^3 - 14*w^2 + 8*w + 5], [729, 3, -3], [739, 739, 4*w^5 - w^4 - 21*w^3 - 5*w^2 + 12*w + 3], [743, 743, -2*w^5 - w^4 + 10*w^3 + 11*w^2 - w - 6], [751, 751, -3*w^5 + w^4 + 17*w^3 + 2*w^2 - 15*w - 2], [751, 751, -2*w^5 + 3*w^4 + 10*w^3 - 11*w^2 - 10*w + 6], [757, 757, w^5 - 3*w^4 - 5*w^3 + 13*w^2 + 7*w - 6], [761, 761, -w^5 + w^4 + 6*w^3 - 3*w^2 - 6*w - 1], [787, 787, -3*w^5 - w^4 + 18*w^3 + 10*w^2 - 15*w - 2], [797, 797, 2*w^5 - 2*w^4 - 11*w^3 + 6*w^2 + 12*w - 1], [797, 797, w^2 - 2*w - 5], [809, 809, 2*w^5 - w^4 - 12*w^3 + 3*w^2 + 12*w - 3], [827, 827, 4*w^5 - 22*w^3 - 8*w^2 + 15*w + 1], [839, 839, 3*w^5 - w^4 - 16*w^3 - 3*w^2 + 10*w + 2], [839, 839, -w^5 + w^4 + 6*w^3 - 4*w^2 - 8*w + 1], [841, 29, -2*w^5 - w^4 + 11*w^3 + 9*w^2 - 7*w - 3], [853, 853, w^5 - 7*w^3 - w^2 + 7*w + 1], [857, 857, -2*w^5 + w^4 + 10*w^3 - w^2 - 6*w + 4], [863, 863, 3*w^5 - w^4 - 17*w^3 - 2*w^2 + 16*w], [877, 877, 2*w^5 - 10*w^3 - 5*w^2 + 3*w + 4], [887, 887, 2*w^5 + w^4 - 13*w^3 - 8*w^2 + 13*w + 5], [887, 887, 3*w^5 - 2*w^4 - 15*w^3 + 3*w^2 + 8*w - 4], [907, 907, 2*w^5 - w^4 - 13*w^3 + 2*w^2 + 17*w], [907, 907, -2*w^5 + 2*w^4 + 10*w^3 - 4*w^2 - 9*w + 1], [919, 919, -3*w^5 + 17*w^3 + 5*w^2 - 11*w], [929, 929, -3*w^5 + 17*w^3 + 6*w^2 - 11*w], [941, 941, 2*w^5 - w^4 - 11*w^3 - w^2 + 9*w + 2], [947, 947, -w^4 + 6*w^2 + 4*w - 4], [947, 947, -2*w^5 + w^4 + 11*w^3 - 11*w - 3], [947, 947, -4*w^5 + w^4 + 22*w^3 + 3*w^2 - 16*w - 1], [953, 953, -w^5 - 2*w^4 + 6*w^3 + 13*w^2 - 3*w - 8], [953, 953, 2*w^5 + 2*w^4 - 13*w^3 - 14*w^2 + 10*w + 6], [961, 31, 2*w^5 + w^4 - 12*w^3 - 10*w^2 + 9*w + 4], [967, 967, 2*w^5 - w^4 - 11*w^3 + w^2 + 11*w + 1], [967, 967, 2*w^5 - 2*w^4 - 11*w^3 + 6*w^2 + 13*w - 3], [983, 983, -4*w^5 + w^4 + 22*w^3 + 4*w^2 - 18*w - 2], [991, 991, 3*w^5 - 16*w^3 - 7*w^2 + 7*w + 2], [991, 991, -2*w^5 - w^4 + 12*w^3 + 8*w^2 - 11*w - 2]]; primes := [ideal : I in primesArray]; heckePol := x^11 + 8*x^10 + 4*x^9 - 105*x^8 - 224*x^7 + 129*x^6 + 446*x^5 - 98*x^4 - 210*x^3 + 8*x^2 + 28*x + 4; K := NumberField(heckePol); heckeEigenvaluesArray := [e, -1487/522*e^10 - 3809/174*e^9 - 1105/261*e^8 + 157915/522*e^7 + 284195/522*e^6 - 293947/522*e^5 - 603587/522*e^4 + 5708/9*e^3 + 124786/261*e^2 - 39649/261*e - 13238/261, 745/174*e^10 + 955/29*e^9 + 602/87*e^8 - 78731/174*e^7 - 71486/87*e^6 + 141803/174*e^5 + 147608/87*e^4 - 2821/3*e^3 - 59201/87*e^2 + 20402/87*e + 6154/87, 2252/261*e^10 + 5831/87*e^9 + 4937/261*e^8 - 237628/261*e^7 - 449093/261*e^6 + 395143/261*e^5 + 911675/261*e^4 - 14944/9*e^3 - 370550/261*e^2 + 103283/261*e + 37636/261, 36/29*e^10 + 280/29*e^9 + 85/29*e^8 - 3786/29*e^7 - 7271/29*e^6 + 6060/29*e^5 + 14835/29*e^4 - 216*e^3 - 6226/29*e^2 + 1591/29*e + 532/29, 206/29*e^10 + 1599/29*e^9 + 430/29*e^8 - 21819/29*e^7 - 40928/29*e^6 + 37364/29*e^5 + 84749/29*e^4 - 1401*e^3 - 35401/29*e^2 + 9618/29*e + 3650/29, 1, -3757/174*e^10 - 9735/58*e^9 - 4211/87*e^8 + 396305/174*e^7 + 751699/174*e^6 - 653753/174*e^5 - 1524289/174*e^4 + 12277/3*e^3 + 309254/87*e^2 - 85823/87*e - 31174/87, -4355/261*e^10 - 11309/87*e^9 - 10451/261*e^8 + 458257/261*e^7 + 878309/261*e^6 - 731719/261*e^5 - 1750787/261*e^4 + 27448/9*e^3 + 693512/261*e^2 - 189914/261*e - 68356/261, -2153/261*e^10 - 5555/87*e^9 - 4145/261*e^8 + 228043/261*e^7 + 422696/261*e^6 - 399184/261*e^5 - 874163/261*e^4 + 15520/9*e^3 + 362288/261*e^2 - 112922/261*e - 37942/261, -66/29*e^10 - 523/29*e^9 - 238/29*e^8 + 6854/29*e^7 + 14263/29*e^6 - 8413/29*e^5 - 26574/29*e^4 + 266*e^3 + 9742/29*e^2 - 1694/29*e - 695/29, 949/174*e^10 + 1213/29*e^9 + 635/87*e^8 - 100997/174*e^7 - 89888/87*e^6 + 193079/174*e^5 + 193193/87*e^4 - 3829/3*e^3 - 82013/87*e^2 + 28820/87*e + 9034/87, 9731/522*e^10 + 12625/87*e^9 + 11374/261*e^8 - 1025083/522*e^7 - 978049/261*e^6 + 1659589/522*e^5 + 1961164/261*e^4 - 31304/9*e^3 - 784999/261*e^2 + 218614/261*e + 79082/261, 4369/522*e^10 + 5627/87*e^9 + 4136/261*e^8 - 461189/522*e^7 - 426836/261*e^6 + 798359/522*e^5 + 870842/261*e^4 - 15547/9*e^3 - 348023/261*e^2 + 108728/261*e + 37204/261, -1921/522*e^10 - 4975/174*e^9 - 2174/261*e^8 + 201827/522*e^7 + 383053/522*e^6 - 325553/522*e^5 - 755437/522*e^4 + 6223/9*e^3 + 141878/261*e^2 - 41903/261*e - 12562/261, -6935/522*e^10 - 9007/87*e^9 - 8455/261*e^8 + 729007/522*e^7 + 700936/261*e^6 - 1152157/522*e^5 - 1394395/261*e^4 + 21407/9*e^3 + 552580/261*e^2 - 147700/261*e - 54614/261, -2441/174*e^10 - 6321/58*e^9 - 2659/87*e^8 + 257809/174*e^7 + 486461/174*e^6 - 431569/174*e^5 - 989363/174*e^4 + 8252/3*e^3 + 202684/87*e^2 - 58939/87*e - 21302/87, -653/261*e^10 - 1679/87*e^9 - 1106/261*e^8 + 69307/261*e^7 + 126311/261*e^6 - 126094/261*e^5 - 264245/261*e^4 + 5086/9*e^3 + 113756/261*e^2 - 39854/261*e - 14422/261, -323/18*e^10 - 418/3*e^9 - 352/9*e^8 + 34069/18*e^7 + 32170/9*e^6 - 56701/18*e^5 - 65245/9*e^4 + 31399/9*e^3 + 26653/9*e^2 - 7864/9*e - 2804/9, -13147/522*e^10 - 16964/87*e^9 - 13061/261*e^8 + 1388663/522*e^7 + 1293560/261*e^6 - 2381375/522*e^5 - 2644190/261*e^4 + 45772/9*e^3 + 1067003/261*e^2 - 319892/261*e - 110116/261, 2239/261*e^10 + 5764/87*e^9 + 4108/261*e^8 - 236807/261*e^7 - 436903/261*e^6 + 415739/261*e^5 + 904822/261*e^4 - 15800/9*e^3 - 364804/261*e^2 + 104032/261*e + 35894/261, 8429/522*e^10 + 10876/87*e^9 + 8428/261*e^8 - 889693/522*e^7 - 829762/261*e^6 + 1518313/522*e^5 + 1692151/261*e^4 - 29093/9*e^3 - 678652/261*e^2 + 203500/261*e + 70148/261, 6461/522*e^10 + 8341/87*e^9 + 6472/261*e^8 - 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200714/261, 731/174*e^10 + 1849/58*e^9 + 169/87*e^8 - 79279/174*e^7 - 132809/174*e^6 + 176779/174*e^5 + 323285/174*e^4 - 3404/3*e^3 - 80728/87*e^2 + 25423/87*e + 9740/87, 4508/261*e^10 + 11522/87*e^9 + 6194/261*e^8 - 478741/261*e^7 - 855278/261*e^6 + 904591/261*e^5 + 1827647/261*e^4 - 35791/9*e^3 - 773405/261*e^2 + 267008/261*e + 87292/261, -103/6*e^10 - 267/2*e^9 - 116/3*e^8 + 10871/6*e^7 + 20623/6*e^6 - 17975/6*e^5 - 41839/6*e^4 + 9926/3*e^3 + 8666/3*e^2 - 2489/3*e - 994/3, 5912/261*e^10 + 15278/87*e^9 + 12206/261*e^8 - 624829/261*e^7 - 1171124/261*e^6 + 1065502/261*e^5 + 2409866/261*e^4 - 40867/9*e^3 - 1015262/261*e^2 + 302000/261*e + 109432/261, -777/29*e^10 - 6053/29*e^9 - 1837/29*e^8 + 81932/29*e^7 + 156611/29*e^6 - 132825/29*e^5 - 315498/29*e^4 + 4995*e^3 + 127964/29*e^2 - 36038/29*e - 12478/29, 1361/29*e^10 + 10534/29*e^9 + 2681/29*e^8 - 143823/29*e^7 - 267780/29*e^6 + 247681/29*e^5 + 550548/29*e^4 - 9466*e^3 - 225084/29*e^2 + 66552/29*e + 23744/29, 3028/87*e^10 + 7816/29*e^9 + 6070/87*e^8 - 319943/87*e^7 - 597436/87*e^6 + 548264/87*e^5 + 1230340/87*e^4 - 20774/3*e^3 - 505960/87*e^2 + 146410/87*e + 52880/87, 5179/174*e^10 + 6706/29*e^9 + 5723/87*e^8 - 546287/174*e^7 - 516965/87*e^6 + 905129/174*e^5 + 1048589/87*e^4 - 17128/3*e^3 - 426287/87*e^2 + 123734/87*e + 43276/87, 865/522*e^10 + 1088/87*e^9 + 386/261*e^8 - 89843/522*e^7 - 77495/261*e^6 + 167165/522*e^5 + 149162/261*e^4 - 4051/9*e^3 - 57272/261*e^2 + 39980/261*e + 2362/261, 8677/522*e^10 + 11147/87*e^9 + 7622/261*e^8 - 915755/522*e^7 - 839588/261*e^6 + 1612511/522*e^5 + 1715738/261*e^4 - 31969/9*e^3 - 683348/261*e^2 + 230366/261*e + 76324/261, -13181/261*e^10 - 34217/87*e^9 - 31121/261*e^8 + 1389184/261*e^7 + 2656706/261*e^6 - 2246980/261*e^5 - 5357921/261*e^4 + 83701/9*e^3 + 2176700/261*e^2 - 584561/261*e - 217480/261, 7445/261*e^10 + 19217/87*e^9 + 14900/261*e^8 - 786412/261*e^7 - 1467434/261*e^6 + 1344814/261*e^5 + 3007928/261*e^4 - 50809/9*e^3 - 1209737/261*e^2 + 341666/261*e + 122236/261, 21217/522*e^10 + 55195/174*e^9 + 26897/261*e^8 - 2228861/522*e^7 - 4314181/522*e^6 + 3466877/522*e^5 + 8544307/522*e^4 - 64327/9*e^3 - 1701290/261*e^2 + 456227/261*e + 162214/261, -97/18*e^10 - 253/6*e^9 - 137/9*e^8 + 10115/18*e^7 + 20023/18*e^6 - 14483/18*e^5 - 38563/18*e^4 + 7286/9*e^3 + 7136/9*e^2 - 1526/9*e - 766/9, 18841/522*e^10 + 24416/87*e^9 + 21308/261*e^8 - 1986293/522*e^7 - 1885871/261*e^6 + 3260057/522*e^5 + 3806480/261*e^4 - 61600/9*e^3 - 1538723/261*e^2 + 436436/261*e + 154924/261, -8071/174*e^10 - 20797/58*e^9 - 7547/87*e^8 + 854489/174*e^7 + 1579831/174*e^6 - 1502555/174*e^5 - 3284515/174*e^4 + 28912/3*e^3 + 687230/87*e^2 - 206639/87*e - 74872/87, -4018/261*e^10 - 10315/87*e^9 - 6769/261*e^8 + 424667/261*e^7 + 774841/261*e^6 - 757544/261*e^5 - 1596847/261*e^4 + 29573/9*e^3 + 624424/261*e^2 - 205054/261*e - 60782/261, 48007/522*e^10 + 124135/174*e^9 + 50885/261*e^8 - 5065097/522*e^7 - 9526681/522*e^6 + 8499977/522*e^5 + 19362619/522*e^4 - 161902/9*e^3 - 3922466/261*e^2 + 1133039/261*e + 403150/261, 421/174*e^10 + 1041/58*e^9 - 85/87*e^8 - 44309/174*e^7 - 68137/174*e^6 + 97529/174*e^5 + 132817/174*e^4 - 2305/3*e^3 - 12740/87*e^2 + 17027/87*e - 68/87, 19619/522*e^10 + 51269/174*e^9 + 27523/261*e^8 - 2057431/522*e^7 - 4049837/522*e^6 + 3048859/522*e^5 + 7865783/522*e^4 - 55799/9*e^3 - 1524568/261*e^2 + 371842/261*e + 147020/261, -2479/522*e^10 - 6325/174*e^9 - 1535/261*e^8 + 263729/522*e^7 + 466159/522*e^6 - 508919/522*e^5 - 1002649/522*e^4 + 10111/9*e^3 + 211430/261*e^2 - 74033/261*e - 22804/261, 3218/261*e^10 + 8387/87*e^9 + 8837/261*e^8 - 336058/261*e^7 - 660704/261*e^6 + 491836/261*e^5 + 1274063/261*e^4 - 17884/9*e^3 - 491891/261*e^2 + 111692/261*e + 51718/261, 3412/261*e^10 + 8731/87*e^9 + 4966/261*e^8 - 361661/261*e^7 - 648874/261*e^6 + 671948/261*e^5 + 1359928/261*e^4 - 26993/9*e^3 - 553888/261*e^2 + 205276/261*e + 56312/261, 19756/261*e^10 + 51172/87*e^9 + 44107/261*e^8 - 2082245/261*e^7 - 3946573/261*e^6 + 3427190/261*e^5 + 7963300/261*e^4 - 129818/9*e^3 - 3206260/261*e^2 + 909958/261*e + 321272/261, -18677/522*e^10 - 48395/174*e^9 - 20797/261*e^8 + 1971559/522*e^7 + 3734507/522*e^6 - 3270247/522*e^5 - 7590251/522*e^4 + 61760/9*e^3 + 1551286/261*e^2 - 443911/261*e - 155420/261, -7499/261*e^10 - 19502/87*e^9 - 18986/261*e^8 + 787393/261*e^7 + 1524494/261*e^6 - 1224187/261*e^5 - 3022220/261*e^4 + 45796/9*e^3 + 1231778/261*e^2 - 331046/261*e - 129472/261, 18073/261*e^10 + 46915/87*e^9 + 42910/261*e^8 - 1902857/261*e^7 - 3641896/261*e^6 + 3058973/261*e^5 + 7300279/261*e^4 - 114833/9*e^3 - 2949139/261*e^2 + 808384/261*e + 298286/261, 10417/174*e^10 + 13496/29*e^9 + 11624/87*e^8 - 1099499/174*e^7 - 1041950/87*e^6 + 1822007/174*e^5 + 2119679/87*e^4 - 34336/3*e^3 - 871616/87*e^2 + 243068/87*e + 89200/87, -31445/522*e^10 - 81233/174*e^9 - 32284/261*e^8 + 3320185/522*e^7 + 6215075/522*e^6 - 5639347/522*e^5 - 12682313/522*e^4 + 108116/9*e^3 + 2580466/261*e^2 - 757465/261*e - 271202/261, -703/9*e^10 - 1819/3*e^9 - 1519/9*e^8 + 74165/9*e^7 + 139915/9*e^6 - 123740/9*e^5 - 284428/9*e^4 + 135760/9*e^3 + 115732/9*e^2 - 32593/9*e - 11888/9, 5605/261*e^10 + 14626/87*e^9 + 15115/261*e^8 - 588797/261*e^7 - 1150396/261*e^6 + 895349/261*e^5 + 2256442/261*e^4 - 33422/9*e^3 - 892618/261*e^2 + 249760/261*e + 82910/261, -182/29*e^10 - 1364/29*e^9 - 6/29*e^8 + 19411/29*e^7 + 31344/29*e^6 - 42720/29*e^5 - 69900/29*e^4 + 1771*e^3 + 26185/29*e^2 - 12076/29*e - 2522/29, 26836/261*e^10 + 69418/87*e^9 + 57334/261*e^8 - 2832074/261*e^7 - 5332408/261*e^6 + 4746242/261*e^5 + 10844629/261*e^4 - 180104/9*e^3 - 4392214/261*e^2 + 1263358/261*e + 451496/261, -623/58*e^10 - 4823/58*e^9 - 578/29*e^8 + 66297/58*e^7 + 122335/58*e^6 - 118985/58*e^5 - 259843/58*e^4 + 2222*e^3 + 55642/29*e^2 - 14554/29*e - 6738/29, 563/522*e^10 + 1349/174*e^9 - 851/261*e^8 - 62539/522*e^7 - 76451/522*e^6 + 192811/522*e^5 + 212063/522*e^4 - 4181/9*e^3 - 23416/261*e^2 + 27298/261*e - 2212/261, -2440/261*e^10 - 6472/87*e^9 - 9196/261*e^8 + 253697/261*e^7 + 530335/261*e^6 - 304748/261*e^5 - 959905/261*e^4 + 10442/9*e^3 + 327844/261*e^2 - 65749/261*e - 21068/261, 17785/522*e^10 + 45685/174*e^9 + 15083/261*e^8 - 1882139/522*e^7 - 3436147/522*e^6 + 3372065/522*e^5 + 7124005/522*e^4 - 65836/9*e^3 - 1432454/261*e^2 + 460352/261*e + 154120/261, 2999/87*e^10 + 7787/29*e^9 + 7172/87*e^8 - 315796/87*e^7 - 605411/87*e^6 + 506620/87*e^5 + 1217435/87*e^4 - 18796/3*e^3 - 494882/87*e^2 + 129416/87*e + 48994/87, -31951/522*e^10 - 41027/87*e^9 - 27116/261*e^8 + 3379847/522*e^7 + 3088040/261*e^6 - 6044471/522*e^5 - 6417518/261*e^4 + 117298/9*e^3 + 2613209/261*e^2 - 823718/261*e - 279526/261, -7481/174*e^10 - 19349/58*e^9 - 8116/87*e^8 + 787849/174*e^7 + 1487987/174*e^6 - 1302415/174*e^5 - 3003797/174*e^4 + 24617/3*e^3 + 600688/87*e^2 - 168511/87*e - 61964/87, -1433/29*e^10 - 11210/29*e^9 - 3779/29*e^8 + 150757/29*e^7 + 293487/29*e^6 - 232918/29*e^5 - 582045/29*e^4 + 8636*e^3 + 235506/29*e^2 - 61614/29*e - 23532/29, 4067/261*e^10 + 10514/87*e^9 + 8669/261*e^8 - 428665/261*e^7 - 807497/261*e^6 + 715951/261*e^5 + 1640285/261*e^4 - 27475/9*e^3 - 683144/261*e^2 + 205142/261*e + 77788/261, 16081/522*e^10 + 20690/87*e^9 + 14618/261*e^8 - 1700441/522*e^7 - 1567052/261*e^6 + 2995499/522*e^5 + 3249260/261*e^4 - 58138/9*e^3 - 1356803/261*e^2 + 424274/261*e + 150094/261, 6161/522*e^10 + 16133/174*e^9 + 9274/261*e^8 - 642301/522*e^7 - 1281317/522*e^6 + 893803/522*e^5 + 2397497/522*e^4 - 16538/9*e^3 - 427984/261*e^2 + 111754/261*e + 37208/261, 2911/522*e^10 + 7909/174*e^9 + 7700/261*e^8 - 299765/522*e^7 - 685651/522*e^6 + 235553/522*e^5 + 1149871/522*e^4 - 3199/9*e^3 - 191018/261*e^2 + 17459/261*e + 13120/261, 7900/261*e^10 + 20518/87*e^9 + 19120/261*e^8 - 831329/261*e^7 - 1597327/261*e^6 + 1325000/261*e^5 + 3205762/261*e^4 - 49190/9*e^3 - 1310362/261*e^2 + 344944/261*e + 131006/261, -21209/522*e^10 - 54779/174*e^9 - 21964/261*e^8 + 2237029/522*e^7 + 4198505/522*e^6 - 3770707/522*e^5 - 8578517/522*e^4 + 70907/9*e^3 + 1738672/261*e^2 - 487396/261*e - 176816/261, 27923/522*e^10 + 36262/87*e^9 + 33682/261*e^8 - 2938507/522*e^7 - 2819635/261*e^6 + 4684441/522*e^5 + 5636104/261*e^4 - 87008/9*e^3 - 2247370/261*e^2 + 600616/261*e + 217406/261, 4264/87*e^10 + 11101/29*e^9 + 10912/87*e^8 - 448334/87*e^7 - 868669/87*e^6 + 698063/87*e^5 + 1723000/87*e^4 - 25910/3*e^3 - 690700/87*e^2 + 180280/87*e + 67820/87, -32021/522*e^10 - 41629/87*e^9 - 40069/261*e^8 + 3364231/522*e^7 + 3248428/261*e^6 - 5258503/522*e^5 - 6435868/261*e^4 + 97739/9*e^3 + 2552467/261*e^2 - 676204/261*e - 249242/261, 27787/522*e^10 + 71977/174*e^9 + 31253/261*e^8 - 2927027/522*e^7 - 5557459/522*e^6 + 4793459/522*e^5 + 11213023/522*e^4 - 90085/9*e^3 - 2259074/261*e^2 + 633545/261*e + 229174/261, -9740/261*e^10 - 25022/87*e^9 - 16556/261*e^8 + 1031032/261*e^7 + 1883045/261*e^6 - 1849111/261*e^5 - 3912617/261*e^4 + 72220/9*e^3 + 1587548/261*e^2 - 516194/261*e - 166678/261, 1665/58*e^10 + 6504/29*e^9 + 2107/29*e^8 - 175465/58*e^7 - 169523/29*e^6 + 277027/58*e^5 + 338829/29*e^4 - 5110*e^3 - 135124/29*e^2 + 35012/29*e + 12520/29, -19178/261*e^10 - 49478/87*e^9 - 37685/261*e^8 + 2027191/261*e^7 + 3773159/261*e^6 - 3498103/261*e^5 - 7771598/261*e^4 + 133864/9*e^3 + 3201083/261*e^2 - 952214/261*e - 339868/261, -15161/522*e^10 - 39389/174*e^9 - 18391/261*e^8 + 1596283/522*e^7 + 3067457/522*e^6 - 2546437/522*e^5 - 6161927/522*e^4 + 46904/9*e^3 + 1248748/261*e^2 - 324955/261*e - 129692/261, -1363/18*e^10 - 3523/6*e^9 - 1427/9*e^8 + 143849/18*e^7 + 270085/18*e^6 - 242627/18*e^5 - 550147/18*e^4 + 134621/9*e^3 + 111944/9*e^2 - 33371/9*e - 11590/9, 1425/29*e^10 + 11064/29*e^9 + 3135/29*e^8 - 150109/29*e^7 - 284038/29*e^6 + 247473/29*e^5 + 573393/29*e^4 - 9354*e^3 - 230288/29*e^2 + 65691/29*e + 22692/29, 2084/261*e^10 + 5534/87*e^9 + 7595/261*e^8 - 219409/261*e^7 - 454766/261*e^6 + 291289/261*e^5 + 884408/261*e^4 - 9505/9*e^3 - 360761/261*e^2 + 68492/261*e + 31828/261]; 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;