/* 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![7, 6, -5, -2, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [4, 2, -w^2 + w + 3], [7, 7, w], [7, 7, -w^2 + 2*w + 1], [7, 7, w^2 - 2], [7, 7, w - 1], [23, 23, -w^3 + w^2 + 3*w - 1], [23, 23, w^3 - 2*w^2 - 2*w + 2], [31, 31, w^2 - 5], [31, 31, -w^2 + 2*w + 4], [41, 41, -w^3 + 2*w^2 + 2*w - 6], [41, 41, w^3 - w^2 - 3*w - 3], [47, 47, w^2 - 2*w - 5], [47, 47, w^2 - 6], [71, 71, -w^3 + 2*w^2 + 3*w - 1], [71, 71, -w^3 + w^2 + 4*w - 3], [79, 79, -w - 3], [79, 79, w - 4], [81, 3, -3], [89, 89, w^2 - 3*w - 2], [89, 89, w^2 + w - 4], [97, 97, 2*w^3 - 5*w^2 - 4*w + 9], [97, 97, -2*w^3 + w^2 + 8*w + 2], [113, 113, w^3 - 2*w^2 - 4*w + 2], [113, 113, w^3 - w^2 - 5*w + 3], [121, 11, 2*w^2 - w - 9], [121, 11, -2*w^2 + 3*w + 8], [127, 127, w^3 - 3*w^2 - 2*w + 5], [127, 127, w^3 - 5*w - 1], [137, 137, 2*w - 1], [167, 167, 2*w^3 - 2*w^2 - 7*w - 2], [167, 167, w^3 - 7*w - 4], [167, 167, 2*w^3 - w^2 - 9*w - 2], [167, 167, 2*w^3 - 4*w^2 - 5*w + 9], [191, 191, 2*w^3 - 4*w^2 - 5*w + 5], [191, 191, -2*w^3 + 2*w^2 + 7*w - 2], [193, 193, -2*w^3 + 3*w^2 + 6*w - 6], [193, 193, -2*w^3 + 2*w^2 + 7*w - 4], [193, 193, -2*w^3 + 4*w^2 + 5*w - 3], [193, 193, 2*w^3 - 3*w^2 - 6*w + 1], [199, 199, -w^3 - 2*w^2 + 6*w + 10], [199, 199, -w^3 + 5*w^2 - w - 13], [223, 223, -w^3 + 2*w^2 + 3*w - 8], [223, 223, w^3 - w^2 - 4*w - 4], [233, 233, -w^2 - 1], [233, 233, w^2 - 2*w + 2], [239, 239, 3*w^2 - 2*w - 13], [239, 239, 3*w^2 - 4*w - 12], [241, 241, w^3 - w^2 - 6*w + 3], [241, 241, w^3 - 2*w^2 - 5*w + 3], [257, 257, -w^3 + 3*w^2 + 4*w - 9], [257, 257, -2*w^3 + 4*w^2 + 8*w - 11], [257, 257, -w^3 + 2*w^2 + 4], [257, 257, w^3 - 7*w - 3], [263, 263, -w^3 + w^2 + 3*w + 5], [263, 263, w^3 - 5*w^2 + 12], [263, 263, -w^3 - 2*w^2 + 7*w + 8], [263, 263, -w^3 + 7*w + 2], [271, 271, -w^3 + 3*w^2 + 2*w - 3], [271, 271, w^3 - 5*w + 1], [289, 17, 3*w^2 - 3*w - 8], [289, 17, 3*w^2 - 3*w - 10], [311, 311, -2*w^3 + 2*w^2 + 7*w - 1], [311, 311, -w^3 + 5*w^2 - w - 11], [311, 311, w^3 + 2*w^2 - 6*w - 8], [311, 311, -2*w^3 + 4*w^2 + 5*w - 6], [337, 337, w^3 - w^2 - 2*w - 4], [337, 337, 2*w^3 - 9*w - 4], [337, 337, 2*w^3 - 6*w^2 - 3*w + 11], [337, 337, -w^3 + 2*w^2 + w - 6], [353, 353, -w^3 + 4*w^2 - 12], [353, 353, w^3 + w^2 - 5*w - 9], [359, 359, 2*w^3 - 4*w^2 - 6*w + 5], [359, 359, 2*w^3 - 2*w^2 - 8*w + 3], [361, 19, -w^3 + 5*w^2 - 13], [361, 19, w^3 + 2*w^2 - 7*w - 9], [367, 367, 2*w^3 - 10*w - 5], [367, 367, w^3 - w^2 - 6*w + 2], [367, 367, -w^3 + 2*w^2 + 5*w - 4], [367, 367, 2*w^3 - 6*w^2 - 4*w + 13], [401, 401, -w^3 + 2*w^2 + 5*w - 5], [401, 401, -w^3 + w^2 + 6*w - 1], [409, 409, 2*w^3 - 3*w^2 - 6*w + 4], [409, 409, 3*w^3 - 3*w^2 - 12*w + 1], [409, 409, -3*w^3 + 6*w^2 + 9*w - 11], [409, 409, -2*w^3 + 3*w^2 + 6*w - 3], [431, 431, 3*w^2 - 5*w - 10], [431, 431, w^3 + w^2 - 8*w - 3], [457, 457, -2*w^3 + 4*w^2 + 4*w - 5], [457, 457, -2*w^3 + 2*w^2 + 6*w - 1], [463, 463, -2*w^3 + 7*w^2 + w - 16], [463, 463, 3*w^3 - 6*w^2 - 8*w + 8], [503, 503, -w^3 + 3*w^2 + 2*w - 12], [503, 503, -w^3 + 4*w^2 + 4*w - 10], [521, 521, w^3 + w^2 - 6*w - 2], [521, 521, -w^3 + 4*w^2 + w - 6], [529, 23, w^2 - w - 8], [569, 569, w^3 - 3*w - 6], [569, 569, -w^3 + 3*w^2 - 8], [577, 577, -w^3 + 4*w^2 - w - 10], [577, 577, w^3 + w^2 - 4*w - 8], [593, 593, -w^3 + 3*w^2 - 12], [593, 593, w^3 - 3*w - 10], [599, 599, w^3 + 3*w^2 - 7*w - 9], [599, 599, -w^3 + 3*w^2 + 3*w - 13], [601, 601, w^2 + 2*w - 4], [601, 601, w^2 - 4*w - 1], [607, 607, 2*w^2 - 13], [607, 607, w^3 + 2*w^2 - 7*w - 5], [607, 607, 2*w^3 - 3*w^2 - 8*w + 8], [607, 607, 2*w^2 - 4*w - 11], [617, 617, w^3 + w^2 - 9*w - 1], [617, 617, w^3 - w^2 - 4*w - 5], [617, 617, -w^3 + 2*w^2 + 3*w - 9], [617, 617, -w^3 + 4*w^2 + 4*w - 8], [625, 5, -5], [631, 631, -w^3 + 5*w^2 - 2*w - 13], [631, 631, w^3 + 2*w^2 - 5*w - 11], [641, 641, w^2 - 2*w + 3], [641, 641, -w^3 + 6*w - 2], [641, 641, -w^3 + 3*w^2 + 3*w - 3], [641, 641, -w^2 - 2], [647, 647, -2*w^3 + 5*w^2 + 4*w - 6], [647, 647, 2*w^3 - 4*w^2 - 5*w - 1], [647, 647, 3*w^3 - 7*w^2 - 7*w + 13], [647, 647, -2*w^3 + w^2 + 8*w - 1], [673, 673, -w^3 + 7*w - 2], [673, 673, -w^3 + 3*w^2 + 4*w - 4], [719, 719, -2*w^3 + 5*w^2 + 3*w - 12], [719, 719, 4*w^2 - 5*w - 10], [719, 719, -4*w^2 + 3*w + 11], [719, 719, 2*w^3 - w^2 - 7*w - 6], [727, 727, -w - 5], [727, 727, w - 6], [743, 743, w^2 + 2*w - 11], [743, 743, w^3 + 4*w^2 - 11*w - 11], [751, 751, -2*w^2 + w + 13], [751, 751, 2*w^2 - 3*w - 12], [769, 769, 2*w^3 - 6*w^2 - 3*w + 17], [769, 769, 3*w^3 - 2*w^2 - 11*w + 2], [809, 809, w^3 + 3*w^2 - 6*w - 11], [809, 809, -3*w^3 + 5*w^2 + 8*w - 11], [823, 823, -w^3 + w^2 + 2*w - 6], [823, 823, 4*w^2 - 5*w - 11], [823, 823, -4*w^2 + 3*w + 12], [823, 823, -w^3 + 4*w^2 + 4*w - 12], [839, 839, -2*w^3 + w^2 + 6*w + 5], [839, 839, -3*w^3 + 8*w^2 + 7*w - 17], [839, 839, 3*w^3 - w^2 - 14*w - 5], [839, 839, -2*w^3 + 5*w^2 + 2*w - 10], [857, 857, w^3 + w^2 - 8*w - 2], [857, 857, -w^3 + 4*w^2 + 3*w - 8], [863, 863, -w^3 + w^2 - w - 1], [863, 863, w^3 - 2*w^2 + 2*w - 2], [911, 911, -w^3 + 3*w^2 - 11], [911, 911, w^3 - 3*w - 9], [919, 919, -2*w^3 + w^2 + 8*w - 2], [919, 919, -2*w^3 + 5*w^2 + 4*w - 5], [929, 929, -2*w^3 + 2*w^2 + 9*w - 4], [929, 929, 2*w^3 - 9*w - 12], [929, 929, 2*w^3 - 6*w^2 - 3*w + 19], [929, 929, -2*w^3 + 4*w^2 + 7*w - 5], [937, 937, -w^3 + 2*w^2 - 6], [937, 937, -w^3 + 3*w^2 - 10], [937, 937, w^3 - 3*w - 8], [937, 937, w^3 - w^2 - w - 5], [953, 953, 2*w^3 - 6*w^2 - 3*w + 18], [953, 953, -3*w^3 + 3*w^2 + 11*w - 5], [961, 31, 4*w^2 - 4*w - 11], [967, 967, -w^3 + 7*w - 3], [967, 967, -2*w^3 + 2*w^2 + 11*w + 2], [967, 967, -w^3 - 4*w^2 + 9*w + 9], [967, 967, -w^3 + 3*w^2 + 4*w - 3], [977, 977, 2*w^2 - 5*w - 8], [977, 977, -w^3 + 2*w^2 - w + 6], [977, 977, -w^3 + w^2 - 6], [977, 977, 2*w^2 + w - 11], [983, 983, -w^3 + 6*w^2 - 2*w - 16], [983, 983, w^3 + 3*w^2 - 7*w - 13]]; primes := [ideal : I in primesArray]; heckePol := x^8 - 2*x^7 - 16*x^6 + 23*x^5 + 78*x^4 - 48*x^3 - 126*x^2 - 25*x - 1; K := NumberField(heckePol); heckeEigenvaluesArray := [e, -247/397*e^7 + 1099/794*e^6 + 3800/397*e^5 - 12917/794*e^4 - 35145/794*e^3 + 29943/794*e^2 + 55843/794*e + 4019/794, 258/397*e^7 - 492/397*e^6 - 4183/397*e^5 + 5748/397*e^4 + 20631/397*e^3 - 12893/397*e^2 - 33098/397*e - 3777/397, -48/397*e^7 + 110/397*e^6 + 769/397*e^5 - 1411/397*e^4 - 3746/397*e^3 + 3959/397*e^2 + 6167/397*e + 278/397, 263/794*e^7 - 260/397*e^6 - 4321/794*e^5 + 6201/794*e^4 + 21931/794*e^3 - 14637/794*e^2 - 36263/794*e - 1812/397, -657/794*e^7 + 728/397*e^6 + 9955/794*e^5 - 16807/794*e^4 - 44971/794*e^3 + 37093/794*e^2 + 70491/794*e + 3565/397, -1, 5/397*e^7 - 28/397*e^6 - 138/397*e^5 + 453/397*e^4 + 1300/397*e^3 - 2141/397*e^2 - 3165/397*e + 1344/397, 271/794*e^7 - 203/397*e^6 - 4383/794*e^5 + 4385/794*e^4 + 20835/794*e^3 - 7423/794*e^2 - 30211/794*e - 2960/397, 618/397*e^7 - 1317/397*e^6 - 9752/397*e^5 + 15338/397*e^4 + 46741/397*e^3 - 34447/397*e^2 - 74785/397*e - 6656/397, -707/397*e^7 + 3075/794*e^6 + 10938/397*e^5 - 35925/794*e^4 - 101253/794*e^3 + 81673/794*e^2 + 153863/794*e + 12391/794, 304/397*e^7 - 1261/794*e^6 - 4738/397*e^5 + 14829/794*e^4 + 44141/794*e^3 - 34929/794*e^2 - 67661/794*e - 3389/794, 297/397*e^7 - 631/397*e^6 - 4783/397*e^5 + 7614/397*e^4 + 23625/397*e^3 - 18715/397*e^2 - 38729/397*e - 3060/397, 1/397*e^7 - 85/397*e^6 + 290/397*e^5 + 964/397*e^4 - 3313/397*e^3 - 1778/397*e^2 + 6116/397*e + 507/397, -37/397*e^7 - 31/397*e^6 + 783/397*e^5 + 459/397*e^4 - 4459/397*e^3 - 2498/397*e^2 + 5159/397*e + 5458/397, 143/397*e^7 - 887/794*e^6 - 1803/397*e^5 + 10111/794*e^4 + 11237/794*e^3 - 20595/794*e^2 - 13901/794*e - 4667/794, 531/397*e^7 - 1068/397*e^6 - 8780/397*e^5 + 12855/397*e^4 + 44765/397*e^3 - 31018/397*e^2 - 74500/397*e - 5507/397, 138/397*e^7 - 217/397*e^6 - 2459/397*e^5 + 2816/397*e^4 + 13251/397*e^3 - 8752/397*e^2 - 20658/397*e + 2079/397, -1109/794*e^7 + 1279/397*e^6 + 17031/794*e^5 - 30127/794*e^4 - 79121/794*e^3 + 68981/794*e^2 + 127935/794*e + 9274/397, -74/397*e^7 + 273/794*e^6 + 1169/397*e^5 - 3325/794*e^4 - 11087/794*e^3 + 8667/794*e^2 + 17857/794*e - 1591/794, 1543/794*e^7 - 1859/397*e^6 - 22975/794*e^5 + 43563/794*e^4 + 101445/794*e^3 - 99831/794*e^2 - 156517/794*e - 6445/397, 659/397*e^7 - 2855/794*e^6 - 10566/397*e^5 + 33897/794*e^4 + 104083/794*e^3 - 77725/794*e^2 - 173289/794*e - 21363/794, -1184/397*e^7 + 2581/397*e^6 + 18307/397*e^5 - 30173/397*e^4 - 84726/397*e^3 + 68145/397*e^2 + 132137/397*e + 12283/397, 14/397*e^7 + 1/397*e^6 - 307/397*e^5 - 2/397*e^4 + 1655/397*e^3 + 119/397*e^2 - 1716/397*e + 746/397, 108/397*e^7 - 49/397*e^6 - 2425/397*e^5 + 892/397*e^4 + 16567/397*e^3 - 3449/397*e^2 - 32634/397*e - 4000/397, 590/397*e^7 - 1319/397*e^6 - 9138/397*e^5 + 15342/397*e^4 + 42240/397*e^3 - 33494/397*e^2 - 63810/397*e - 8545/397, -395/794*e^7 + 312/397*e^6 + 7329/794*e^5 - 7997/794*e^4 - 42753/794*e^3 + 21455/794*e^2 + 78531/794*e + 4477/397, 865/397*e^7 - 2065/397*e^6 - 13155/397*e^5 + 24377/397*e^4 + 59748/397*e^3 - 55969/397*e^2 - 92980/397*e - 9261/397, -340/397*e^7 + 713/397*e^6 + 5414/397*e^5 - 8572/397*e^4 - 26071/397*e^3 + 20930/397*e^2 + 40937/397*e + 4285/397, 376/397*e^7 - 994/397*e^6 - 5296/397*e^5 + 11913/397*e^4 + 21139/397*e^3 - 29358/397*e^2 - 29583/397*e + 2057/397, -45/397*e^7 + 107/794*e^6 + 845/397*e^5 - 1405/794*e^4 - 9505/794*e^3 + 4793/794*e^2 + 13697/794*e - 2357/794, 162/397*e^7 - 941/794*e^6 - 1851/397*e^5 + 11013/794*e^4 + 7619/794*e^3 - 27021/794*e^2 - 2225/794*e + 10629/794, -679/397*e^7 + 3079/794*e^6 + 10324/397*e^5 - 35933/794*e^4 - 93839/794*e^3 + 80561/794*e^2 + 146205/794*e + 17757/794, -685/397*e^7 + 1454/397*e^6 + 10966/397*e^5 - 17200/397*e^4 - 53442/397*e^3 + 39634/397*e^2 + 86627/397*e + 10402/397, -97/794*e^7 - 46/397*e^6 + 3233/794*e^5 + 581/794*e^4 - 27205/794*e^3 - 229/794*e^2 + 54255/794*e + 4590/397, 881/397*e^7 - 1837/397*e^6 - 14073/397*e^5 + 21539/397*e^4 + 69069/397*e^3 - 49481/397*e^2 - 113827/397*e - 9883/397, 2651/794*e^7 - 2897/397*e^6 - 41487/794*e^5 + 68359/794*e^4 + 196583/794*e^3 - 157903/794*e^2 - 317167/794*e - 15377/397, -1788/397*e^7 + 3899/397*e^6 + 27752/397*e^5 - 45513/397*e^4 - 129018/397*e^3 + 102314/397*e^2 + 198060/397*e + 19288/397, -1648/397*e^7 + 3512/397*e^6 + 25476/397*e^5 - 40769/397*e^4 - 117232/397*e^3 + 90800/397*e^2 + 179312/397*e + 16823/397, -835/794*e^7 + 750/397*e^6 + 13915/794*e^5 - 16895/794*e^4 - 72989/794*e^3 + 32007/794*e^2 + 129173/794*e + 14022/397, -591/794*e^7 + 702/397*e^6 + 8451/794*e^5 - 16703/794*e^4 - 34957/794*e^3 + 42815/794*e^2 + 52533/794*e - 5112/397, 417/397*e^7 - 906/397*e^6 - 6507/397*e^5 + 10546/397*e^4 + 31402/397*e^3 - 22856/397*e^2 - 55536/397*e - 5740/397, -382/397*e^7 + 710/397*e^6 + 5938/397*e^5 - 7772/397*e^4 - 27066/397*e^3 + 14221/397*e^2 + 38542/397*e + 6811/397, -798/397*e^7 + 1531/397*e^6 + 13132/397*e^5 - 17751/397*e^4 - 65751/397*e^3 + 38078/397*e^2 + 103767/397*e + 16234/397, -1417/397*e^7 + 6263/794*e^6 + 22197/397*e^5 - 73267/794*e^4 - 212403/794*e^3 + 164883/794*e^2 + 349239/794*e + 38811/794, -1655/397*e^7 + 3710/397*e^6 + 25431/397*e^5 - 43547/397*e^4 - 117067/397*e^3 + 99673/397*e^2 + 182155/397*e + 14862/397, -576/397*e^7 + 2243/794*e^6 + 9625/397*e^5 - 27115/794*e^4 - 97447/794*e^3 + 65241/794*e^2 + 149993/794*e + 11039/794, 847/397*e^7 - 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405/397*e^6 - 5087/397*e^5 + 5177/397*e^4 + 26857/397*e^3 - 17229/397*e^2 - 37485/397*e + 15073/397, -1507/397*e^7 + 3040/397*e^6 + 23887/397*e^5 - 34664/397*e^4 - 116302/397*e^3 + 71156/397*e^2 + 192882/397*e + 31936/397, -400/397*e^7 + 907/794*e^6 + 7864/397*e^5 - 11739/794*e^4 - 97237/794*e^3 + 34885/794*e^2 + 182845/794*e + 24351/794, -17/397*e^7 + 111/794*e^6 - 166/397*e^5 - 619/794*e^4 + 5849/794*e^3 - 289/794*e^2 - 2695/794*e + 10949/794, 67/794*e^7 - 664/397*e^6 + 1565/794*e^5 + 16551/794*e^4 - 21089/794*e^3 - 48063/794*e^2 + 36195/794*e + 14404/397, -1010/397*e^7 + 2480/397*e^6 + 14775/397*e^5 - 29177/397*e^4 - 63703/397*e^3 + 66845/397*e^2 + 99410/397*e + 9588/397, -983/794*e^7 + 688/397*e^6 + 17047/794*e^5 - 14265/794*e^4 - 92413/794*e^3 + 20427/794*e^2 + 158543/794*e + 22953/397, 2297/794*e^7 - 2541/397*e^6 - 35369/794*e^5 + 59789/794*e^4 + 160917/794*e^3 - 140665/794*e^2 - 243151/794*e + 2471/397, 613/397*e^7 - 892/397*e^6 - 11202/397*e^5 + 10121/397*e^4 + 63703/397*e^3 - 18808/397*e^2 - 107350/397*e - 25865/397]; 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;