/* 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![29, 11, -10, -2, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [11, 11, -w - 1], [11, 11, w^2 - 5], [11, 11, -w^2 + 2*w + 4], [11, 11, w - 2], [16, 2, 2], [19, 19, w^2 - 2*w - 5], [19, 19, -w^2 + 6], [25, 5, -2*w^2 + 2*w + 11], [29, 29, w], [29, 29, 2*w^2 - w - 10], [29, 29, -2*w^2 + 3*w + 9], [29, 29, w - 1], [31, 31, w^3 - 6*w - 6], [31, 31, -w^3 + 3*w^2 + 3*w - 11], [41, 41, w^3 - 4*w^2 - 2*w + 16], [41, 41, w^3 - 5*w^2 - 2*w + 24], [59, 59, w^3 - w^2 - 5*w - 2], [59, 59, 2*w^2 - w - 13], [61, 61, w^3 - w^2 - 6*w + 3], [61, 61, -w^3 + 2*w^2 + 5*w - 3], [71, 71, 2*w^2 - w - 16], [71, 71, -w^3 + 3*w^2 + 4*w - 10], [71, 71, -w^3 + 7*w + 4], [71, 71, w^3 - 5*w^2 - 2*w + 23], [79, 79, -w^3 + 4*w^2 + 3*w - 16], [79, 79, w^3 + w^2 - 8*w - 10], [81, 3, -3], [131, 131, 4*w^2 - 5*w - 24], [131, 131, 4*w^2 - 3*w - 25], [139, 139, w^2 + w - 9], [139, 139, -w^3 + w^2 + 6*w - 5], [149, 149, w^3 - w^2 - 4*w + 2], [149, 149, w^3 - 2*w^2 - 3*w + 2], [151, 151, -w^3 + 7*w + 3], [151, 151, -w^3 + 3*w^2 + 4*w - 9], [179, 179, -w^3 - 2*w^2 + 10*w + 16], [179, 179, -w^3 + 3*w^2 + 3*w - 6], [179, 179, w^3 - 6*w - 1], [179, 179, w^3 - 5*w^2 - 3*w + 23], [191, 191, w^3 - 6*w^2 - w + 27], [191, 191, w^3 + 3*w^2 - 10*w - 21], [199, 199, -w^3 + 2*w^2 + 6*w - 4], [199, 199, -4*w^2 + 3*w + 22], [199, 199, 4*w^2 - 5*w - 21], [199, 199, 3*w^2 - 4*w - 12], [239, 239, -w^3 + w^2 + 4*w - 6], [239, 239, 2*w^3 - 5*w^2 - 9*w + 17], [239, 239, 2*w^3 - w^2 - 13*w - 5], [239, 239, w^3 - 8*w^2 + 2*w + 41], [241, 241, -w^3 + 5*w^2 + 4*w - 25], [241, 241, w^3 - 6*w - 8], [241, 241, -w^3 + 3*w^2 + 3*w - 13], [241, 241, w^3 - 5*w^2 + w + 19], [269, 269, w^2 - 2*w - 10], [269, 269, w^2 - 11], [271, 271, w^3 + 2*w^2 - 8*w - 18], [271, 271, 2*w^3 - 13*w - 10], [271, 271, 2*w^3 + w^2 - 15*w - 16], [271, 271, w^3 - 5*w^2 - w + 23], [289, 17, 2*w^2 - 11], [289, 17, -2*w^2 + 4*w + 9], [311, 311, w^3 - 4*w^2 - 3*w + 10], [311, 311, w^3 - 7*w^2 - w + 34], [331, 331, w^3 - w^2 - 7*w + 4], [331, 331, w^3 - 2*w^2 - 6*w + 3], [349, 349, w^3 + w^2 - 6*w - 12], [349, 349, -w^3 + 4*w^2 + w - 16], [361, 19, 4*w^2 - 4*w - 21], [379, 379, w^3 - 3*w^2 - 3*w + 15], [379, 379, 2*w^3 - 2*w^2 - 12*w + 1], [379, 379, -2*w^3 + 4*w^2 + 10*w - 11], [379, 379, w^3 - 6*w - 10], [401, 401, -2*w^3 + 7*w^2 + 8*w - 30], [401, 401, -w^3 + 2*w^2 + 6*w - 2], [419, 419, 5*w^2 - 4*w - 26], [419, 419, 5*w^2 - 6*w - 25], [421, 421, 3*w^2 - w - 20], [421, 421, -2*w^3 + 2*w^2 + 10*w + 3], [431, 431, w^2 - 3*w - 5], [431, 431, w^2 + w - 7], [449, 449, 2*w^3 - 13*w - 7], [449, 449, -2*w^3 + 6*w^2 + 7*w - 18], [461, 461, w^3 - w^2 - 8*w + 3], [461, 461, -w^3 + 2*w^2 + 7*w - 5], [491, 491, w^3 - 3*w^2 - 6*w + 15], [491, 491, w^3 - 9*w - 7], [499, 499, 4*w^2 - 6*w - 23], [499, 499, -4*w^2 + 2*w + 25], [509, 509, 2*w^3 - 3*w^2 - 11*w + 6], [521, 521, -w^3 + 3*w^2 + 2*w - 12], [521, 521, -w^3 + 4*w^2 + 2*w - 19], [529, 23, w^3 - 7*w^2 - w + 35], [529, 23, -w^3 + 3*w^2 + 4*w - 5], [541, 541, 2*w^3 - w^2 - 12*w - 5], [541, 541, w^3 - 2*w^2 - 7*w + 9], [569, 569, w^3 - 8*w - 2], [569, 569, -w^3 + 3*w^2 + 5*w - 9], [599, 599, -w^3 - 4*w^2 + 10*w + 25], [599, 599, -w^3 + 7*w^2 - 36], [601, 601, 5*w^2 - 6*w - 26], [601, 601, -2*w^3 + 8*w^2 + 6*w - 33], [601, 601, -2*w^3 - 2*w^2 + 16*w + 21], [601, 601, 5*w^2 - 4*w - 27], [619, 619, w^3 + 4*w^2 - 11*w - 26], [619, 619, w^3 - 3*w^2 - 5*w + 18], [641, 641, -w^3 + 3*w^2 + 6*w - 14], [641, 641, -w^3 + 9*w + 6], [659, 659, -w^3 - 4*w^2 + 10*w + 28], [659, 659, -w^3 + 6*w^2 - w - 27], [659, 659, -w^3 - 3*w^2 + 8*w + 23], [659, 659, -w^3 + 7*w^2 - w - 33], [691, 691, w^3 - w^2 - 8*w + 2], [691, 691, 2*w^3 - 2*w^2 - 11*w + 1], [701, 701, -w^3 - w^2 + 7*w + 15], [701, 701, -w^3 + 4*w^2 + 2*w - 20], [709, 709, -2*w^3 - w^2 + 15*w + 15], [709, 709, 2*w^3 - 7*w^2 - 7*w + 27], [719, 719, -w^3 - 4*w^2 + 13*w + 26], [719, 719, w^3 - 7*w^2 - 2*w + 34], [739, 739, 2*w^3 + 2*w^2 - 15*w - 24], [739, 739, -2*w^3 + 8*w^2 + 5*w - 35], [751, 751, -w^3 - 3*w^2 + 9*w + 25], [751, 751, -w^3 + 6*w^2 - 30], [761, 761, 3*w^3 - 2*w^2 - 16*w - 8], [761, 761, -3*w^3 + 7*w^2 + 11*w - 23], [769, 769, -2*w^3 + 4*w^2 + 11*w - 10], [769, 769, w^3 + 2*w^2 - 9*w - 12], [809, 809, -3*w^3 + 3*w^2 + 16*w + 4], [809, 809, w^3 + 5*w^2 - 12*w - 30], [811, 811, -w^3 - 4*w^2 + 10*w + 27], [811, 811, -2*w^3 + 5*w^2 + 9*w - 14], [811, 811, -2*w^3 + w^2 + 13*w + 2], [811, 811, -w^3 + 7*w^2 - w - 32], [821, 821, 2*w^3 - 8*w^2 - 7*w + 35], [821, 821, -2*w^3 - 2*w^2 + 17*w + 22], [829, 829, -3*w^3 + 20*w + 13], [829, 829, -w^3 + 7*w^2 - w - 36], [829, 829, w^3 + 4*w^2 - 10*w - 31], [829, 829, 2*w^3 - 3*w^2 - 11*w + 14], [859, 859, -w^3 + 7*w^2 - 3*w - 31], [859, 859, 7*w^2 - 9*w - 39], [859, 859, 7*w^2 - 5*w - 41], [859, 859, 3*w^3 - 4*w^2 - 17*w + 2], [929, 929, w^3 + 5*w^2 - 14*w - 31], [929, 929, -2*w^3 + 4*w^2 + 10*w - 5], [929, 929, w^3 - 9*w^2 + 4*w + 44], [929, 929, -w^3 + 8*w^2 + w - 39], [941, 941, -w^3 + 7*w^2 - 34], [941, 941, w^3 + 4*w^2 - 11*w - 28], [961, 31, -5*w^2 + 5*w + 28], [971, 971, -3*w^3 + w^2 + 19*w + 14], [971, 971, -3*w^3 + 8*w^2 + 12*w - 31], [991, 991, -w^3 - 5*w^2 + 11*w + 35], [991, 991, w^3 - 8*w^2 + 2*w + 40]]; primes := [ideal : I in primesArray]; heckePol := x^11 + 13*x^10 + 35*x^9 - 180*x^8 - 1093*x^7 - 1581*x^6 + 704*x^5 + 4129*x^4 + 4587*x^3 + 2413*x^2 + 628*x + 65; K := NumberField(heckePol); heckeEigenvaluesArray := [e, -235376386/7292459*e^10 - 49618000/123601*e^9 - 6592024481/7292459*e^8 + 46065393491/7292459*e^7 + 231327572217/7292459*e^6 + 242171639117/7292459*e^5 - 301129631936/7292459*e^4 - 801878655850/7292459*e^3 - 629978008875/7292459*e^2 - 215991738661/7292459*e - 27602323077/7292459, -117406043/7292459*e^10 - 25025013/123601*e^9 - 3481338756/7292459*e^8 + 22628200318/7292459*e^7 + 118757219839/7292459*e^6 + 134916146210/7292459*e^5 - 141194260795/7292459*e^4 - 425587604917/7292459*e^3 - 355538934734/7292459*e^2 - 128863978047/7292459*e - 17347790571/7292459, 88373725/7292459*e^10 + 18535270/123601*e^9 + 2407519314/7292459*e^8 - 17431373092/7292459*e^7 - 85722394569/7292459*e^6 - 85783562476/7292459*e^5 + 117205428914/7292459*e^4 + 292461826508/7292459*e^3 + 220067716219/7292459*e^2 + 71656865703/7292459*e + 8580612002/7292459, 82731016/7292459*e^10 + 17646995/123601*e^9 + 2461552827/7292459*e^8 - 15936413209/7292459*e^7 - 83852246973/7292459*e^6 - 95587117531/7292459*e^5 + 99587508159/7292459*e^4 + 301206811627/7292459*e^3 + 251323767564/7292459*e^2 + 90651904568/7292459*e + 12097818728/7292459, 141685076/7292459*e^10 + 29556038/123601*e^9 + 3748926541/7292459*e^8 - 28130964500/7292459*e^7 - 135443866473/7292459*e^6 - 129673772761/7292459*e^5 + 191888967872/7292459*e^4 + 453776502525/7292459*e^3 + 331556412082/7292459*e^2 + 105212504024/7292459*e + 12305220412/7292459, -46755998/7292459*e^10 - 9931123/123601*e^9 - 1361584535/7292459*e^8 + 9059993591/7292459*e^7 + 46874463578/7292459*e^6 + 51864009924/7292459*e^5 - 57676803190/7292459*e^4 - 166384397739/7292459*e^3 - 135717905410/7292459*e^2 - 47846607250/7292459*e - 6219306685/7292459, 77461999/7292459*e^10 + 16334739/123601*e^9 + 2175848274/7292459*e^8 - 15124494044/7292459*e^7 - 76161437471/7292459*e^6 - 80532096852/7292459*e^5 + 97075951689/7292459*e^4 + 264205993351/7292459*e^3 + 212263833401/7292459*e^2 + 75298031293/7292459*e + 10045534239/7292459, 1, 428691953/7292459*e^10 + 90484018/123601*e^9 + 12088441482/7292459*e^8 - 83731195125/7292459*e^7 - 422690409569/7292459*e^6 - 447374647411/7292459*e^5 + 543228610518/7292459*e^4 + 1470910532707/7292459*e^3 + 1167812605114/7292459*e^2 + 405259441569/7292459*e + 52521266553/7292459, 113849917/7292459*e^10 + 24120785/123601*e^9 + 3272579040/7292459*e^8 - 22136373811/7292459*e^7 - 113379749764/7292459*e^6 - 123172976991/7292459*e^5 + 142205455587/7292459*e^4 + 399077530818/7292459*e^3 + 321681901165/7292459*e^2 + 112894982920/7292459*e + 14771514471/7292459, -261398299/7292459*e^10 - 54968310/123601*e^9 - 7226671790/7292459*e^8 + 51321753569/7292459*e^7 + 255237565735/7292459*e^6 + 262163839327/7292459*e^5 - 338331715353/7292459*e^4 - 877868273498/7292459*e^3 - 680465045397/7292459*e^2 - 230652819186/7292459*e - 29224700408/7292459, 36125214/7292459*e^10 + 7858203/123601*e^9 + 1182348241/7292459*e^8 - 6759386880/7292459*e^7 - 38472423788/7292459*e^6 - 49670180488/7292459*e^5 + 38097946963/7292459*e^4 + 145624959591/7292459*e^3 + 133490253321/7292459*e^2 + 52118337024/7292459*e + 7478265565/7292459, -217689051/7292459*e^10 - 45528023/123601*e^9 - 5843038192/7292459*e^8 + 43063315297/7292459*e^7 + 209523270304/7292459*e^6 + 205422952967/7292459*e^5 - 290394037343/7292459*e^4 - 708020327145/7292459*e^3 - 528306716113/7292459*e^2 - 171897352735/7292459*e - 20760084304/7292459, 137954022/7292459*e^10 + 29432558/123601*e^9 + 4112158913/7292459*e^8 - 26530201271/7292459*e^7 - 139852200370/7292459*e^6 - 160395535358/7292459*e^5 + 163504719053/7292459*e^4 + 502454286065/7292459*e^3 + 425193701599/7292459*e^2 + 156688541320/7292459*e + 21562395223/7292459, -211188369/7292459*e^10 - 44429453/123601*e^9 - 5851200737/7292459*e^8 + 41450657601/7292459*e^7 + 206460979272/7292459*e^6 + 212572975253/7292459*e^5 - 273390778874/7292459*e^4 - 710903627389/7292459*e^3 - 551085672890/7292459*e^2 - 186856510042/7292459*e - 23733077415/7292459, -314445190/7292459*e^10 - 66834137/123601*e^9 - 9194920856/7292459*e^8 + 60799866338/7292459*e^7 + 315682081527/7292459*e^6 + 352475568919/7292459*e^5 - 381360305287/7292459*e^4 - 1121580233589/7292459*e^3 - 930388182316/7292459*e^2 - 337192564922/7292459*e - 45671588389/7292459, 192495529/7292459*e^10 + 40898898/123601*e^9 + 5619296052/7292459*e^8 - 37226830677/7292459*e^7 - 193048529764/7292459*e^6 - 215234778211/7292459*e^5 + 233121189804/7292459*e^4 + 685020691676/7292459*e^3 + 569079821834/7292459*e^2 + 207070236347/7292459*e + 28225830007/7292459, 568672652/7292459*e^10 + 119673996/123601*e^9 + 15783199822/7292459*e^8 - 111555846460/7292459*e^7 - 556397593882/7292459*e^6 - 574582826336/7292459*e^5 + 734325059801/7292459*e^4 + 1918272624067/7292459*e^3 + 1491209614270/7292459*e^2 + 506192275071/7292459*e + 64028956015/7292459, -4119214/123601*e^10 - 50774386/123601*e^9 - 109923345/123601*e^8 + 815982750/123601*e^7 + 3953330158/123601*e^6 + 3840709417/123601*e^5 - 5521116536/123601*e^4 - 13307847770/123601*e^3 - 9865488097/123601*e^2 - 3197682195/123601*e - 387384240/123601, -1446614/7292459*e^10 + 245324/123601*e^9 + 348618678/7292459*e^8 + 1016273505/7292459*e^7 - 5267479601/7292459*e^6 - 27403244551/7292459*e^5 - 22125310742/7292459*e^4 + 46018274454/7292459*e^3 + 83725841379/7292459*e^2 + 45466382100/7292459*e + 8057369550/7292459, 658432901/7292459*e^10 + 138727713/123601*e^9 + 18391306465/7292459*e^8 - 128937278607/7292459*e^7 - 646211208438/7292459*e^6 - 674063081031/7292459*e^5 + 843758900985/7292459*e^4 + 2236485387539/7292459*e^3 + 1753679986837/7292459*e^2 + 600500266323/7292459*e + 76649345455/7292459, -513187785/7292459*e^10 - 107938527/123601*e^9 - 14200732053/7292459*e^8 + 100759307288/7292459*e^7 + 501407475954/7292459*e^6 + 515252118226/7292459*e^5 - 665495254781/7292459*e^4 - 1725856654615/7292459*e^3 - 1334886742111/7292459*e^2 - 450242666931/7292459*e - 56531363197/7292459, -88111617/7292459*e^10 - 18699008/123601*e^9 - 2557284985/7292459*e^8 + 17063336763/7292459*e^7 + 88097786471/7292459*e^6 + 97499415618/7292459*e^5 - 107228212905/7292459*e^4 - 311801345330/7292459*e^3 - 257740718282/7292459*e^2 - 93060236586/7292459*e - 12524316107/7292459, -21298640/7292459*e^10 - 4636957/123601*e^9 - 698725226/7292459*e^8 + 3993228880/7292459*e^7 + 22751417593/7292459*e^6 + 29248084739/7292459*e^5 - 23250181506/7292459*e^4 - 86604607260/7292459*e^3 - 77352953256/7292459*e^2 - 28635132641/7292459*e - 3773437423/7292459, 587111083/7292459*e^10 + 123313851/123601*e^9 + 16125132864/7292459*e^8 - 115491575454/7292459*e^7 - 571518820116/7292459*e^6 - 580637600432/7292459*e^5 + 766975734837/7292459*e^4 + 1958543616269/7292459*e^3 + 1501462776700/7292459*e^2 + 501601708058/7292459*e + 62191713591/7292459, 154849058/7292459*e^10 + 32989246/123601*e^9 + 4578465884/7292459*e^8 - 29881710632/7292459*e^7 - 156445472383/7292459*e^6 - 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35388062424/7292459*e^7 - 152924699349/7292459*e^6 - 107451449462/7292459*e^5 + 270047536670/7292459*e^4 + 465192708530/7292459*e^3 + 246983204576/7292459*e^2 + 40057393711/7292459*e - 1376504140/7292459, 567465631/7292459*e^10 + 120007911/123601*e^9 + 16161654767/7292459*e^8 - 110578163582/7292459*e^7 - 562414397162/7292459*e^6 - 603408016719/7292459*e^5 + 713809721057/7292459*e^4 + 1968816691780/7292459*e^3 + 1575508272627/7292459*e^2 + 549670316532/7292459*e + 71540012498/7292459, -2969849996/7292459*e^10 - 625340503/123601*e^9 - 82679689374/7292459*e^8 + 582078755784/7292459*e^7 + 2909989021201/7292459*e^6 + 3020100254874/7292459*e^5 - 3819485673709/7292459*e^4 - 10051358460912/7292459*e^3 - 7849319705063/7292459*e^2 - 2676981964694/7292459*e - 340311557975/7292459, 287997647/7292459*e^10 + 61746676/123601*e^9 + 8774752744/7292459*e^8 - 55250023973/7292459*e^7 - 295985285535/7292459*e^6 - 345605999117/7292459*e^5 + 347737769176/7292459*e^4 + 1079783205832/7292459*e^3 + 897018352028/7292459*e^2 + 315333234776/7292459*e + 40082792255/7292459, -1990114919/7292459*e^10 - 419201011/123601*e^9 - 55514797894/7292459*e^8 + 389846769845/7292459*e^7 + 1951908860782/7292459*e^6 + 2032002090558/7292459*e^5 - 2553819992662/7292459*e^4 - 6750222487323/7292459*e^3 - 5284470797185/7292459*e^2 - 1806541776220/7292459*e - 230329838789/7292459, -1220247228/7292459*e^10 - 255389144/123601*e^9 - 32882083328/7292459*e^8 + 241152265642/7292459*e^7 + 1176709906987/7292459*e^6 + 1160972010875/7292459*e^5 - 1621403593741/7292459*e^4 - 3985394624589/7292459*e^3 - 2989504605553/7292459*e^2 - 979243072755/7292459*e - 119520394130/7292459, 1529029963/7292459*e^10 + 320664613/123601*e^9 + 41656931118/7292459*e^8 - 301370397018/7292459*e^7 - 1482463562897/7292459*e^6 - 1487757719348/7292459*e^5 + 2011947103703/7292459*e^4 + 5055791445128/7292459*e^3 + 3840729212172/7292459*e^2 + 1271302295673/7292459*e + 156140872106/7292459, -1197790931/7292459*e^10 - 253212162/123601*e^9 - 34043197421/7292459*e^8 + 233554271593/7292459*e^7 + 1185982818496/7292459*e^6 + 1268212341333/7292459*e^5 - 1511710749961/7292459*e^4 - 4147578704163/7292459*e^3 - 3307302087593/7292459*e^2 - 1148023294849/7292459*e - 148020039984/7292459, -3012280/7292459*e^10 - 759145/123601*e^9 - 163251157/7292459*e^8 + 525112608/7292459*e^7 + 4606136865/7292459*e^6 + 7699203717/7292459*e^5 - 5969306234/7292459*e^4 - 23098235815/7292459*e^3 - 13177036834/7292459*e^2 + 1454522881/7292459*e + 1784159707/7292459, 1301543333/7292459*e^10 + 275938517/123601*e^9 + 37551410586/7292459*e^8 - 252744803754/7292459*e^7 - 1298374474634/7292459*e^6 - 1419375293303/7292459*e^5 + 1614509994615/7292459*e^4 + 4580219430000/7292459*e^3 + 3717443758275/7292459*e^2 + 1313098570821/7292459*e + 172562308183/7292459, 439925124/7292459*e^10 + 92184059/123601*e^9 + 11940935081/7292459*e^8 - 86705940244/7292459*e^7 - 425480656817/7292459*e^6 - 426090460728/7292459*e^5 + 575070572583/7292459*e^4 + 1447339327393/7292459*e^3 + 1107853610750/7292459*e^2 + 371930976264/7292459*e + 46435156755/7292459, -267073678/7292459*e^10 - 57253635/123601*e^9 - 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173570027/123601*e^9 - 21649106146/7292459*e^8 + 166618513849/7292459*e^7 + 790528963368/7292459*e^6 + 731615893036/7292459*e^5 - 1152105467121/7292459*e^4 - 2615807345210/7292459*e^3 - 1858085922795/7292459*e^2 - 570256911324/7292459*e - 64135191736/7292459, 2172011748/7292459*e^10 + 458663314/123601*e^9 + 61396898861/7292459*e^8 - 423984937528/7292459*e^7 - 2144285849809/7292459*e^6 - 2277249575811/7292459*e^5 + 2746904323624/7292459*e^4 + 7472681401895/7292459*e^3 + 5945623637638/7292459*e^2 + 2066735524419/7292459*e + 268288831364/7292459, 1209400716/7292459*e^10 + 253611200/123601*e^9 + 32929773056/7292459*e^8 - 238441878543/7292459*e^7 - 1172330245731/7292459*e^6 - 1174741947567/7292459*e^5 + 1594356863250/7292459*e^4 + 3995371653824/7292459*e^3 + 3028836148450/7292459*e^2 + 1002460940928/7292459*e + 123791592049/7292459, 2441896654/7292459*e^10 + 514893504/123601*e^9 + 68479435060/7292459*e^8 - 477770871400/7292459*e^7 - 2401576559947/7292459*e^6 - 2518503653043/7292459*e^5 + 3121932234814/7292459*e^4 + 8330963936429/7292459*e^3 + 6550485687067/7292459*e^2 + 2247978126870/7292459*e + 287660301140/7292459, -249204556/7292459*e^10 - 50939178/123601*e^9 - 5856879448/7292459*e^8 + 50843691174/7292459*e^7 + 225486446043/7292459*e^6 + 173727218064/7292459*e^5 - 374356044202/7292459*e^4 - 701408656795/7292459*e^3 - 420361683418/7292459*e^2 - 99440391295/7292459*e - 6958462075/7292459, 1344915481/7292459*e^10 + 286523756/123601*e^9 + 39782047796/7292459*e^8 - 259355733391/7292459*e^7 - 1358584293204/7292459*e^6 - 1538858050197/7292459*e^5 + 1619658505097/7292459*e^4 + 4861519279836/7292459*e^3 + 4056539237953/7292459*e^2 + 1470296759110/7292459*e + 198317509611/7292459, -1266107802/7292459*e^10 - 265555855/123601*e^9 - 34522554389/7292459*e^8 + 249428310686/7292459*e^7 + 1227810998676/7292459*e^6 + 1235039084668/7292459*e^5 - 1659540111660/7292459*e^4 - 4187908237478/7292459*e^3 - 3196645574539/7292459*e^2 - 1067618127940/7292459*e - 133105977937/7292459, -102335825/7292459*e^10 - 21678977/123601*e^9 - 2943032914/7292459*e^8 + 19863963997/7292459*e^7 + 101819166002/7292459*e^6 + 111244681008/7292459*e^5 - 125221292969/7292459*e^4 - 357006766873/7292459*e^3 - 294200157491/7292459*e^2 - 108790632769/7292459*e - 15540735732/7292459, -305548146/7292459*e^10 - 66836380/123601*e^9 - 10267039034/7292459*e^8 + 56631828978/7292459*e^7 + 329867079307/7292459*e^6 + 440484626016/7292459*e^5 - 305787889713/7292459*e^4 - 1266078262650/7292459*e^3 - 1194393314844/7292459*e^2 - 477840292456/7292459*e - 69994422359/7292459, 687564074/7292459*e^10 + 146385171/123601*e^9 + 20265914959/7292459*e^8 - 132771884798/7292459*e^7 - 693473378332/7292459*e^6 - 780708021828/7292459*e^5 + 835151266610/7292459*e^4 + 2477595884583/7292459*e^3 + 2051131580975/7292459*e^2 + 734900891600/7292459*e + 97257369048/7292459, -1333361920/7292459*e^10 - 280608846/123601*e^9 - 37006132584/7292459*e^8 + 261640941035/7292459*e^7 + 1304841132998/7292459*e^6 + 1346069989951/7292459*e^5 - 1727625762088/7292459*e^4 - 4500603492759/7292459*e^3 - 3484361875917/7292459*e^2 - 1173097875374/7292459*e - 146485737542/7292459, 1328831192/7292459*e^10 + 281266595/123601*e^9 + 38013485368/7292459*e^8 - 258673400217/7292459*e^7 - 1320055946090/7292459*e^6 - 1424718313852/7292459*e^5 + 1666390617947/7292459*e^4 + 4633718105502/7292459*e^3 + 3719939054423/7292459*e^2 + 1299664284971/7292459*e + 168742279458/7292459, 1161687975/7292459*e^10 + 242299989/123601*e^9 + 30706842069/7292459*e^8 - 230782134475/7292459*e^7 - 1110245774942/7292459*e^6 - 1059776256514/7292459*e^5 + 1580561668318/7292459*e^4 + 3718690328286/7292459*e^3 + 2699805661057/7292459*e^2 + 846960023560/7292459*e + 97466360595/7292459]; 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;