/* 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![44, 0, -14, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [4, 2, -1/2*w^3 + w^2 + 4*w - 8], [11, 11, 1/2*w^3 - 3/2*w^2 - 4*w + 11], [11, 11, -1/2*w^2 - w + 2], [11, 11, -1/2*w^2 + w + 2], [19, 19, 1/2*w^3 - 1/2*w^2 - 4*w + 5], [19, 19, 1/2*w^3 + w^2 - 4*w - 9], [19, 19, 1/2*w^3 - w^2 - 4*w + 9], [19, 19, 1/2*w^3 + 1/2*w^2 - 4*w - 5], [25, 5, 1/2*w^2 - 1], [29, 29, 1/2*w^3 - 4*w - 1], [29, 29, -1/2*w^3 + 4*w - 1], [31, 31, -w - 1], [31, 31, w - 1], [41, 41, 1/2*w^3 + 1/2*w^2 - 4*w - 2], [41, 41, -1/2*w^3 + 1/2*w^2 + 4*w - 2], [49, 7, -1/2*w^3 + 1/2*w^2 + 5*w - 7], [49, 7, 3/2*w^3 - 7/2*w^2 - 13*w + 29], [59, 59, -1/2*w^3 - 3/2*w^2 + 3*w + 10], [59, 59, 1/2*w^3 - 3/2*w^2 - 3*w + 10], [61, 61, 1/2*w^2 + w - 6], [61, 61, 1/2*w^2 - w - 6], [71, 71, w^3 - 5/2*w^2 - 8*w + 20], [71, 71, -w^3 + 5/2*w^2 + 10*w - 24], [81, 3, -3], [101, 101, -1/2*w^3 + 1/2*w^2 + 4*w - 1], [101, 101, 1/2*w^3 + 1/2*w^2 - 4*w - 1], [109, 109, 3/2*w^2 + w - 12], [109, 109, 3/2*w^2 - w - 12], [149, 149, 1/2*w^3 + 2*w^2 - 4*w - 15], [149, 149, -1/2*w^3 + 2*w^2 + 4*w - 15], [151, 151, 1/2*w^3 - 1/2*w^2 - 4*w + 8], [151, 151, -w^3 + 5/2*w^2 + 8*w - 18], [151, 151, -w^3 + 3/2*w^2 + 10*w - 18], [151, 151, -1/2*w^3 - 1/2*w^2 + 4*w + 8], [179, 179, 1/2*w^3 + w^2 - 4*w - 5], [179, 179, -1/2*w^3 + w^2 + 4*w - 5], [181, 181, 1/2*w^3 - 5*w - 3], [181, 181, w^3 - 1/2*w^2 - 7*w - 1], [181, 181, -1/2*w^3 - 5/2*w^2 + 4*w + 15], [181, 181, 1/2*w^3 - 5*w + 3], [191, 191, w^3 - 2*w^2 - 7*w + 13], [191, 191, -w^3 - 2*w^2 + 7*w + 13], [199, 199, -5/2*w^2 - 2*w + 18], [199, 199, -5/2*w^2 + 2*w + 18], [241, 241, -3/2*w^3 + 7/2*w^2 + 14*w - 30], [241, 241, -3/2*w^3 + 5/2*w^2 + 16*w - 30], [251, 251, 3/2*w^2 - w - 13], [251, 251, 3/2*w^2 + w - 13], [269, 269, 1/2*w^3 + 1/2*w^2 - 3*w - 10], [269, 269, -w^3 + 1/2*w^2 + 7*w - 8], [269, 269, w^3 + 1/2*w^2 - 7*w - 8], [269, 269, -1/2*w^3 + 1/2*w^2 + 3*w - 10], [271, 271, -w^3 + 8*w + 5], [271, 271, -2*w^2 + 2*w + 15], [271, 271, 2*w^2 + 2*w - 15], [271, 271, w^3 - 8*w + 5], [281, 281, -2*w + 3], [281, 281, w^3 - 2*w^2 - 12*w + 25], [311, 311, 1/2*w^3 + 5/2*w^2 - 3*w - 17], [311, 311, -1/2*w^3 + 5/2*w^2 + 3*w - 17], [331, 331, -1/2*w^3 + 3/2*w^2 + 5*w - 10], [331, 331, 1/2*w^3 + 3/2*w^2 - 5*w - 10], [349, 349, -1/2*w^2 + 3*w - 3], [349, 349, -w^3 + 5/2*w^2 + 11*w - 25], [379, 379, 5/2*w^2 - w - 21], [379, 379, 5/2*w^2 + w - 21], [389, 389, 3/2*w^2 - w - 14], [389, 389, 1/2*w^3 - 5/2*w^2 - 3*w + 16], [389, 389, -1/2*w^3 - 5/2*w^2 + 3*w + 16], [389, 389, 3/2*w^2 + w - 14], [409, 409, w^3 + w^2 - 8*w - 7], [409, 409, -w^3 + w^2 + 8*w - 7], [419, 419, 3/2*w^3 - 5/2*w^2 - 15*w + 28], [419, 419, 3/2*w^3 - 7/2*w^2 - 13*w + 28], [439, 439, 1/2*w^2 + 2*w - 9], [439, 439, -3/2*w^2 + 2*w + 6], [439, 439, -3/2*w^2 - 2*w + 6], [439, 439, 1/2*w^2 - 2*w - 9], [449, 449, -w^3 + 1/2*w^2 + 6*w - 1], [449, 449, 1/2*w^2 + 2*w - 5], [449, 449, 1/2*w^2 - 2*w - 5], [449, 449, w^3 + 1/2*w^2 - 6*w - 1], [461, 461, w^3 + 3/2*w^2 - 7*w - 9], [461, 461, -w^3 + 3/2*w^2 + 7*w - 9], [491, 491, -1/2*w^3 - 1/2*w^2 + 6*w + 7], [491, 491, -5/2*w^2 + w + 16], [491, 491, 5/2*w^2 + w - 16], [491, 491, 1/2*w^3 - 1/2*w^2 - 6*w + 7], [499, 499, -1/2*w^3 + 5/2*w^2 + 4*w - 19], [499, 499, 1/2*w^3 + 5/2*w^2 - 4*w - 19], [541, 541, -1/2*w^3 + 3/2*w^2 + 4*w - 7], [541, 541, 1/2*w^3 + 3/2*w^2 - 4*w - 7], [569, 569, 1/2*w^2 + 2*w - 2], [569, 569, 1/2*w^2 - 2*w - 2], [571, 571, -2*w^3 + 9/2*w^2 + 17*w - 39], [571, 571, 1/2*w^3 - 1/2*w^2 - 5*w - 2], [571, 571, -1/2*w^3 - 1/2*w^2 + 5*w - 2], [571, 571, w^3 - 5/2*w^2 - 11*w + 27], [599, 599, -w^3 + 2*w^2 + 8*w - 13], [599, 599, w^3 + 2*w^2 - 8*w - 13], [601, 601, 1/2*w^2 + 2*w - 4], [601, 601, 1/2*w^2 - 2*w - 4], [619, 619, -1/2*w^3 + 2*w^2 + 6*w - 13], [619, 619, 1/2*w^3 + 2*w^2 - 6*w - 13], [631, 631, -w^3 - 1/2*w^2 + 6*w - 2], [631, 631, w^3 - 1/2*w^2 - 6*w - 2], [641, 641, w^3 - 8*w - 1], [641, 641, -1/2*w^3 - 1/2*w^2 + 5*w - 1], [641, 641, 1/2*w^3 - 1/2*w^2 - 5*w - 1], [641, 641, w^3 - 8*w + 1], [661, 661, -1/2*w^3 + 7/2*w^2 + 2*w - 25], [661, 661, 3/2*w^3 - 9/2*w^2 - 11*w + 32], [661, 661, -5/2*w^3 + 13/2*w^2 + 23*w - 56], [661, 661, 1/2*w^3 + 7/2*w^2 - 2*w - 25], [691, 691, 1/2*w^3 + 7/2*w^2 - 3*w - 28], [691, 691, -1/2*w^3 + 7/2*w^2 + 3*w - 28], [701, 701, -1/2*w^3 + 1/2*w^2 + 5*w - 10], [701, 701, 3/2*w^3 - 5/2*w^2 - 17*w + 34], [709, 709, 1/2*w^3 - 5/2*w^2 - 6*w + 17], [709, 709, w^3 + 5/2*w^2 - 7*w - 14], [709, 709, -w^3 + 5/2*w^2 + 7*w - 14], [709, 709, -1/2*w^3 - 5/2*w^2 + 6*w + 17], [719, 719, -w^3 + 7/2*w^2 + 8*w - 23], [719, 719, w^3 + 7/2*w^2 - 8*w - 23], [751, 751, 7/2*w^2 + 2*w - 27], [751, 751, 7/2*w^2 - 2*w - 27], [761, 761, -1/2*w^3 + 3/2*w^2 + 3*w - 13], [761, 761, 1/2*w^3 + 3/2*w^2 - 3*w - 13], [769, 769, -w^3 + 7/2*w^2 + 8*w - 25], [769, 769, w^3 + 7/2*w^2 - 8*w - 25], [809, 809, 2*w^3 - 7/2*w^2 - 20*w + 40], [809, 809, -1/2*w^2 + 2*w - 4], [811, 811, -1/2*w^2 - 3*w + 8], [811, 811, 1/2*w^3 + 3/2*w^2 - 3*w - 18], [811, 811, -1/2*w^3 + 3/2*w^2 + 3*w - 18], [811, 811, 1/2*w^2 - 3*w - 8], [821, 821, 3/2*w^3 - 7/2*w^2 - 16*w + 37], [821, 821, -3/2*w^3 + 7/2*w^2 + 12*w - 29], [839, 839, 3*w^2 - 2*w - 23], [839, 839, 3*w^2 + 2*w - 23], [841, 29, -5/2*w^2 + 16], [859, 859, 5/2*w^2 + 3*w - 16], [859, 859, 5/2*w^2 - 3*w - 16], [911, 911, 3*w^2 - 2*w - 17], [911, 911, 3*w^2 + 2*w - 17], [919, 919, 3*w^2 - w - 23], [919, 919, -2*w^2 + 3*w + 15], [919, 919, -2*w^2 - 3*w + 15], [919, 919, 3*w^2 + w - 23], [941, 941, 3/2*w^2 - 3*w - 2], [941, 941, 2*w^3 - 11/2*w^2 - 17*w + 42], [961, 31, -w^2 + 13], [971, 971, 1/2*w^3 + 1/2*w^2 - 6*w - 5], [971, 971, -1/2*w^3 + 1/2*w^2 + 6*w - 5], [991, 991, w^3 + 3/2*w^2 - 8*w - 8], [991, 991, -w^3 + 3/2*w^2 + 8*w - 8]]; primes := [ideal : I in primesArray]; heckePol := x^18 - 40*x^16 + 616*x^14 - 4678*x^12 + 18495*x^10 - 36933*x^8 + 34206*x^6 - 15148*x^4 + 3066*x^2 - 225; K := NumberField(heckePol); heckeEigenvaluesArray := [e, 15461/454128*e^16 - 155189/113532*e^14 + 9621143/454128*e^12 - 18463901/113532*e^10 + 24758759/37844*e^8 - 202726725/151376*e^6 + 95258835/75688*e^4 - 219572465/454128*e^2 + 4435105/75688, 3661769/13623840*e^17 - 28992919/2724768*e^15 + 2195969249/13623840*e^13 - 16225458137/13623840*e^11 + 2035065799/454128*e^9 - 36751464769/4541280*e^7 + 26899602733/4541280*e^5 - 23948006177/13623840*e^3 + 778354223/4541280*e, 477311/3027520*e^17 - 3799543/605504*e^15 + 290177671/3027520*e^13 - 2172841533/3027520*e^11 + 836912061/302752*e^9 - 15828879173/3027520*e^7 + 12866736571/3027520*e^5 - 4377783723/3027520*e^3 + 499061661/3027520*e, 3752327/27247680*e^17 - 29615179/5449536*e^15 + 2232671447/27247680*e^13 - 16380784601/27247680*e^11 + 2030651419/908256*e^9 - 35870410687/9082560*e^7 + 25077314269/9082560*e^5 - 22342061171/27247680*e^3 + 869490059/9082560*e, -202039/908256*e^16 + 7992901/908256*e^14 - 120891847/908256*e^12 + 890344615/908256*e^10 - 554631709/151376*e^8 + 1974315807/302752*e^6 - 1394592907/302752*e^4 + 1193529991/908256*e^2 - 38537697/302752, -1, 69227/756880*e^17 - 69729/18922*e^15 + 43368877/756880*e^13 - 83391999/189220*e^11 + 67082071/37844*e^9 - 2739006161/756880*e^7 + 1282546121/378440*e^5 - 995037811/756880*e^3 + 60329641/378440*e, -36227/302752*e^16 + 1428039/302752*e^14 - 21508323/302752*e^12 + 157671297/302752*e^10 - 293055893/151376*e^8 + 1034860081/302752*e^6 - 719902639/302752*e^4 + 204690311/302752*e^2 - 20773249/302752, 352133/851490*e^17 - 22348559/1362384*e^15 + 424457521/1702980*e^13 - 12605731807/6811920*e^11 + 1594543943/227064*e^9 - 1830448559/141915*e^7 + 22289110283/2270640*e^5 - 10393865411/3405960*e^3 + 702930283/2270640*e, -1016779/13623840*e^17 + 8150561/2724768*e^15 - 629105539/13623840*e^13 + 4790005927/13623840*e^11 - 632684177/454128*e^9 + 12623011379/4541280*e^7 - 11457403763/4541280*e^5 + 13501738507/13623840*e^3 - 562343353/4541280*e, 627245/1816512*e^17 - 24808061/1816512*e^15 + 375343445/1816512*e^13 - 2768997707/1816512*e^11 + 1732528469/302752*e^9 - 6230141333/605504*e^7 + 4504878367/605504*e^5 - 3884156921/1816512*e^3 + 116705185/605504*e, -1088947/6811920*e^17 + 8518583/1362384*e^15 - 632956447/6811920*e^13 + 4528540201/6811920*e^11 - 534671555/227064*e^9 + 8420521547/2270640*e^7 - 3955347329/2270640*e^5 + 746986051/6811920*e^3 + 58356581/2270640*e, 175447/1513760*e^17 - 1406861/302752*e^15 + 108624687/1513760*e^13 - 827270331/1513760*e^11 + 327843959/151376*e^9 - 6540352861/1513760*e^7 + 5932147317/1513760*e^5 - 2325344551/1513760*e^3 + 294471647/1513760*e, 886417/3405960*e^17 - 870457/85149*e^15 + 520755667/3405960*e^13 - 470592377/425745*e^11 + 226614553/56766*e^9 - 7479204257/1135320*e^7 + 2112043597/567660*e^5 - 2285107261/3405960*e^3 + 20413307/567660*e, 1924829/6811920*e^17 - 15278245/1362384*e^15 + 1162513709/6811920*e^13 - 8663228447/6811920*e^11 + 1104598117/227064*e^9 - 20639506309/2270640*e^7 + 16336338283/2270640*e^5 - 15946835417/6811920*e^3 + 567525653/2270640*e, 152513/756880*e^17 - 1202477/151376*e^15 + 90488023/756880*e^13 - 661600349/756880*e^11 + 244275157/75688*e^9 - 4237673579/756880*e^7 + 2781725873/756880*e^5 - 665140279/756880*e^3 + 38863093/756880*e, -54959/605504*e^16 + 2148295/605504*e^14 - 31952879/605504*e^12 + 229682057/605504*e^10 - 412381749/302752*e^8 + 1351763381/605504*e^6 - 750486455/605504*e^4 + 118500283/605504*e^2 + 2369471/605504, -50835/302752*e^16 + 2008461/302752*e^14 - 30354611/302752*e^12 + 223739079/302752*e^10 - 419914063/151376*e^8 + 1513510481/302752*e^6 - 1110631873/302752*e^4 + 344604907/302752*e^2 - 38356171/302752, -117403/605504*e^16 + 4621491/605504*e^14 - 69466683/605504*e^12 + 507698429/605504*e^10 - 938725913/302752*e^8 + 3278214489/605504*e^6 - 2207121571/605504*e^4 + 580092855/605504*e^2 - 50431701/605504, 196159/605504*e^16 - 7744835/605504*e^14 + 117023735/605504*e^12 - 863366813/605504*e^10 + 1625579793/302752*e^8 - 5903725445/605504*e^6 + 4396240139/605504*e^4 - 1337353243/605504*e^2 + 136379869/605504, -50597/908256*e^16 + 2087669/908256*e^14 - 33487409/908256*e^12 + 268302167/908256*e^10 - 190391209/151376*e^8 + 846477293/302752*e^6 - 899259047/302752*e^4 + 1149497789/908256*e^2 - 48730949/302752, 191725/908256*e^16 - 7593049/908256*e^14 + 115206733/908256*e^12 - 854836063/908256*e^10 + 541256905/151376*e^8 - 1997785653/302752*e^6 + 1540946555/302752*e^4 - 1455858889/908256*e^2 + 48622485/302752, 257413/1816512*e^16 - 10141021/1816512*e^14 + 152225629/1816512*e^12 - 1105364587/1816512*e^10 + 669398949/302752*e^8 - 2234218861/605504*e^6 + 1307336175/605504*e^4 - 786827953/1816512*e^2 + 13068753/605504, 2510377/13623840*e^17 - 20039153/2724768*e^15 + 1536619057/13623840*e^13 - 11576172331/13623840*e^11 + 1501215689/454128*e^9 - 28936198577/4541280*e^7 + 24526622159/4541280*e^5 - 26863834381/13623840*e^3 + 1168682809/4541280*e, -5884379/27247680*e^17 + 46430971/5449536*e^15 - 3494674259/27247680*e^13 + 25517893577/27247680*e^11 - 3126594931/908256*e^9 + 53610522259/9082560*e^7 - 34071981013/9082560*e^5 + 23861026007/27247680*e^3 - 519812723/9082560*e, -26587/302752*e^16 + 1044773/302752*e^14 - 15718423/302752*e^12 + 115746491/302752*e^10 - 218824611/151376*e^8 + 810174249/302752*e^6 - 645598681/302752*e^4 + 232618983/302752*e^2 - 29015519/302752, 595225/1816512*e^16 - 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42248275379/9082560*e^3 + 1408087851/3027520*e, -3339311/5449536*e^17 + 131403551/5449536*e^15 - 1971160943/5449536*e^13 + 14323785185/5449536*e^11 - 8707181183/908256*e^9 + 29380773607/1816512*e^7 - 17761192789/1816512*e^5 + 10859689931/5449536*e^3 - 199200995/1816512*e, -154748/141915*e^17 + 9817445/227064*e^15 - 93189968/141915*e^13 + 5532914617/1135320*e^11 - 699579589/37844*e^9 + 1604991223/47305*e^7 - 9745781313/378440*e^5 + 4486943951/567660*e^3 - 290657893/378440*e, -296761/1362384*e^17 + 11746705/1362384*e^15 - 177903265/1362384*e^13 + 1313982871/1362384*e^11 - 823371313/227064*e^9 + 2966942945/454128*e^7 - 2148260459/454128*e^5 + 1801802965/1362384*e^3 - 43855861/454128*e, 11286757/27247680*e^17 - 88736789/5449536*e^15 + 6646827997/27247680*e^13 - 48213633271/27247680*e^11 + 5843465453/908256*e^9 - 97795156397/9082560*e^7 + 56986318139/9082560*e^5 - 27745119001/27247680*e^3 - 154013651/9082560*e, -30251/151376*e^16 + 1184573/151376*e^14 - 17624479/151376*e^12 + 126244203/151376*e^10 - 223960131/75688*e^8 + 709615193/151376*e^6 - 348582289/151376*e^4 + 43128655/151376*e^2 + 2318545/151376, 917729/1816512*e^16 - 36518885/1816512*e^14 + 557297441/1816512*e^12 - 4163504963/1816512*e^10 + 2659821941/302752*e^8 - 9963449161/605504*e^6 + 7945161743/605504*e^4 - 8009382557/1816512*e^2 + 295634065/605504, -684569/681192*e^17 + 27009389/681192*e^15 - 407003879/681192*e^13 + 2981725793/681192*e^11 - 1841549603/113532*e^9 + 6442236553/227064*e^7 - 4339254535/227064*e^5 + 3358019495/681192*e^3 - 89509187/227064*e, -15924211/27247680*e^17 + 125465171/5449536*e^15 - 9434706451/27247680*e^13 + 68940458713/27247680*e^11 - 8480231747/908256*e^9 + 146906533931/9082560*e^7 - 95417883197/9082560*e^5 + 62645294023/27247680*e^3 - 629332387/9082560*e, 519415/454128*e^17 - 20657407/454128*e^15 + 315270967/454128*e^13 - 2359091233/454128*e^11 + 1513769295/75688*e^9 - 5726452431/151376*e^7 + 4660190373/151376*e^5 - 4770251059/454128*e^3 + 180001987/151376*e, -1027159/567660*e^17 + 16225165/227064*e^15 - 153000706/141915*e^13 + 8982903839/1135320*e^11 - 1112710699/37844*e^9 + 9785473319/189220*e^7 - 13382503501/378440*e^5 + 5362843987/567660*e^3 - 312917211/378440*e, -1578965/1816512*e^16 + 62442689/1816512*e^14 - 944648189/1816512*e^12 + 6968215007/1816512*e^10 - 4359533929/302752*e^8 + 15675926045/605504*e^6 - 11345303475/605504*e^4 + 9899754953/1816512*e^2 - 325195445/605504, -80201/454128*e^16 + 801173/113532*e^14 - 48883049/454128*e^12 + 180482605/227064*e^10 - 27823796/9461*e^8 + 761079785/151376*e^6 - 234177799/75688*e^4 + 303924551/454128*e^2 - 1626999/37844, -782267/908256*e^16 + 30696281/908256*e^14 - 459110219/908256*e^12 + 3327789443/908256*e^10 - 2019947489/151376*e^8 + 6821303651/302752*e^6 - 4161457415/302752*e^4 + 2665944347/908256*e^2 - 56912085/302752, -503585/454128*e^16 + 9945565/227064*e^14 - 300285281/454128*e^12 + 275835115/56766*e^10 - 685305551/37844*e^8 + 4852449201/151376*e^6 - 843058565/37844*e^4 + 2747288483/454128*e^2 - 40320337/75688, 123067/302752*e^16 - 4823141/302752*e^14 + 72173539/302752*e^12 - 525715495/302752*e^10 + 971505119/151376*e^8 - 3412313033/302752*e^6 + 2351225817/302752*e^4 - 632347371/302752*e^2 + 54799243/302752, -25337/56766*e^16 + 7982707/454128*e^14 - 29915491/113532*e^12 + 865791751/454128*e^10 - 520060707/75688*e^8 + 106417971/9461*e^6 - 938998963/151376*e^4 + 260312567/227064*e^2 - 12987011/151376, -113875/454128*e^16 + 284626/28383*e^14 - 70241455/454128*e^12 + 268075319/227064*e^10 - 44634668/9461*e^8 + 1450334243/151376*e^6 - 676462159/75688*e^4 + 1574593609/454128*e^2 - 15868995/37844, 337795/454128*e^16 - 3338005/113532*e^14 + 201905863/454128*e^12 - 744604241/227064*e^10 + 233098701/18922*e^8 - 3362987283/151376*e^6 + 1230074505/75688*e^4 - 2202907897/454128*e^2 + 4769282/9461, 57231877/27247680*e^17 - 452879297/5449536*e^15 + 34278628117/27247680*e^13 - 253092408811/27247680*e^11 + 31716266129/908256*e^9 - 571840988717/9082560*e^7 + 416128625879/9082560*e^5 - 361089657481/27247680*e^3 + 10930879009/9082560*e, -4078429/3027520*e^17 + 32269129/605504*e^15 - 2441099149/3027520*e^13 + 17996666947/3027520*e^11 - 6743214731/302752*e^9 + 120712414287/3027520*e^7 - 86540213389/3027520*e^5 + 25179894337/3027520*e^3 - 2444806299/3027520*e, 43513/227064*e^17 - 226124/28383*e^15 + 29455105/227064*e^13 - 121252427/113532*e^11 + 45015955/9461*e^9 - 858045937/75688*e^7 + 500393969/37844*e^5 - 1374915271/227064*e^3 + 7664575/9461*e, -1791331/908256*e^17 + 70830877/908256*e^15 - 1071001399/908256*e^13 + 7890248059/908256*e^11 - 4922670465/151376*e^9 + 17592692907/302752*e^7 - 12545087707/302752*e^5 + 10779200215/908256*e^3 - 341000981/302752*e, -3177551/9082560*e^17 + 25302931/1816512*e^15 - 1932050471/9082560*e^13 + 14446125593/9082560*e^11 - 1847383739/302752*e^9 + 34621044871/3027520*e^7 - 27570938197/3027520*e^5 + 27466509683/9082560*e^3 - 866401227/3027520*e, 14105317/13623840*e^17 - 112266251/2724768*e^15 + 8575552957/13623840*e^13 - 64275069301/13623840*e^11 + 8273571803/454128*e^9 - 157475743997/4541280*e^7 + 130188298889/4541280*e^5 - 137655391261/13623840*e^3 + 5368772659/4541280*e, -1930573/1816512*e^16 + 76443973/1816512*e^14 - 1159803061/1816512*e^12 + 8608256083/1816512*e^10 - 5455648765/302752*e^8 + 20183656181/605504*e^6 - 15666752039/605504*e^4 + 15070113481/1816512*e^2 - 529431689/605504, 305577/605504*e^16 - 12144225/605504*e^14 + 185308417/605504*e^12 - 1388180295/605504*e^10 + 2682185435/302752*e^8 - 10233440531/605504*e^6 + 8463653201/605504*e^4 - 2871561637/605504*e^2 + 310606255/605504, -1643491/1816512*e^16 + 65066839/1816512*e^14 - 986034499/1816512*e^12 + 7293629905/1816512*e^10 - 4585594823/302752*e^8 + 16655215771/605504*e^6 - 12349161557/605504*e^4 + 11104132327/1816512*e^2 - 355902427/605504, -80257/302752*e^16 + 3142089/302752*e^14 - 46801345/302752*e^12 + 336647239/302752*e^10 - 603819323/151376*e^8 + 1969679803/302752*e^6 - 1077845625/302752*e^4 + 185826069/302752*e^2 - 10935023/302752, 47983/151376*e^16 - 1902287/151376*e^14 + 28879643/151376*e^12 - 214126861/151376*e^10 + 405199449/75688*e^8 - 1478439141/151376*e^6 + 1098206079/151376*e^4 - 311178703/151376*e^2 + 22281045/151376, -269251/605504*e^16 + 10609763/605504*e^14 - 159603291/605504*e^12 + 1166447637/605504*e^10 - 2153160897/302752*e^8 + 7482794705/605504*e^6 - 4988177315/605504*e^4 + 1346899927/605504*e^2 - 136122845/605504, 166373/113532*e^16 - 6587651/113532*e^14 + 99746279/113532*e^12 - 735583115/113532*e^10 + 458981907/18922*e^8 - 1637282361/37844*e^6 + 1157993913/37844*e^4 - 970992791/113532*e^2 + 28754101/37844, 182257/454128*e^16 - 7280383/454128*e^14 + 111483865/454128*e^12 - 834120421/454128*e^10 + 531293887/75688*e^8 - 1963406873/151376*e^6 + 1496226505/151376*e^4 - 1345610641/454128*e^2 + 37563691/151376, 3090287/1702980*e^17 - 97943605/1362384*e^15 + 1857237527/1702980*e^13 - 55043474009/6811920*e^11 + 6942557281/227064*e^9 - 31730965537/567660*e^7 + 95821406161/2270640*e^5 - 44739005377/3405960*e^3 + 3059738861/2270640*e, -194579/151376*e^16 + 1915715/37844*e^14 - 115217939/151376*e^12 + 420988639/75688*e^10 - 194396702/9461*e^8 + 5413361145/151376*e^6 - 1804277295/75688*e^4 + 932548269/151376*e^2 - 20457483/37844, 347481/605504*e^16 - 13655817/605504*e^14 + 204493657/605504*e^12 - 1482468343/605504*e^10 + 2693534747/302752*e^8 - 9023164067/605504*e^6 + 5344270057/605504*e^4 - 1048047325/605504*e^2 + 47580399/605504, 2304553/27247680*e^17 - 18302921/5449536*e^15 + 1386178753/27247680*e^13 - 10157985619/27247680*e^11 + 1239928097/908256*e^9 - 20677060433/9082560*e^7 + 11132436071/9082560*e^5 - 328991629/27247680*e^3 - 934199039/9082560*e, 178429/454128*e^16 - 1742959/113532*e^14 + 103774993/454128*e^12 - 374302163/227064*e^10 + 113111581/18922*e^8 - 1523666477/151376*e^6 + 468934999/75688*e^4 - 670110943/454128*e^2 + 2474515/18922, 1273885/1816512*e^16 - 50538769/1816512*e^14 + 766712101/1816512*e^12 - 5662244863/1816512*e^10 + 3534307737/302752*e^8 - 12585408869/605504*e^6 + 8833040339/605504*e^4 - 7237679809/1816512*e^2 + 206336037/605504, 261395/908256*e^16 - 10166243/908256*e^14 + 149782067/908256*e^12 - 1056816605/908256*e^10 + 607716227/151376*e^8 - 1793028523/302752*e^6 + 607645761/302752*e^4 + 289318081/908256*e^2 - 36541065/302752, 187997/227064*e^16 - 14802253/454128*e^14 + 55590193/113532*e^12 - 1621471723/454128*e^10 + 993583079/75688*e^8 - 1710001945/75688*e^6 + 2208623345/151376*e^4 - 817724333/227064*e^2 + 45265839/151376, -220909/454128*e^16 + 8768041/454128*e^14 - 133499569/454128*e^12 + 996299299/454128*e^10 - 637348941/75688*e^8 + 2401747173/151376*e^6 - 1946757851/151376*e^4 + 2012509717/454128*e^2 - 75919633/151376, -339619/1362384*e^17 + 3335077/340596*e^15 - 199593535/1362384*e^13 + 722178197/681192*e^11 - 217723489/56766*e^9 + 2875533827/454128*e^7 - 788788945/227064*e^5 + 475362625/1362384*e^3 + 5225827/56766*e, -1268863/681192*e^17 + 100366727/1362384*e^15 - 759070105/681192*e^13 + 11190935927/1362384*e^11 - 6989172743/227064*e^9 + 12513897011/227064*e^7 - 17912635987/454128*e^5 + 955191116/85149*e^3 - 458867039/454128*e, 71485/1362384*e^17 - 3477265/1362384*e^15 + 68538553/1362384*e^13 - 704556451/1362384*e^11 + 669993037/227064*e^9 - 4139561813/454128*e^7 + 6151690307/454128*e^5 - 9611874205/1362384*e^3 + 455367841/454128*e, -1640449/1362384*e^17 + 64999729/1362384*e^15 - 986167513/1362384*e^13 + 7307209687/1362384*e^11 - 4607257681/227064*e^9 + 16827200393/454128*e^7 - 12640747211/454128*e^5 + 11577058525/1362384*e^3 - 365599285/454128*e, 164731/605504*e^16 - 6327311/605504*e^14 + 92019563/605504*e^12 - 642637161/605504*e^10 + 1105076333/302752*e^8 - 3303748121/605504*e^6 + 1223628599/605504*e^4 + 214997297/605504*e^2 - 91301015/605504, 6939017/27247680*e^17 - 53084185/5449536*e^15 + 3800910257/27247680*e^13 - 25433460851/27247680*e^11 + 2597209057/908256*e^9 - 25109755057/9082560*e^7 - 20033976521/9082560*e^5 + 63473272579/27247680*e^3 - 3048929791/9082560*e, -6051413/4541280*e^17 + 47957089/908256*e^15 - 3639857693/4541280*e^13 + 27012169379/4541280*e^11 - 3418893385/151376*e^9 + 62969076413/1513760*e^7 - 48310247351/1513760*e^5 + 45560511809/4541280*e^3 - 1490850121/1513760*e, -17653/113532*e^16 + 2759851/454128*e^14 - 20462147/227064*e^12 + 291167917/454128*e^10 - 169402093/75688*e^8 + 127379551/37844*e^6 - 180010499/151376*e^4 - 30289673/113532*e^2 + 14476387/151376, -9241897/27247680*e^17 + 73866941/5449536*e^15 - 5685810457/27247680*e^13 + 43221447151/27247680*e^11 - 5713093037/908256*e^9 + 114591975857/9082560*e^7 - 105579388139/9082560*e^5 + 127719433741/27247680*e^3 - 5437398589/9082560*e, -12531491/6811920*e^17 + 49422167/681192*e^15 - 7443017711/6811920*e^13 + 13619748977/1702980*e^11 - 3359429723/113532*e^9 + 117125454691/2270640*e^7 - 19531993453/567660*e^5 + 59251594733/6811920*e^3 - 751470181/1135320*e, 227755/908256*e^16 - 8870479/908256*e^14 + 130711699/908256*e^12 - 919153249/908256*e^10 + 522042743/151376*e^8 - 1473965475/302752*e^6 + 335012445/302752*e^4 + 660460697/908256*e^2 - 55656949/302752, -1249445/5449536*e^17 + 48676025/5449536*e^15 - 717341501/5449536*e^13 + 5043050279/5449536*e^11 - 2862837329/908256*e^9 + 8094713581/1816512*e^7 - 1965664075/1816512*e^5 - 2745919015/5449536*e^3 + 132010531/1816512*e, -10493/340596*e^17 + 2157989/1362384*e^15 - 2797796/85149*e^13 + 480305801/1362384*e^11 - 471836621/227064*e^9 + 743841331/113532*e^7 - 4438836349/454128*e^5 + 3342725035/681192*e^3 - 286879829/454128*e, 2186603/3405960*e^17 - 17381815/681192*e^15 + 1325706173/3405960*e^13 - 9918162239/3405960*e^11 + 1273473241/113532*e^9 - 24128734123/1135320*e^7 + 19722785161/1135320*e^5 - 20378630069/3405960*e^3 + 837806621/1135320*e, -6024677/6811920*e^17 + 47778103/1362384*e^15 - 3626450537/6811920*e^13 + 26870744621/6811920*e^11 - 3384200395/227064*e^9 + 61540400317/2270640*e^7 - 45592074949/2270640*e^5 + 40102328921/6811920*e^3 - 1108080899/2270640*e]; 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;