// Make newform 546.8.a.g in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_546_a();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_546_8_a_g();". // This may take a long time! To see verbose output, uncomment the SetVerbose lines below. // The precision argument determines an initial guess on how many Fourier coefficients to use. // This guess is increased enough to uniquely determine the newform. // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_546_8_a_g();". // This may take a long time! To see verbose output, uncomment the SetVerbose line below. // The default sign is -1. You can change this with the optional parameter "sign". function ConvertToHeckeField(input: pass_field := false, Kf := []) if not pass_field then poly := [421951600, 704960, -51426, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-782880, 39536, 11, -1], [-7188770, -28066, 299, 1]]; Rf_basisdens := [1, 1, 9300, 4650]; Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf); Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]]; inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf); return Eltseq(inp_vec); end function; // To make the character of type GrpDrchElt, type "MakeCharacter_546_a();" function MakeCharacter_546_a() N := 546; order := 1; char_gens := [365, 157, 379]; v := [1, 1, 1]; // chi(gens[i]) = zeta^v[i] assert UnitGenerators(DirichletGroup(N)) eq char_gens; F := CyclotomicField(order); chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]); return MinimalBaseRingCharacter(chi); end function; function MakeCharacter_546_a_Hecke(Kf) return MakeCharacter_546_a(); end function; function ExtendMultiplicatively(weight, aps, character) prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec primes := PrimesUpTo(prec); prime_powers := primes; assert #primes eq #aps; log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1) F := Universe(aps); FXY := PolynomialRing(F, 2); // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ... R := PowerSeriesRing(FXY : Precision := log_prec + 1); recursion := Coefficients(1/(1 - X*T + Y*T^2)); coeffs := [F!0: i in [1..(prec+1)]]; coeffs[1] := 1; //a_1 for i := 1 to #primes do p := primes[i]; coeffs[p] := aps[i]; b := p^(weight - 1) * F!character(p); r := 2; p_power := p * p; //deals with powers of p while p_power le prec do Append(~prime_powers, p_power); coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]); p_power *:= p; r +:= 1; end while; end for; Sort(~prime_powers); for pp in prime_powers do for k := 1 to Floor(prec/pp) do if GCD(k, pp) eq 1 then coeffs[pp*k] := coeffs[pp]*coeffs[k]; end if; end for; end for; return coeffs; end function; function qexpCoeffs() // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" weight := 8; raw_aps := [[8, 0, 0, 0], [27, 0, 0, 0], [-82, 0, 1, 0], [343, 0, 0, 0], [-1987, 1, -9, 1], [-2197, 0, 0, 0], [-6789, -111, -31, -9], [-8717, 72, 31, -3], [-5696, 392, -201, 38], [-39551, -556, 27, 145], [-68737, 543, -138, -121], [-79718, 371, -31, -334], [-177595, -1372, -64, 249], [-222977, -1560, 805, 217], [-140079, -2283, -1200, -265], [-546425, 1335, 1062, -851], [11048, 6876, -10098, 1066], [-607924, 7391, -85, 2404], [-1227178, 1928, 7058, -1066], [-1382337, -19326, 12762, 1113], [1528231, 8398, -8251, -3393], [1637639, -14917, 5300, -6089], [1313827, 587, -3210, 4585], [-1163832, -22879, 16228, -472], [3135121, 18311, 33906, 3671], [-2817403, 57336, 10918, -5121], [-3614864, -63555, -4099, -3402], [-9863035, 2336, -8542, 4777], [135822, 34083, -9433, 9310], [-3645341, 69565, 43778, -10623], [6304242, -5110, 32586, 12358], [-19992956, 18247, 74555, -25706], [-31878304, -195237, 92219, 12182], [-9875958, -43030, 127252, 526], [-11273526, 57974, -18388, 13218], [6336286, -34925, 39307, -18756], [8911492, 332809, -215753, -7328], [-20911358, -20270, -130686, 46694], [12585591, 106166, -124065, -16329], [-22339337, -227264, 242536, 46293], [1594280, 51527, 325462, -54544], [57726916, -185696, 57280, -63962], [-46349495, -305485, -85851, -57721], [17344372, 413062, -149474, -6314], [-15748958, -451878, -13588, 6750], [19208182, 188911, -297217, 107664], [70213173, -114602, 41915, -34709], [16102153, -1207813, -313064, -20827], [-78798300, 325898, -105840, -1046], [-76608516, -451056, -488368, -13702], [-78473449, -339055, -229896, -43447], [-97494951, 1463790, 417650, 44323], [-29947429, -865797, -428758, -93655], [11712516, 1177565, -154149, 181206], [3935859, 1393478, 207472, 106597], [-41885010, -1009279, 1652734, -47970], [-116824479, 360432, -308616, -8873], [-114758248, -921878, 57282, 8684], [-98812677, 1790089, 569018, -19063], [63111558, 1160944, -732794, -463530], [-25825494, -1222700, 633644, -223722], [-135609852, -1403913, 157668, 295092], [-122638835, 1918253, -104358, 284453], [-322544416, 1574320, -2169206, 123352], [-202149452, -816726, -994758, -43126], [355474552, -2508972, 1567984, -106044], [-46888794, -4241228, 1956866, 118878], 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[234489701945, 3759504815, -687270629, -474903941], [160036613462, -1688330858, 1204437154, -384338558], [-351398305410, -3545220646, 1995321110, 36351422], [34171880605, -2311162128, 1289737354, -450932771], [-150059008954, -2359428399, 1561538373, 116270426], [64379512490, 3740217618, -2331361302, -234415850], [581796636337, -1688380545, -133758204, -12207553], [-126874667595, 1032505138, -2879980524, 146322775], [691079305055, -148293085, 1852213398, 293069383], [-160795929936, -1537923134, 560347262, -47505618], [312390515460, -3032789012, -135280688, -350327316], [9312086456, -2576757714, 548102056, 140712704], [28234765330, -1818302452, 2216616192, 968567504], [-169538451231, 102847151, -100883267, 887357337], [459833952131, 2431297306, 461407525, -122253865], [-363551656625, -1500504269, 1985903274, -151547695], [-745865819390, 1962505260, 970637536, -120557268], [674483963089, 4844507992, -939962054, -187618713], [507724746454, 113786275, 1543262791, 43316370], [-88093137265, -2492362761, 859741985, 598601771], [-450812563426, 3229890976, 216615823, -146502812], [454128718383, -1118165314, -2530594054, 222030561], [-164231787694, 4754532556, -2753778298, 246946122], [-179225992875, 909315247, 482805726, -808112265], [-563941013318, -780487994, 120232947, -213141448], [-141867050388, 1468369111, 1617549897, 183583950], [162905391925, -631178574, 1319288613, 179380953], [-323442438789, -1279560088, 1172620214, 255514991], [299510861928, -5555424766, 61338356, -466278588], [-216583322981, 1850024006, -1695955434, 491614141], [55748407195, -1625500028, -376695045, 695312719], [-567582427252, 3274442943, -2225386893, 582696702], [247297166701, -1543561947, -762840420, 496915815], [337448254606, -1167695252, 3638582938, 130295636], [1011635105283, -2094835883, -1225201725, 543692525], [-16876978626, 2222806697, 83896369, -235055830], [465357586139, -2036682515, -2048274039, -350388161], [-378238589376, 4790926996, 798730104, 724879686], [474039058088, 2666739111, 873242288, 571959832], [-687128742664, 232119775, 487117481, 933534042], [284952068012, -1769725955, -168909627, 64451322], [-155418340514, -504540782, 3054203130, -567269678], [4150754949, -4496911303, -2187057038, -473143727], [549373140203, -1545058196, 3672246099, -193875489], [689754081932, -2905499163, -2632861539, 872161226], [-197986916504, 543011801, 2171347351, -1148415992], [-263018245706, 3038316244, -3231097136, 1123499086], [740429271178, 2188344334, -2635454146, -143927510], [611833714256, -2951977144, 593354340, -880767662], [-366442399440, 1021145309, -1929697208, 57519720], [93541121227, -4703884834, -2352572382, -316178223], [-994823028566, 1652237724, -4868475504, -228400722], [-687951742881, -4164944602, 3355495070, -69224599], [-35153099085, 2219799145, -2497666998, -70262487], [-23231788468, -3364457442, 5120583294, 281515080], [-357553218559, -2460328289, 2365695097, 821832109], [318065597685, 378797494, -1879240195, -1310310217], [370594300434, 5088075282, -1078322150, 598094026], [26197615275, 4460739467, 531642056, -179603787], [-1246938606683, -735960553, -2324793846, -336033609], [777230953834, 2006774133, -125316173, 717391004], [-685958446008, -5672342709, 3522001470, -263105868], [-656930779842, 6502109063, -1466714080, 1183684162], [17635852258, -1803184987, 1210076609, 104986056], [962549757298, 1325287426, -710126192, 951848704], [329924030758, 850420711, -1468925713, 351683800], [-983399646627, -2443226670, -1282633517, -551923021], [570053087165, -166693215, 460193108, -936456529], [499156313880, 7319915578, -2855267540, -410246082], [-844341973197, -648376170, -1739489589, -779079433], [393666498122, -3324917911, 2614977838, -647700350], [-40007358738, -9853266383, -90372559, -314154360], [206505688853, 5170994494, -3000315775, 764879049], [-303000109907, 93971854, -634377814, -1388076933], [-436779586035, 6461315060, -1782588688, -323465245], [-608077124020, 5049116163, 1775737495, -589252274], [1025090230159, 1148646042, -1688600946, 652009235], [-1230483347874, 4586889452, 1882482088, -592794142], [419779008389, 7479920746, -3578864931, -268773039], [155897280916, 3180088656, 2058038332, 982286038], [414074760367, -1189052582, -1214572498, 1406081017], [271495993664, -846186736, 31913004, -1084102432], [292708545264, -896201765, 3751404379, 680630696], [-164770648100, -11000108013, -10983908, 128686116], [-1075848028008, 5451208117, -2106793652, -168441968], [98959031344, -1515545254, -2166850954, -236016556], [-378499736035, -1208233258, -6679462549, 399648607], [-88584433232, 2766226910, 1178072988, 1568233176]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_546_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_546_8_a_g();". // This may take a long time! To see verbose output, uncomment the SetVerbose lines below. // The precision argument determines an initial guess on how many Fourier coefficients to use. // This guess is increased enough to uniquely determine the newform. function MakeNewformModFrm_546_8_a_g(:prec:=4) chi := MakeCharacter_546_a(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 8)); S := BaseChange(S, Kf); maxprec := NextPrime(2999) - 1; while true do trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]); B := Basis(S, prec + 1); S_basismat := Matrix([AbsEltseq(g): g in B]); if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then S_basismat := ChangeRing(S_basismat,Kf); f_lincom := Solution(S_basismat,trunc_vec); f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]]; return f; end if; error if prec eq maxprec, "Unable to distinguish newform within newspace"; prec := Min(Ceiling(1.25 * prec), maxprec); end while; end function; // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_546_8_a_g();". // This may take a long time! To see verbose output, uncomment the SetVerbose line below. // The default sign is -1. You can change this with the optional parameter "sign". function MakeNewformModSym_546_8_a_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_546_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,8,sign))); Vf := Kernel([<5,R![75954600, -3512580, -70133, 328, 1]>],Snew); return Vf; end function;