// Make newform 8034.2.a.z in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_8034_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_8034_2_a_z();". // 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_8034_2_a_z();". // 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 := [1552, -6316, 1118, 15763, -7015, -14618, 6361, 6204, -2404, -1239, 434, 108, -36, -3, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-592135085106752, 2735906855139534, -235038449926693, -5100601073725863, 1186311372117602, 2912651300509245, -742563932810444, -670560099753944, 176417731545861, 62687524315698, -17247628357060, -1741789540612, 518140367105, -3341759347], [-7510828872730, 305776841980152, -429175712907053, -446376121324599, 575804728969528, 223588882118493, -252692469808438, -43074872724922, 49208409954315, 1787398276572, -4283106154742, 241219668796, 121522864993, -12670008395], [41261138165888, 16523568578403, -160392768597467, -94789427122899, 143064053475349, 103511839862979, -45172405213102, -36307339325161, 5955785366799, 5237503644027, -355914392252, -328982202812, 8250683722, 7437129421], [67234006456384, -164021431732794, 22338499957991, 146667384042234, -121000025836105, -20121761021580, 81401298589306, -13884105134675, -20391279195918, 4893999455973, 2013117619607, -534709503235, -59503252771, 16735095548], [-696096892188, 18990965483608, 16363934034728, -27525433033153, -50057127693710, 19719493652720, 33626274944798, -9159648834785, -8331580582394, 2167385042979, 818192200816, -220735533930, -24088984468, 6829796404], [321698456894260, -873824979116130, 18632735153039, 1021694063716269, -772353790449958, -284435428081239, 574313684971036, -37437704187596, -146139006849039, 24846996819930, 14443924013984, -3096334884004, -428736219223, 101168043581], [93265012848748, 72415118017302, 267983097437951, -538687991427303, -853332240720538, 636877173580161, 563752133186128, -280935521431316, -137259949634175, 54759751083666, 13089742387748, -4698358621144, -378058949095, 132195076697], [-240724229362315, 498400052484456, 529906004653921, -910338490594662, -440040849178856, 566623348152159, 170536386071396, -162281355426796, -31415177667555, 22989677068086, 2517262146709, -1543899870542, -65377606061, 37454660035], [-113491089496292, 23740354322750, 482165035857659, -144254361560619, -659138629686466, 171611111684913, 350770178370856, -88069692253324, -80361026660595, 20353862368234, 7666899083920, -2016688298880, -224068680263, 61385752497], [-52594046507132, 23740354322750, 472015528692799, -144254361560619, -659138629686466, 171611111684913, 350770178370856, -88069692253324, -80361026660595, 20353862368234, 7666899083920, -2016688298880, -224068680263, 61385752497], [-91513305290492, 106258840381992, 413593508460356, -434168909972403, -496358883141133, 378111654620451, 231962318420908, -135044548880321, -47813470495515, 22844790904641, 4177793518358, -1783176710104, -114887884225, 47799908567], [280944669298364, -506871319390622, -853786741692003, 1197390391324439, 1007892134519590, -932125820605101, -513376507176776, 325219663165232, 114757728296943, -55483675391710, -10647593900116, 4434352237832, 303334971739, -120974609621]]; Rf_basisdens := [1, 1, 30448521494580, 15224260747290, 7612130373645, 7612130373645, 2537376791215, 30448521494580, 30448521494580, 7612130373645, 10149507164860, 10149507164860, 7612130373645, 10149507164860]; 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_8034_a();" function MakeCharacter_8034_a() N := 8034; order := 1; char_gens := [5357, 1237, 5773]; 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_8034_a_Hecke(Kf) return MakeCharacter_8034_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 := 2; raw_aps := [[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1], [0, 1, 0, 0, 0, 0, 0, 1, -1, 0, 1, 0, 0, 0], [2, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1], [0, 1, 0, 0, 1, 0, 1, -1, 1, 0, 0, 1, 0, 0], [-1, 0, 0, -1, 1, -1, 1, 0, 1, 0, -1, 1, 1, -1], [0, 1, 0, 0, -1, 0, -1, 1, 0, 0, 0, 0, 0, 1], [-1, -1, 0, 0, 0, 0, 0, 0, -1, -1, -1, 0, 0, 0], [0, -1, -1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1], [1, -1, 0, 1, -2, -1, -1, 2, -1, 0, 0, -1, 0, 0], [1, -1, 1, 1, 0, -1, 0, 1, -1, -1, 0, 0, 0, -1], [1, -2, -1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1], [1, -1, 0, -1, 1, 2, 1, -1, 0, 0, -1, -1, 0, -1], [0, 0, 0, 0, -1, 0, 1, 1, 1, 0, -1, 0, 0, 1], [5, -1, 0, 0, -1, 0, -1, -1, 0, -1, 0, 0, 0, -1], [1, -1, 0, 1, 0, -1, 0, -1, 1, 0, 0, 0, 0, -2], [0, 1, 1, -1, 2, 1, 2, 0, 1, 2, 0, 0, 2, 1], [-1, 0, 0, 0, 1, -2, 1, 1, 0, 1, 0, 2, 0, 2], [-1, 2, 0, -1, 1, -1, 1, 1, 1, 2, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -1, -1, -1, 2, 2, 2, -3, 2, 1, 0, 0, 1, 0], [2, 1, -1, 0, 0, -1, 0, -1, 2, 1, 0, 1, -1, -1], [0, 0, 0, 0, -1, -1, -1, 1, -1, 0, 1, -1, -1, 1], [1, -2, 0, 2, 0, 1, 0, 0, 0, -1, 1, 0, 0, 1], [2, -1, 1, -1, -2, 1, -2, -1, -1, -2, 0, -1, -2, -2], [-3, 0, 1, -2, 2, 1, 3, -1, 1, 2, 0, -1, 1, 0], [2, -2, 1, 0, 3, 0, 1, 0, 1, 1, 1, 0, 1, 2], [1, -2, 0, 2, -1, -2, -1, 1, 0, 1, 0, 1, 2, -1], [5, -1, 0, -1, -2, -1, 0, 2, 0, 0, 0, 0, 0, 1], [1, -1, 0, 1, 4, -4, 0, -1, 1, 0, 1, 1, 3, 0], [3, -3, 0, 2, 0, 0, 0, -1, -1, -1, 2, 0, 2, 1], [-2, -1, 1, 1, 1, -4, 1, 2, 0, 0, 0, 1, 3, 1], [-1, -1, 0, 0, 1, 3, 1, -1, -1, 1, 0, 0, 1, 2], [3, 1, -1, 0, -3, -1, -3, 1, 0, 0, 0, 0, -1, 0], [4, 0, 1, 0, -3, 3, 1, 2, -1, -1, 0, -2, -1, 1], [0, -1, -1, -2, 1, 1, -1, -1, 2, 1, 0, 1, -1, -1], [4, 1, 1, 0, -2, 2, -1, 1, -3, -1, 2, -3, -2, 2], [-4, 2, 0, 0, -3, 0, -3, 3, -1, 0, -1, -3, -2, -2], [2, -1, -2, 0, 1, 0, -2, -1, 2, 0, 0, 0, 0, 1], [0, 1, 0, 2, -1, 2, -1, 2, -3, 0, 0, -1, -2, 2], [-1, -1, 0, 3, 2, -1, 0, -1, 2, 0, 1, 2, 2, 0], [2, -2, 0, 0, 0, -2, -2, 0, -1, 0, -2, 0, 0, 0], [1, 1, -1, 0, 1, -1, 1, 1, 1, 0, -1, 1, 2, 1], [-4, 0, -1, -2, 4, 0, 4, -4, 2, 1, -1, 3, 1, 1], [5, 0, -1, -1, -2, 1, -2, 0, 0, 1, 0, -3, 0, 1], [1, -1, 0, 1, -1, -1, -1, -1, -1, -2, -1, -1, 1, -2], [-2, 2, -1, -1, 2, -4, 0, -2, 2, 0, -2, 3, 1, -1], [-1, -3, 1, 3, 3, -2, 0, 0, -2, -1, 2, 2, 1, 2], [-3, 7, 0, -4, -1, 2, 1, -2, 3, 1, 0, -1, 0, -2], [-1, -1, 1, 2, 2, 1, 0, -1, 0, 0, 0, 1, 1, -2], [3, 3, 1, -2, 1, -3, -1, 0, 2, 0, 0, 0, -1, 0], [0, 0, 0, 0, 3, 0, 0, -3, 4, 0, -2, 1, 4, -3], [-2, 0, 1, 2, -1, 1, 1, 0, 0, -3, 0, -1, -1, 0], [2, 1, 0, 3, -2, 0, -2, 3, -3, 1, 0, -2, -1, 0], [-5, 3, 0, -4, 0, 4, 2, -1, 3, 1, -1, 0, -1, -3], [2, 1, 2, -2, 0, 3, 0, 0, 1, 0, -1, -2, -2, -2], [-3, -3, -2, -1, 2, 1, 2, -3, 1, 0, -2, 2, 0, 2], [7, 3, 0, -2, -2, 2, -1, -2, -2, -1, -2, -1, -2, -4], [0, 1, -1, -1, -4, 2, -2, 2, 1, 0, 0, -2, -1, -2], [6, -3, -1, -1, 3, 0, 1, -1, 4, 2, -1, 3, 1, 2], [0, 0, 1, 0, -3, 2, 1, 2, -3, -3, 1, -2, -5, -2], [1, -4, -2, -1, 3, 0, 1, -1, 0, 2, 0, 1, 1, 4], [2, 1, -1, 1, -2, 2, -2, 3, -1, 0, 0, -2, -3, 2], [-7, -3, 0, -3, 3, 3, 3, -4, 1, 0, -2, 1, 2, -2], [4, 2, 1, -1, 1, -1, 5, 0, -1, 0, -2, 2, 2, 2], [8, 1, 1, -3, -6, 2, -4, -2, 1, -2, -4, -2, -3, -4], [3, -3, 1, -1, 0, 2, 0, 2, -3, -1, 0, -4, -1, -2], [0, -3, -2, 1, 1, 0, -2, 1, -1, 1, 0, 1, 1, 1], [1, 3, 1, 1, 1, -3, 2, 2, 0, 1, 0, 1, 2, -4], [-2, 0, -1, 3, -1, -3, -1, 3, 0, 0, 0, 0, 2, 6], [-3, 3, -1, 4, 0, -2, -2, 2, 0, 2, 2, 0, 0, -1], [1, 3, -1, 1, 0, -1, 3, -2, 2, -1, 0, 1, 2, -4], [5, -2, -1, -1, 2, -1, 2, 0, 0, 3, 2, 0, 0, 4], [-3, -1, 1, 2, 5, -2, 3, -3, 1, 0, 2, 3, 2, 0], [-1, 3, 0, -2, 3, -1, 1, 2, 4, 3, -1, 3, 3, 2], [1, 4, 2, 2, -1, 1, -1, 1, -1, -1, 2, 0, -1, -1], [7, 0, 0, 0, 0, 0, -2, -2, -2, -1, 0, -1, -2, 1], [10, 0, -2, -2, 1, -1, -2, 0, 2, 4, 0, 0, 1, 1], [2, 2, 0, 1, -2, 1, -2, -1, 0, 1, -1, 2, 0, 1], [-3, 1, 0, -3, 1, 0, 3, -4, 2, -2, 0, 3, 1, -3], [-1, -5, 0, 0, 2, 4, 4, -2, -1, 1, 1, 0, 2, 2], [9, -3, 0, -1, -1, 4, 3, -1, -2, 0, 0, -2, -1, -2], [9, 1, 0, 0, 1, -2, 1, 2, 0, -1, 0, 0, 2, -5], [-1, 1, 2, 1, 2, 1, 4, 1, 0, 2, 0, 2, 2, 3], [4, -6, -2, 0, -1, 5, -2, -1, -2, 0, 0, -3, -1, 2], [8, -3, 0, 1, 5, -2, 5, -4, 4, 1, 0, 1, 5, 1], [5, 3, 2, 0, 1, 1, 3, -1, 3, -1, -2, -1, 1, -5], [8, 3, 0, -1, 1, 3, 1, 2, -2, 1, -1, 2, -1, -1], [7, 2, 1, 0, -1, -1, -1, 0, -2, 0, 0, 1, 3, 1], [-1, -2, -3, 4, -4, -4, -6, 2, -3, -2, 0, 1, 2, 1], [6, 2, -2, 0, -1, 0, -2, -3, 2, 0, -4, -1, 0, -3], [1, 1, 1, 1, 5, -2, -1, -4, 0, -1, 1, 4, 1, -1], [8, 1, -1, -4, 1, 2, 1, 1, 2, 1, 0, 1, -2, 1], [10, 0, 1, -1, 6, -1, 2, -1, 0, 2, 0, 1, 4, 0], [-1, -4, 0, 1, 0, 0, 0, 1, -4, 0, 0, 0, 1, 7], [-4, 2, 2, 0, 1, 0, 5, 2, -2, 0, -2, 1, 2, 0], [2, 1, 1, 0, 1, 2, -2, -3, 1, -1, 2, 2, -4, 1], [0, -4, -3, 1, 4, -2, 2, -1, 0, 0, -3, 4, 3, 2], [3, 3, 2, -1, -3, -3, -1, 3, -1, -2, -2, -1, -2, 1], [-5, 1, 1, 0, 1, 3, 3, -2, 1, 2, 1, -4, -1, -3], [-2, 7, -1, -6, -1, 1, 1, -5, 5, -1, -4, 3, 1, -3], [8, 3, 2, -3, -2, 3, -2, -5, 2, -3, -2, -3, -4, -6], [7, -2, -1, -2, 0, 0, 2, -1, 2, 4, -2, 1, 2, 1], [8, -1, -2, 2, -3, 1, -1, 2, 0, 0, -4, 2, -1, 0], [-5, 0, -2, -1, 1, 1, 1, -5, 2, 0, -1, 1, 1, 1], [-7, 10, 1, -4, 0, -3, 2, 5, 2, 2, 0, 2, 1, 1], [18, -3, 0, 1, 4, -1, 0, -1, 0, 1, 2, -1, 0, 0], [2, -1, 3, 1, 0, 1, -2, -3, 3, -2, 2, -2, -2, 1], [9, 7, -2, 0, -3, 2, 1, 1, 0, -1, 2, 1, -2, 2], [10, -5, 1, 1, 2, -1, 0, -4, -2, -2, 2, 1, 0, 6], [5, 8, 1, -1, 1, 6, 6, 0, 2, 1, 0, 1, 1, -4], [-7, 0, -1, 5, 1, -1, 1, 4, -1, 1, 2, 1, 0, 9], [10, 0, 2, -4, 2, 3, 0, 0, -1, 2, -2, 1, -1, 2], [3, -5, 0, 4, 1, -4, -1, -1, -3, -3, 0, -2, 2, -3], [3, 3, -1, -2, -4, 4, -2, 2, 0, 2, 2, -4, -4, 2], [5, 2, 1, 1, -2, -4, -5, 3, -2, -1, 2, -3, -3, 2], [-12, -7, -1, 1, 6, -1, 6, -4, 3, 0, 0, 3, 4, 2], [8, -4, -2, 4, -9, 1, -7, 1, -1, -2, 2, -2, -3, 0], [4, 2, 1, -4, 0, 8, 0, -3, -1, 1, -2, -2, -4, -1], [-2, -1, -1, -2, 3, -3, -3, -4, 0, 1, -2, 3, 1, -2], [13, 0, -2, 2, 0, -4, -4, -1, -1, 1, 2, 2, 0, -2], [6, -4, -3, 0, 3, -3, 2, -3, 0, 1, -4, 4, 3, -5], [-2, 4, 2, -3, 3, 1, 5, -3, 2, 1, -2, -3, 0, -5], [7, 5, 1, -1, 2, 1, 4, 0, 0, 3, 0, 1, 2, 2], [-3, -1, 0, -1, 3, -2, 2, 1, 3, 0, 2, 0, -1, 4], [4, 5, -1, -1, -2, -5, 0, 3, 1, 2, -2, 1, 0, 4], [-4, 6, 0, -6, -6, 5, -4, 4, -3, -2, 0, -5, -7, -6], [0, -3, 2, 1, -6, 5, 0, -1, -5, -3, -2, -5, -6, -2], [-7, -7, 1, 6, 4, -2, 3, -2, -2, 2, 2, 1, 4, 4], [6, -2, 2, 6, -3, -4, -3, 4, -7, -4, 0, -3, 2, -1], [2, -1, -2, 0, -5, -1, -7, 3, 1, -2, 0, -1, -1, -3], [-1, 11, 0, -3, 1, 2, 3, 0, 0, 0, -2, 1, -5, -4], [-5, 0, 1, 4, -3, -1, -5, 1, -1, -4, 0, -2, -1, -2], [6, 3, -2, 0, 2, -6, 0, -1, 3, 0, -2, 6, 6, -4], [0, -3, -1, 0, 1, -2, -1, 1, 1, 3, 2, 2, 2, 3], [7, 7, 0, 3, -1, 3, -1, 1, -2, 0, 4, 0, 0, 0], [10, -1, 1, -2, 0, 0, 0, 1, 1, 1, 0, -3, 2, 0], [7, 0, 0, 0, -4, 4, -4, 0, -1, 1, 2, -1, -2, 6], [-6, 0, 2, 3, 6, -5, 8, 3, 0, -1, 2, 6, 6, 6], [5, 4, 3, 1, 3, -1, 1, -2, 0, -1, 0, 2, 4, -3], [-4, 0, 0, 1, -1, 0, -3, 2, 3, -1, 0, -2, 1, -1], [8, -2, 2, -6, -2, 7, 6, -5, 0, 0, -2, -3, -1, -2], [-2, 4, -1, -1, 2, 1, 0, 3, 0, 0, -2, -1, -2, -1], [8, -9, 1, 1, 0, 4, 2, -3, 1, 0, 0, 1, 3, 5], [4, 6, 2, 0, 0, -1, 0, 3, -3, 0, 3, 1, 2, 2], [3, 0, 2, 0, 1, 1, 1, 2, -3, 1, 3, -1, 0, 2], [-1, 0, -3, -1, -3, 2, 3, 3, -3, -1, 1, 0, 1, -2], [4, 5, 1, 0, -1, -2, -1, -1, 1, 1, -3, 0, -2, 0], [6, -3, 0, 3, 6, -4, 0, -6, 0, -1, 2, 3, 5, 1], [5, -6, -2, 6, -1, -2, -5, 1, -1, -1, -2, 0, 0, 7], [-3, 0, -2, -1, 3, 1, 1, 2, 0, 2, -3, 2, 0, 5], [-4, -5, -3, 0, 1, 2, -3, 1, -3, 1, 2, 0, -2, 3], [6, -5, 0, 0, 4, 1, 2, -2, -1, 0, 4, 1, 5, -1], [-18, 2, 1, -1, 1, -2, 1, 6, -3, 2, 0, 0, -1, 3], [19, 0, 1, 2, 2, 0, 0, 1, 2, 2, 2, 2, 4, 0], [7, 0, -1, 7, 2, -5, 0, 4, -3, 1, 2, 1, 6, -1], [12, -3, -2, 7, 2, -5, -5, 0, -3, 1, 4, 3, 4, 6], [-8, 6, 1, -5, 1, 0, 7, 0, 3, 0, -1, 0, 2, -1], [2, 2, 0, -4, 1, 4, 1, -1, 4, 4, 0, 0, 0, 8], [12, 3, 3, -6, 1, 3, 5, 3, -2, 1, 0, -3, -1, 1], [6, -12, 1, 5, 4, -1, -2, 1, -2, -2, 2, 2, 2, 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-2, 0, 1, -6, 1, -2, 7, 0, -2, 2, -3, -2, -3], [-22, 9, 4, 5, -3, 4, 3, 9, -7, -5, 3, -5, -6, -1], [-13, -8, -3, 5, -7, 2, 1, 9, -5, 1, 0, -2, 3, -1], [3, -9, -3, -6, 10, 3, 6, -14, 6, -2, -4, 4, -1, -4], [2, 4, 0, -2, 7, 6, 7, 1, 2, 4, 2, -3, -2, -5], [-4, -2, 1, 6, 2, -7, 0, 8, -2, -1, 2, -4, 5, 5], [7, -3, -2, 7, 1, -7, -13, 1, -1, -2, 4, -2, 4, 1], [-1, -2, 1, 4, 6, 1, 3, -4, 0, 0, 4, 3, 1, 2], [-7, 3, -5, 1, -3, 2, -4, 3, -3, -3, 2, 1, 1, 5], [-2, -3, -2, 2, 8, -3, 2, -7, 4, 4, 2, 5, -1, 2], [1, -9, 2, 1, 7, -2, 3, -2, 0, 0, 0, 9, 3, 0], [-3, -1, -1, 2, 3, -3, 5, 9, 5, 2, 2, 5, 5, 9], [27, -8, -5, 9, -2, -5, -8, -2, -1, 5, 5, 0, 4, 5], [-10, 2, -3, -5, 0, 0, -8, 7, -2, 2, -5, 0, -2, 3], [11, -9, -4, 4, 0, -6, -7, -2, -4, 1, 0, 3, 4, 0], [4, -2, 4, 1, 1, -11, -1, 4, 0, -5, 0, -2, 2, 4], [-18, 3, -3, -1, 3, -1, -5, -7, 0, 2, -1, -1, -3, -6], [0, 12, -1, -7, -2, 10, -2, -4, 2, 0, -1, -3, -6, -2], [4, 1, -4, 3, 0, -5, 0, -2, 2, 1, 2, 8, 2, 2], [3, -3, 2, 8, -3, 1, -3, -6, -6, -5, 1, -1, -5, -8], [5, -8, 3, 4, 8, 2, 2, 7, -6, 0, 4, 5, 6, 9], [7, 1, -1, 5, 3, -2, 3, -2, 2, 5, 0, 0, 1, -4], [2, 0, 3, -3, 16, 0, 8, -4, 8, 2, 4, 7, 9, 6], [16, 3, -5, 8, 1, -6, -9, 0, -4, -5, 2, 5, 0, 6], [-5, -9, -1, -3, 19, 2, 15, -4, 6, 5, 1, 6, 7, 7], [-13, 3, 1, -3, 0, -1, -8, -5, 2, -1, 2, 0, -6, -4], [-2, -2, 0, -5, -1, 8, -3, 1, -4, -3, 2, -5, -7, 4], [-4, -3, 6, -5, 0, 2, 2, 2, -8, -1, 0, -2, -5, 6], [-4, 8, -6, 6, -4, -8, -2, 6, 1, 6, 2, 0, 2, 0], [7, -14, -4, 8, -5, -2, -4, 2, -6, -5, 2, -3, 2, -7], [-4, -3, -1, 6, -7, -7, -7, 5, -3, -1, -2, 0, -1, 1], [-13, -5, -7, 11, 7, -7, 1, -3, 3, 1, 0, 7, 4, 6], [-14, 2, 0, 1, 7, 12, 5, -6, 4, -1, 4, -1, -1, 3], [0, -2, 0, 4, -13, 0, -5, 5, -2, -4, 2, -6, -2, 4], [14, -3, -2, 0, 10, -3, 2, 0, 0, 8, 1, 4, 4, 6], [9, 0, 5, -3, 9, 2, 7, 1, 4, 5, -2, 2, 7, 5], [1, 3, -3, -2, -8, 1, -6, 0, -5, -6, -5, -3, -10, -7], [8, 6, -1, 0, -3, -1, -15, -2, -4, 1, 1, -5, -7, 1], [-4, -3, -1, -3, 3, -3, 5, -3, 8, -2, -3, 4, -4, -10], [5, 0, 0, 0, -1, 4, 3, 4, 1, 3, -9, 0, 3, -6], [2, 3, 3, 4, 2, -2, -2, 6, -4, -3, 3, -1, 5, 2], [11, -5, 1, 7, 3, -7, 1, 2, -4, 3, 0, 5, 14, 5], [0, 0, 0, -1, 2, 6, 0, -14, 4, 3, 0, 0, 9, -7], [-10, -1, 1, 3, -5, 3, -1, 0, -3, -4, 2, -8, 4, 0], [5, 0, -3, 5, 1, -1, -4, -3, 0, -7, -2, 8, 0, 2], [6, 0, 1, 3, -2, 3, 2, -6, -4, 6, 8, -2, -6, 0], [-10, 9, -2, -10, 4, 2, 6, -1, 9, 8, 0, 2, 6, 1], [1, -9, 1, -3, -5, -7, -5, -1, -1, -1, 1, -4, -2, -2], [14, 10, 1, -2, 2, -1, -2, -3, 3, -1, 2, 1, -1, -2], [-15, 5, 2, 5, -1, 1, -3, 4, -10, -6, 3, -4, -4, -10], [1, -19, 2, 10, 3, -3, 3, 3, -8, -1, 3, 0, 6, 10], [-17, 8, 2, -9, 8, -3, 6, -3, 8, 2, -2, 6, 6, -6], [8, 3, 2, -5, 7, 4, 1, -13, 3, -1, -4, 5, -1, -7], [1, 2, -5, -6, -9, -1, -3, 1, 1, 2, -4, -6, 1, 0], [-12, 5, 3, -1, -3, 6, -2, 5, -2, -6, 0, -1, -1, -3], [8, -4, -6, 1, -1, -3, -6, -4, 4, 3, -8, 2, 6, 0], [28, -5, -2, -1, -4, 3, -8, -7, 4, 1, 0, -3, -4, -2], [4, -2, 2, -1, -3, 8, -7, -2, -3, -3, 0, 2, -7, 3], [3, 7, -2, -2, 3, 1, 3, -10, 5, 3, -4, 0, 3, 6], [-7, 15, 0, -4, 7, -4, 7, 0, 0, 1, 9, 4, 3, 1], [12, -9, -3, 4, -7, -4, -1, -2, -1, 3, -5, -2, 3, -1], [-21, 7, 2, -2, 4, 5, -1, -1, 1, 3, 6, -4, 1, 6], [10, -3, 0, -1, 7, 7, 5, 2, 1, -1, 2, -6, 4, 5], [13, -8, -1, -2, 6, 0, 7, 9, 2, 6, 2, 2, 6, 6], [-2, 9, 0, -8, 11, 5, 7, -7, 5, -4, 2, 6, 5, 0], [18, 1, 5, -3, 6, 10, 6, -2, 1, 4, 2, -4, 3, -2], [2, 4, 5, 2, 1, 1, 0, 7, -5, 1, 4, -2, -7, -1], [-13, -1, 5, -1, 10, -1, 4, -3, 3, -1, 0, 7, 6, 0], [13, -14, -6, 8, 2, -9, -10, 0, 3, -1, 4, 5, 5, 4], [-5, -2, -1, -4, 5, 5, 3, -11, 12, 2, 1, 3, 7, 1], [6, 5, -1, 1, 12, -4, 6, -5, 14, 2, -3, 6, 9, -1], [34, -2, -2, 1, -8, -2, -14, 2, -7, -1, -2, -9, -3, -7], [18, -3, -2, 1, -2, -3, -8, -9, 0, -3, 6, 1, -6, 0], [-5, 5, 1, -10, 7, 1, 11, 2, 5, 8, -6, -2, 3, 6], [21, -3, 2, 1, -6, 0, -12, 1, -9, -4, 2, -11, -3, -9], [2, -9, 0, 4, 5, -3, 7, 4, 0, 4, -4, 6, 9, -2], [-12, -9, -2, 0, 1, 4, 1, -7, 3, 2, -4, -1, -2, -3], [8, 1, 1, -2, -11, -4, -7, 4, -9, -7, -4, -5, -2, -3], [29, -6, 2, 3, -1, -4, 3, -4, -4, 0, -4, 0, -1, 0], [-11, 10, -4, 9, 4, 1, -2, 4, 2, -2, 5, 6, 4, 9], [3, 11, -1, -4, -2, -3, -6, 3, -2, 4, 1, 1, -5, -4], [18, 5, -4, 0, 0, -5, -2, 6, 0, 8, 0, 2, 3, 10], [4, 11, 3, 3, 3, 1, -5, 2, -6, -2, 9, 1, 3, 0], [9, 7, -1, -1, 1, -10, -4, -2, 3, -1, 2, -3, -1, -11], [9, -9, 1, 1, 1, -9, 0, -9, 0, 1, -6, 0, 2, -3], [2, -1, 1, -1, -3, 4, -1, 0, -7, -2, 0, -1, -7, 3], [-14, 5, 1, -6, 6, 5, 6, 1, -1, -1, 0, 4, -1, 4], [10, 1, 2, -4, -8, 5, 2, 1, 3, -4, 4, 0, -5, 4], [13, -2, -4, 3, -6, 4, -8, -11, 0, 2, 2, -6, -1, 2], [12, 5, -2, -3, -7, 2, -1, -1, 8, -1, -4, 1, -5, -15], [-11, -4, -3, 2, 8, 0, -2, -4, -3, 2, 0, 2, 2, -2], [16, 0, -6, 5, -1, -8, -9, 1, 3, 3, 0, 0, -3, 2], [-8, -8, -2, -1, 9, -1, 3, -4, 1, 3, 8, 1, 8, 4], [-5, 5, 1, 1, -6, -2, 0, -1, -2, -7, -6, 5, -9, -5], [-9, 2, -1, -7, -4, 1, 6, -4, -4, -5, 0, -2, -4, 2], [-18, 4, 0, 4, -3, -8, 1, 8, -6, -2, 2, -1, 4, 5], [26, 10, -3, -3, -2, -1, -6, 1, 5, 0, -2, -5, 0, -3], [-22, 6, 6, -6, -2, -5, 2, 0, -4, -2, -2, 3, -5, -2], [15, 1, 1, 0, -5, 2, -9, 4, 5, 2, 2, -8, 2, -9], [-6, 3, 4, 2, 1, 0, 1, 1, -6, 2, 6, -1, 2, -2], [37, -5, 0, -4, 5, 4, 1, -14, 1, -1, -3, 1, -2, -9], [-15, -2, 3, 1, 1, 1, -1, 4, -3, -7, -2, 0, -4, 2], [2, -1, 0, -5, -4, 6, 2, 0, -1, -5, -4, -8, -1, -10], [-5, -7, -9, 7, -5, -8, -7, 4, 2, -3, -3, 13, 6, 11], [-6, 12, 1, 2, -4, 2, 0, 1, -4, 3, 1, -1, -2, -10], [-3, 2, 3, 2, -6, -1, -2, 9, -6, -8, -1, 1, -4, 2], [-3, -11, 1, 5, 1, -7, -4, -4, 3, 5, 0, -1, 6, 6], [-11, 4, 0, -6, -1, 0, -1, -9, 4, -3, -1, 5, 2, -6], [-4, -1, -2, 1, -6, 4, 0, -1, 1, -1, -8, -1, -1, -3], [-8, 5, 2, -2, 0, 3, 8, -3, 9, -2, -4, 6, 3, -11], [0, 10, -2, -6, -6, 6, 0, 4, 8, 2, -2, -7, -10, -9], [10, 0, 0, 0, -4, -4, -4, 4, 0, -2, -4, -10, 2, -6], [10, 3, 3, -5, 1, 3, 7, -15, 0, -4, -4, 4, -2, 3], [-16, -6, 1, 5, -1, 2, 7, 0, -7, 0, 2, 1, 3, -7], [10, 3, 2, 4, -6, 7, 0, 7, -4, -4, -3, -3, 1, 2], [6, -7, -1, 4, -9, 1, -3, 0, -3, 1, -6, -2, 3, 8], [24, -3, -2, 8, 1, 1, -7, 1, -4, 4, 0, 2, 7, 1], [17, 15, 3, -1, -15, 7, -7, 1, -9, -3, 1, -7, -6, -3], [-21, 8, 0, 8, 8, -9, 8, -2, -1, -1, 8, 7, 9, 2], [-17, -15, -4, -4, 0, -1, 0, -3, 9, 9, -3, -1, 1, 2], [23, -11, 2, 4, -6, 2, -8, 1, -1, -5, 1, -2, -9, 3], [-15, -2, 5, 2, 2, -4, 4, 5, -4, 2, -6, -1, 4, 1], [0, -2, 2, 1, -11, 10, -1, 7, -5, -3, -2, -5, -9, -4], [10, 14, 3, 4, -5, -3, 1, 9, -7, -1, 5, 2, -5, 8], [-8, 6, -2, -3, 3, -4, 7, 8, 5, 7, 4, 3, 1, -2]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_8034_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_8034_2_a_z();". // 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_8034_2_a_z(:prec:=14) chi := MakeCharacter_8034_a(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 2)); 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_8034_2_a_z();". // 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_8034_2_a_z( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_8034_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![1552, 6316, 1118, -15763, -7015, 14618, 6361, -6204, -2404, 1239, 434, -108, -36, 3, 1]>,<7,R![108608, 657008, 964156, 49673, -640804, -170741, 171823, 45906, -23659, -4948, 1761, 235, -67, -4, 1]>],Snew); return Vf; end function;