// Make newform 294.4.e.j in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_294_e();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_294_e_Hecke();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_294_4_e_j();". // 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_294_4_e_j();". // 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 := [1, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]]; 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_294_e();" function MakeCharacter_294_e() N := 294; order := 3; char_gens := [197, 199]; v := [3, 2]; // 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; // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_294_e_Hecke();" function MakeCharacter_294_e_Hecke(Kf) N := 294; order := 3; char_gens := [197, 199]; char_values := [[1, 0], [0, -1]]; assert UnitGenerators(DirichletGroup(N)) eq char_gens; values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field F := Universe(values);// the Hecke field chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values); return chi; 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 := 4; raw_aps := [[0, 2], [3, -3], [0, 8], [0, 0], [-40, 40], [-4, 0], [-84, 84], [0, 148], [0, -84], [58, 0], [-136, 136], [0, 222], [-420, 0], [-164, 0], [0, 488], [-478, 478], [548, -548], [0, 692], [908, -908], [-524, 0], [440, -440], [0, -1216], [684, 0], [0, 604], [832, 0], [464, -464], [0, -632], [0, 160], [2198, -2198], [770, 0], [-184, 0], [0, -1452], [-646, 646], [3012, 0], [0, 3170], [1880, -1880], [604, -604], [0, -1116], [1784, 0], [0, -344], [-1392, 1392], [-4052, 0], [0, 3108], [-50, 50], [-162, 0], [1544, -1544], [-1204, 0], [-2000, 0], [388, -388], [0, 4180], [0, 1322], [2412, 0], [-4336, 4336], [-764, 0], [0, 4300], [3860, -3860], [-2800, 2800], [0, -4880], [6674, -6674], [-9402, 0], [-9100, 9100], [-5952, 0], [3004, 0], [688, -688], [0, 5592], [0, 2922], [0, 7492], [10766, 0], [3984, -3984], [180, 0], [-10428, 10428], [0, -8684], [5648, -5648], [0, 2546], [8268, 0], [0, -10872], [-10434, 10434], [0, -3044], [0, -8910], [5616, -5616], [8932, 0], [-5538, 0], [6700, -6700], [5048, 0], [0, -1344], [0, 4392], [3666, 0], [0, -26], [-7656, 0], [12608, 0], [0, -3068], [6456, -6456], [-11896, 11896], [-264, 0], [0, 2628], [13568, 0], [0, 20656], [3628, -3628], [0, 4852], [0, 7130], [-12788, 0], [-2406, 2406], [-25412, 25412], [0, 9690], [-5604, 5604], [-21568, 21568], [-20300, 0], [0, 13812], [21996, -21996], [-8368, 0], [0, 21504], [10270, -10270], [28358, 0], [16292, -16292], [11256, 0], [-15518, 15518], [10452, 0], [72, -72], [0, -11962], [-6016, 0], [26068, -26068], [-20530, 0], [0, -10056], [-6152, 6152], [0, -14716], [28202, 0], [0, -22114], [0, -9288], [-23848, 0], [0, -34756], [-26044, 26044], [36204, 0], [0, 11424], [-16622, 0], [0, -38524], [18440, 0], [-13968, 13968], [10916, -10916], [-12360, 0], [-3402, 3402], [-292, 0], [0, -6910], [-568, 568], [-12144, 0], [14828, -14828], [22824, 0], [41780, 0], [-21420, 21420], [0, -18132], [0, -24036], [0, 4374], [46348, 0], [-20660, 0], [0, -1800], [41996, -41996], [-41308, 0], [0, -3936], [0, -7212], [38976, 0], [53544, -53544], [0, 21392], [21162, 0], [8224, 0], [0, -8140], [-32158, 32158], [-41416, 41416], [-12296, 12296], [57652, -57652]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_294_e_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_294_4_e_j();". // 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_294_4_e_j(:prec:=2) chi := MakeCharacter_294_e(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 4)); S := BaseChange(S, Kf); maxprec := NextPrime(997) - 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_294_4_e_j();". // 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_294_4_e_j( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_294_e(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<5,R![64, -8, 1]>,<11,R![1600, 40, 1]>],Snew); return Vf; end function;