// Make newform 324.3.f.f in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_324_f();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_324_f_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_324_3_f_f();". // 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_324_3_f_f();". // 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_324_f();" function MakeCharacter_324_f() N := 324; order := 6; char_gens := [163, 245]; v := [3, 4]; // 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_324_f_Hecke();" function MakeCharacter_324_f_Hecke(Kf) N := 324; order := 6; char_gens := [163, 245]; char_values := [[-1, 0], [-1, 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 := 3; raw_aps := [[0, 2], [0, 0], [-8, 8], [0, 0], [0, 0], [10, -10], [-16, 0], [0, 0], [0, 0], [0, 40], [0, 0], [-70, 0], [-80, 80], [0, 0], [0, 0], [56, 0], [0, 0], [0, 22], [0, 0], [0, 0], [110, 0], [0, 0], [0, 0], [-160, 0], [0, 130], [0, 40], [0, 0], [0, 0], [182, 0], [-224, 224], [0, 0], [0, 0], [0, -176], [0, 0], [280, -280], [0, 0], [-170, 170], [0, 0], [0, 0], [0, -104], [0, 0], [38, 0], [0, 0], [190, -190], [56, 0], [0, 0], [0, 0], [0, 0], [0, 0], [442, -442], [416, 0], [0, 0], [0, 418], [0, 0], [64, -64], [0, 0], [-520, 0], [0, 0], [0, -230], [0, -320], [0, 0], [136, -136], [0, 0], [0, 0], [0, -50], [0, 616], [0, 0], [-350, 350], [0, 0], [0, 598], [0, 544], [0, 0], [0, 0], [550, -550], [0, 0], [0, 0], [0, -680], [650, 0], [-80, 80], [-782, 782], [0, 0], [0, 58], [0, 0], [290, 0], [0, 0], [0, 0], [560, 0], [0, 850], [0, 760], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-440, 440], [-880, 0], [0, 0], [-682, 0], [0, 0], [1064, 0], [0, 0], [0, -1040], [0, 0], [-1150, 0], [0, 0], [-736, 0], [0, 0], [0, 1102], [0, 0], [-70, 0], [1216, -1216], [0, 0], [0, 0], [0, 400], [0, 0], [0, 0], [1144, -1144], [0, 0], [-1178, 1178], [0, -770], [0, -104], [0, 0], [0, 0], [-520, 0], [0, -518], [0, 0], [0, 0], [1450, -1450], [0, 0], [0, 0], [0, 0], [1190, 0], [-1520, 1520], [-962, 962], [1496, 0], [0, 0], [1144, -1144], [560, 0], [0, 0], [0, -1400], [0, 0], [0, 0], [-1258, 0], [0, 0], [0, -410], [0, -464], [0, 0], [0, 0], [-1610, 1610], [-1600, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 1840], [-430, 0], [-1160, 1160], [0, 0], [-1456, 0], [0, 0], [0, 0], [496, -496], [0, 0], [0, 0], [0, -1850]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_324_f_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_324_3_f_f();". // 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_324_3_f_f(:prec:=2) chi := MakeCharacter_324_f(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 3)); 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_324_3_f_f();". // 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_324_3_f_f( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_324_f(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign))); Vf := Kernel([<5,R![64, 8, 1]>,<7,R![0, 1]>,<11,R![0, 1]>],Snew); return Vf; end function;