// Make newform 4056.2.c.e in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4056_c();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4056_c_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_4056_2_c_e();". // 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_4056_2_c_e();". // 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, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0], [0, 2]]; Rf_basisdens := [1, 1]; 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_4056_c();" function MakeCharacter_4056_c() N := 4056; order := 2; char_gens := [1015, 2029, 2705, 3889]; v := [2, 2, 2, 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; // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4056_c_Hecke();" function MakeCharacter_4056_c_Hecke(Kf) N := 4056; order := 2; char_gens := [1015, 2029, 2705, 3889]; char_values := [[1, 0], [1, 0], [1, 0], [-1, 0]]; 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 := 2; raw_aps := [[0, 0], [-1, 0], [0, 1], [0, 0], [0, 2], [0, 0], [-2, 0], [0, 2], [8, 0], [6, 0], [0, -4], [0, 3], [0, 3], [-4, 0], [0, 0], [-2, 0], [0, 2], [-2, 0], [0, 2], [0, -4], [0, 5], [-8, 0], [0, 2], [0, -3], [0, -1], [18, 0], [-16, 0], [-12, 0], [0, 1], [18, 0], [8, 0], [-4, 0], [0, -3], [-12, 0], [0, -7], [0, -8], [-2, 0], [0, 6], [0, 12], [-6, 0], [-12, 0], [-6, 0], [0, 0], [0, 1], [0, 9], [-16, 0], [-20, 0], [0, 4], [0, -6], [0, 11], [-10, 0], [0, 8], [0, 9], [-20, 0], [-2, 0], [-8, 0], [-10, 0], [0, 4], [26, 0], [0, 13], [28, 0], [0, -9], [0, 6], [24, 0], [-6, 0], [0, -3], [0, -10], [-18, 0], [-12, 0], [0, 15], [0, -1], [0, -12], [-8, 0], [-10, 0], [0, -10], [0, 0], [2, 0], [0, 7], [0, -15], [0, 3], [12, 0], [0, 5], [0, -16], [14, 0], [0, 0], [20, 0], [0, -7], [0, 11], [0, 13], [0, 4], [36, 0], [0, -8], [0, 16], [12, 0], [0, -6], [24, 0], [0, -3], [26, 0], [4, 0], [0, -9], [44, 0], [0, -13], [-28, 0], [-10, 0], [-36, 0], [0, -1], [0, 22], [0, -7], [24, 0], [-38, 0], [-40, 0], [0, -19], [0, -21], [0, -22], [0, 8], [14, 0], [0, -6], [-8, 0], [6, 0], [12, 0], [0, -5], [-34, 0], [-2, 0], [0, 2], [0, 2], [-6, 0], [0, -5], [32, 0], [-48, 0], [0, -7], [0, -2], [0, 4], [-24, 0], [38, 0], [0, -11], [0, -1], [0, 9], [0, 14], [-22, 0], [26, 0], [0, -2], [0, -15], [16, 0], [0, -14], [50, 0], [0, -12], [0, -5], [-42, 0], [-12, 0], [0, 16], [0, 9], [-50, 0], [4, 0], [8, 0], [-4, 0], [16, 0], [16, 0], [0, -25], [42, 0], [0, -3], [0, 6], [54, 0], [0, 8], [36, 0], [0, 15], [0, -12], [40, 0], [-26, 0], [0, -23], [50, 0], [0, -18], [0, 15], [40, 0], [0, -5], [40, 0], [26, 0], [0, -14], [0, 15], [-16, 0], [14, 0], [0, 4], [52, 0], [54, 0], [0, 27], [0, 24], [18, 0], [50, 0], [0, 18], [0, 5], [0, 0], [-62, 0], [0, 6], [-20, 0], [0, 11], [36, 0], [22, 0], [0, 7], [-46, 0], [0, 1], [24, 0], [0, 19], [8, 0], [0, -19], [34, 0], [0, -30], [6, 0], [0, -28], [12, 0], [0, 11], [-4, 0], [-18, 0], [-18, 0], [-32, 0], [0, -14], [0, -4], [0, -11], [-40, 0], [-62, 0], [0, 12], [0, 27], [-10, 0], [0, 0], [0, -9], [0, -4], [-60, 0], [10, 0], [-6, 0], [48, 0], [-48, 0], [0, 26], [66, 0], [28, 0], [0, 20], [-10, 0], [-60, 0], [0, 32], [0, 25], [0, -25], [-20, 0], [40, 0], [0, 18], [28, 0], [64, 0], [0, -23], [0, -9], [40, 0], [0, -28], [0, 30], [0, -26], [16, 0], [0, 39], [0, 15], [0, 4], [-10, 0], [54, 0], [0, 6], [38, 0], [0, -2], [-78, 0], [0, -29], [8, 0], [60, 0], [0, -11], [-66, 0], [0, -7], [-4, 0], [0, -19], [0, -13], [0, -38], [-62, 0], [50, 0], [0, 26], [0, -19], [-40, 0], [-14, 0], [0, 32], [0, 6], [0, 31], [0, 5], [-76, 0], [-80, 0], [0, 8], [-8, 0], [0, 29], [0, -14], [-16, 0], [50, 0], [0, 9], [0, -8], [-2, 0], [-26, 0], [12, 0], [0, 19], [0, -6], [-18, 0], [74, 0], [-40, 0], [66, 0], [-12, 0], [0, 22], [-74, 0], [0, -29], [40, 0], [-20, 0], [84, 0], [0, -1], [-36, 0], [-34, 0], [0, 28], [58, 0], [48, 0], [0, -15], [-46, 0], [-20, 0], [0, -4], [-22, 0], [0, 42], [0, -24], [0, 1], [78, 0], [20, 0], [0, 19], [-58, 0], [0, -28], [0, 29], [18, 0], [0, 6], [0, -42], [32, 0], [30, 0], [0, -1], [-42, 0], [24, 0], [0, -18], [0, 6], [0, -42], [0, -1], [0, 33], [0, 11], [-72, 0], [0, -11], [-54, 0], [0, 39], [48, 0], [0, 13], [20, 0], [-42, 0], [0, 26], [0, 0], [34, 0], [0, 42], [0, -11], [0, -11], [24, 0], [10, 0], [26, 0], [0, 32], [0, 38], [-34, 0], [0, 12], [0, -19], [-42, 0], [48, 0], [0, -10], [68, 0], [42, 0], [0, -21], [0, -16], [38, 0], [-52, 0], [-36, 0], [0, 40], [-2, 0], [16, 0], [-82, 0], [0, -30], [96, 0], [0, -17], [66, 0], [-26, 0], [0, -21], [0, -35], [0, 24], [0, 15], [0, -10], [0, 12], [0, -12], [58, 0], [0, -2], [-48, 0], [0, -31], [0, 49], [0, -6], [60, 0], [0, 36], [58, 0], [0, -12], [6, 0], [-28, 0], [0, 23], [0, -31], [-18, 0], [0, 20], [0, -11], [0, 23], [64, 0], [0, 1], [0, -33], [0, -50], [0, 6], [46, 0], [46, 0], [-76, 0], [-76, 0], [-42, 0], [22, 0], [0, 0], [32, 0], [0, -15], [40, 0], [10, 0], [0, 29], [0, 8], [-76, 0], [0, -5], [0, 5], [-28, 0], [0, 3], [0, -6], [88, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4056_c_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4056_2_c_e();". // 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_4056_2_c_e(:prec:=2) chi := MakeCharacter_4056_c(); 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_4056_2_c_e();". // 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_4056_2_c_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4056_c(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![4, 0, 1]>,<7,R![0, 1]>,<11,R![16, 0, 1]>],Snew); return Vf; end function;