// Make newform 4032.2.c.l in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4032_c();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4032_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_4032_2_c_l();". // 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_4032_2_c_l();". // 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, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 2, 0, -1], [0, 0, 0, 2], [-1, 0, 2, -1]]; Rf_basisdens := [1, 1, 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_4032_c();" function MakeCharacter_4032_c() N := 4032; order := 2; char_gens := [127, 3781, 1793, 577]; v := [2, 1, 2, 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_4032_c_Hecke();" function MakeCharacter_4032_c_Hecke(Kf) N := 4032; order := 2; char_gens := [127, 3781, 1793, 577]; char_values := [[1, 0, 0, 0], [-1, 0, 0, 0], [1, 0, 0, 0], [1, 0, 0, 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, 0, 0], [0, 0, 0, 0], [0, 0, 0, -1], [-1, 0, 0, 0], [0, 0, -1, -1], [0, 0, -2, 0], [1, -3, 0, 0], [0, 0, -1, -2], [3, 1, 0, 0], [0, 0, 3, 2], [2, 0, 0, 0], [0, 0, -1, 0], [-1, -5, 0, 0], [0, 0, -3, -4], [-4, 0, 0, 0], [0, 0, 1, 2], [0, 0, 4, 0], [0, 0, 2, 6], [0, 0, -1, -6], [5, -5, 0, 0], [-12, 2, 0, 0], [-2, -6, 0, 0], [0, 0, -2, 2], [-7, -3, 0, 0], [8, -6, 0, 0], [0, 0, 6, -1], [-14, 0, 0, 0], [0, 0, -3, -3], [0, 0, -4, -4], [4, -6, 0, 0], [2, 10, 0, 0], [0, 0, 0, -6], [0, 2, 0, 0], [0, 0, -4, -4], [0, 0, 1, -6], [-8, 4, 0, 0], [0, 0, -2, -6], [0, 0, -3, 2], [6, 6, 0, 0], [0, 0, -4, -11], [0, 0, -7, -9], [0, 0, 8, 4], [-23, -1, 0, 0], [-14, -6, 0, 0], [0, 0, 9, 12], [-8, 0, 0, 0], [0, 0, -3, 8], [12, -4, 0, 0], [0, 0, -2, -10], [0, 0, 4, 4], [-6, -4, 0, 0], [-19, -5, 0, 0], [-4, -2, 0, 0], [0, 0, -8, -10], [-9, 3, 0, 0], [1, 3, 0, 0], [0, 0, -10, 1], [22, -4, 0, 0], [0, 0, 7, 4], [22, 4, 0, 0], [0, 0, 1, -6], [0, 0, 8, -3], [0, 0, 9, 10], [-4, -16, 0, 0], [6, -4, 0, 0], [0, 0, -7, -6], [0, 0, 5, 4], [-6, -16, 0, 0], [0, 0, 7, -1], [0, 0, 4, 10], [-27, 5, 0, 0], [27, -3, 0, 0], [0, 8, 0, 0], [0, 0, 6, 0], [0, 0, -9, 4], [18, 6, 0, 0], [0, 0, 13, 2], [0, 0, 4, 10], [-4, 10, 0, 0], [-16, -14, 0, 0], [0, 0, -10, -4], [0, 0, -15, -12], [1, 3, 0, 0], [14, 12, 0, 0], [0, -8, 0, 0], [0, 0, 1, 11], [20, -2, 0, 0], [10, -12, 0, 0], [0, 0, -10, -1], [-10, 18, 0, 0], [0, 0, 8, 6], [4, -20, 0, 0], [8, 0, 0, 0], [0, 0, -11, -7], [0, 0, -3, -8], [6, 6, 0, 0], [0, 0, -12, -9], [-9, 3, 0, 0], [0, 0, 6, 4], [0, 0, -13, 0], [0, 0, -17, -14], [0, 0, -13, 2], [0, 0, 16, 20], [-22, -8, 0, 0], [0, 0, -3, 6], [10, 20, 0, 0], [0, 0, 10, 10], [7, -5, 0, 0], [-3, 3, 0, 0], [-2, 0, 0, 0], [-12, -20, 0, 0], [0, 0, -10, 8], [32, -2, 0, 0], [0, 0, -2, -8], [26, 2, 0, 0], [40, 2, 0, 0], [0, 0, 1, -14], [-6, -22, 0, 0], [0, 0, -13, -8], [0, 0, 7, 7], [0, 0, 12, -6], [-6, 18, 0, 0], [0, 0, 16, -1], [0, 0, 9, 19], [0, 0, 8, 12], [0, 0, 11, 16], [0, 0, 0, 4], [-14, -2, 0, 0], [-10, 20, 0, 0], [0, 0, -22, -4], [0, 0, 5, 6], [35, -3, 0, 0], [-12, 4, 0, 0], [0, 0, 8, 0], [-15, 1, 0, 0], [10, 0, 0, 0], [0, 0, -6, -21], [0, 0, -8, 4], [0, 0, -4, 5], [-24, 10, 0, 0], [0, 0, 6, 12], [0, 0, -23, -10], [-10, 6, 0, 0], [0, 0, -3, 15], [0, 0, -12, -12], [-6, 22, 0, 0], [0, 0, 4, 14], [1, 1, 0, 0], [0, 0, -19, -18], [39, -7, 0, 0], [0, 0, -6, 4], [-21, 7, 0, 0], [0, 0, 5, 4], [-18, 14, 0, 0], [0, 0, -5, -8], [1, 19, 0, 0], [4, 4, 0, 0], [29, 9, 0, 0], [-26, 8, 0, 0], [0, 0, -18, -29], [0, 0, -1, 9], [6, 12, 0, 0], [38, -2, 0, 0], [0, 0, 0, 12], [6, 28, 0, 0], [16, -12, 0, 0], [32, 12, 0, 0], [0, 0, -12, -10], [34, 14, 0, 0], [0, 0, -22, -11], [0, 0, -11, -11], [0, 0, 0, 6], [-27, -13, 0, 0], [50, 6, 0, 0], [-4, -12, 0, 0], [45, -7, 0, 0], [0, 0, 7, -2], [0, 0, 1, 16], [46, 0, 0, 0], [0, 0, 4, -14], [-6, -18, 0, 0], [0, 0, -2, -16], [0, 0, 25, 16], [1, 21, 0, 0], [-27, 19, 0, 0], [0, 0, -6, -15], [0, 0, -12, 8], [0, 0, -4, -8], [-14, -22, 0, 0], [10, -6, 0, 0], [-4, 18, 0, 0], [0, 0, -3, 11], [0, 0, 21, 20], [0, 0, -22, -1], [0, 0, -31, -7], [21, -7, 0, 0], [-18, -26, 0, 0], [0, 0, -9, -8], [-27, 5, 0, 0], [-12, -24, 0, 0], [0, 0, -3, 12], [28, 12, 0, 0], [0, 0, 12, 4], [28, -18, 0, 0], [0, 0, -2, -12], [0, 0, -20, -21], [32, 0, 0, 0], [0, 0, 19, 5], [6, -16, 0, 0], [0, 0, 21, 14], [-34, 8, 0, 0], [0, 0, -10, 1], [-8, -28, 0, 0], [0, 0, -18, -20], [-12, -20, 0, 0], [14, -4, 0, 0], [-24, -12, 0, 0], [-37, 3, 0, 0], [-43, 7, 0, 0], [0, 0, 11, -10], [0, 0, 4, -8], [-26, 20, 0, 0], [-24, -6, 0, 0], [4, 12, 0, 0], [0, 0, 2, 2], [0, 0, 7, 28], [-21, -5, 0, 0], [19, 25, 0, 0], [30, 8, 0, 0], [0, 0, 11, -13], [0, 0, -8, 8], [0, 0, 11, 10], [-8, 0, 0, 0], [-52, 10, 0, 0], [0, 0, -19, -6], [-24, -16, 0, 0], [-12, -14, 0, 0], [0, 0, 9, 8], [0, 0, -23, -13], [-38, 18, 0, 0], [0, 0, -21, 13], [0, 0, -18, 12], [12, -12, 0, 0], [0, 0, 21, 8], [23, 3, 0, 0], [10, -18, 0, 0], [26, 16, 0, 0], [0, 0, -26, 2], [0, 0, 1, -22], [-35, 15, 0, 0], [0, 0, 12, 16], [27, -1, 0, 0], [-9, 37, 0, 0], [-28, -14, 0, 0], [0, 0, -26, -7], [0, 0, -31, -5], [0, 0, -11, -32], [0, 0, 9, -2], [0, 0, 20, 1], [2, 0, 0, 0], [-54, 2, 0, 0], [0, 0, -5, -21], [0, 0, -8, -24], [0, 0, -20, -26], [-57, -1, 0, 0], [0, 0, -6, -28], [0, 0, -1, -8], [31, 15, 0, 0], [0, 0, -31, -14], [0, 0, 13, 32], [0, 0, 20, -14], [0, 0, -3, 20], [20, 6, 0, 0], [8, 4, 0, 0], [12, 2, 0, 0], [-26, 4, 0, 0], [0, 0, 11, 9], [0, 0, 25, 24], [34, 26, 0, 0], [0, 0, 30, 6], [28, -8, 0, 0], [-2, 2, 0, 0], [-28, -32, 0, 0], [0, 0, 14, 18], [0, 0, -10, -16], [-3, -25, 0, 0], [2, 28, 0, 0], [0, 0, -5, -12], [-70, 4, 0, 0], [3, -41, 0, 0], [0, 0, 25, 0], [0, 0, -8, -26], [-44, -14, 0, 0], [0, 0, -32, -16], [0, 0, 14, 8], [0, 0, 16, -19], [8, 16, 0, 0], [0, 0, 6, 27], [0, 0, -26, -30], [0, 0, -30, 8], [-22, 20, 0, 0], [0, 0, 1, -10], [42, -22, 0, 0], [0, 0, 7, 33], [0, 0, -3, -26], [10, 0, 0, 0], [0, 0, -17, 5], [0, 0, -14, -12], [-25, 1, 0, 0], [0, 0, -25, 4], [-8, -20, 0, 0], [0, 0, 31, -4], [22, 8, 0, 0], [0, 0, 17, 12], [21, 15, 0, 0], [-20, 30, 0, 0], [0, 0, -4, -32], [5, -9, 0, 0], [16, 34, 0, 0], [-36, -18, 0, 0], [0, 0, -13, -2], [6, -22, 0, 0], [0, 0, 4, 11], [52, -16, 0, 0], [-12, -30, 0, 0], [-32, -2, 0, 0], [0, 0, -37, -10], [0, 0, -33, -22], [67, -3, 0, 0], [0, 0, -17, -28], [0, 0, -16, -24], [0, 0, 11, -8], [6, 28, 0, 0], [0, 0, -10, 4], [0, 0, 13, 10], [0, 0, -2, -8], [0, 0, 26, 24], [-17, -5, 0, 0], [-8, -38, 0, 0], [-22, 16, 0, 0], [0, 0, -6, 8], [-18, -12, 0, 0], [0, 0, 14, 7], [12, -24, 0, 0], [0, 0, 17, 10], [0, 0, -17, -31], [0, 0, 22, 16], [0, 0, -9, 12], [-36, 20, 0, 0], [0, 0, -22, -1], [0, 0, 21, 14], [46, 10, 0, 0], [0, 0, -23, -16], [48, -8, 0, 0], [0, 0, 6, -4], [29, 1, 0, 0], [-30, 6, 0, 0], [0, 0, 8, 8], [-8, -18, 0, 0], [33, -5, 0, 0], [0, 0, -22, -28], [63, -9, 0, 0], [-47, -5, 0, 0], [0, 0, 9, -15], [0, 0, 20, 8], [10, -26, 0, 0], [0, 0, 26, 13], [-70, -14, 0, 0], [22, -6, 0, 0], [0, 0, -15, 9], [0, 0, 3, 26], [-31, -1, 0, 0], [0, 0, 13, 0], [-6, -20, 0, 0], [0, 0, 7, 4], [0, 0, -4, -26], [15, 21, 0, 0], [-10, 16, 0, 0], [11, 3, 0, 0], [48, -10, 0, 0], [0, 0, 12, -11], [48, 18, 0, 0], [44, 0, 0, 0], [-18, 16, 0, 0], [0, 0, -14, 4], [82, 10, 0, 0], [-16, 36, 0, 0], [0, 0, 40, 28], [0, 0, 9, 42], [12, -28, 0, 0], [14, -38, 0, 0], [0, 0, -36, 3], [0, 0, -27, -31], [0, 0, 31, 10], [29, -33, 0, 0], [34, -20, 0, 0], [32, 8, 0, 0], [-3, 13, 0, 0], [0, 0, -19, -6], [0, 0, -39, -2], [0, 0, 2, 30], [44, 2, 0, 0], [-10, -18, 0, 0], [-9, 35, 0, 0], [0, 0, -6, 7], [-16, 16, 0, 0], [0, 0, 13, 16], [-30, 24, 0, 0], [0, 0, -34, -4], [0, 0, -10, -24], [-66, 12, 0, 0], [0, 0, 9, -18], [0, 0, -17, 21], [0, 0, -29, 16], [26, 6, 0, 0], [0, 0, 30, 9], [-39, -21, 0, 0], [-42, 36, 0, 0], [-13, 19, 0, 0], [4, 12, 0, 0], [0, 0, -11, 2], [0, 0, 0, 14], [-13, 17, 0, 0], [0, 0, -12, 30], [-58, 12, 0, 0], [0, 0, 8, 19], [0, 0, -45, 1], [-10, 0, 0, 0], [0, 0, 24, 24], [-46, 34, 0, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4032_c_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4032_2_c_l();". // 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_4032_2_c_l(:prec:=4) chi := MakeCharacter_4032_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_4032_2_c_l();". // 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_4032_2_c_l( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4032_c(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![4, 0, 8, 0, 1]>,<11,R![4, 0, 8, 0, 1]>,<13,R![16, 0, 1]>,<17,R![-26, -2, 1]>,<23,R![6, -6, 1]>,<31,R![-2, 1]>],Snew); return Vf; end function;