// Make newform 4020.2.q.i in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4020_q();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4020_q_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_4020_2_q_i();". // 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_4020_2_q_i();". // 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 := [9, -3, -2, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [-3, -2, 2, 1], [0, 5, 1, -1], [-9, 2, 1, 2]]; Rf_basisdens := [1, 6, 3, 3]; 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_4020_q();" function MakeCharacter_4020_q() N := 4020; order := 3; char_gens := [2011, 2681, 3217, 1141]; v := [3, 3, 3, 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_4020_q_Hecke();" function MakeCharacter_4020_q_Hecke(Kf) N := 4020; order := 3; char_gens := [2011, 2681, 3217, 1141]; char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [1, 0, 0, 0], [0, -1, 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], [1, 0, 0, 0], [-1, 0, 0, 0], [0, -1, 0, 1], [0, 1, 0, 1], [1, -1, 0, 0], [-4, 5, 1, -1], [-4, 3, -1, 1], [-4, 5, 1, -1], [0, -1, 0, -1], [0, -1, 0, -2], [-2, -1, -3, 3], [0, 7, 0, 1], [1, 0, 1, 0], [0, 5, 0, -1], [-4, 0, -2, 0], [2, 0, -2, 0], [-11, 11, 0, 0], [6, -6, -1, -1], [0, -3, 0, -3], [13, -13, 0, 0], [0, 11, 0, -2], [0, -3, -3, 3], [-2, 0, -4, 0], [1, -1, 0, 0], [0, -1, 0, 5], [0, -5, 0, -3], [0, 0, 0, 0], [5, 0, 3, 0], [0, 3, 0, 3], [0, 11, 0, -2], [8, 0, 4, 0], [-16, 0, -2, 0], [3, 0, -1, 0], [-4, 0, -2, 0], [1, -1, 0, 0], [-6, 7, 1, -1], [0, 7, 0, 0], [0, 17, 0, -1], [-6, 9, 3, -3], [-10, 0, -2, 0], [0, -7, 0, -2], [-12, 9, -3, 3], [-7, 0, 3, 0], [0, 7, 0, 1], [-1, 3, 2, -2], [-1, 3, 2, -2], [-13, 0, 3, 0], [0, 5, 0, 5], [-5, -1, -6, 6], [0, 11, 0, -1], [0, 21, 0, -3], [17, 0, 3, 0], [-12, 9, -3, 3], [0, 7, 0, 1], [-4, 0, 4, 0], [16, 0, 2, 0], [-19, 0, -3, 0], [3, 0, 5, 0], [-16, 17, 1, -1], [4, 0, 4, 0], [-6, 0, 0, 0], [1, -1, 0, 0], [6, 0, 6, 0], [10, 0, 4, 0], [-6, 9, 3, -3], [0, 11, 0, -8], [1, -1, 0, 0], [0, 9, 0, -3], [-5, 0, 1, 0], [12, -3, 9, -9], [0, 0, 0, 0], [0, -5, 0, -3], [0, 1, 0, 6], [21, -17, 4, -4], [24, -27, -3, 3], [-8, 13, 5, -5], [-19, 0, 3, 0], [-8, 0, -10, 0], [0, -3, 0, -7], [6, -15, -9, 9], [23, -21, 2, -2], [0, 1, 0, -11], [0, -19, 0, 1], [1, -1, 0, 0], [0, 1, 0, -5], [0, 23, 0, 5], [0, 9, 0, 8], [10, 0, -4, 0], [4, -1, 3, -3], [-8, 13, 5, -5], [-12, 21, 9, -9], [17, -21, -4, 4], [-2, 0, 2, 0], [-20, 23, 3, -3], [0, -15, 0, 3], [16, 0, 2, 0], [-8, 0, -10, 0], [16, -13, 3, -3], [13, 0, 7, 0], [0, 43, 0, 0], [0, 11, 0, -1], [-2, 0, -10, 0], [0, -25, 0, 5], [0, 27, 0, -4], [0, -23, 0, -6], [16, -23, -7, 7], [20, -19, 1, -1], [0, -3, 0, 3], [-19, 15, -4, 4], [0, -13, 0, -8], [19, -13, 6, -6], [14, 0, -8, 0], [-29, 35, 6, -6], [-16, 3, -13, 13], [0, 15, 0, 9], [8, 0, 12, 0], [-8, 1, -7, 7], [36, -39, -3, 3], [0, 17, 0, -1], [23, 0, -3, 0], [13, 0, 7, 0], [0, 15, 0, 9], [36, -39, -3, 3], [0, 7, 0, 9], [0, -17, 0, 1], [26, -33, -7, 7], [-42, 45, 3, -3], [7, -1, 6, -6], [0, -35, 0, -3], [5, -9, -4, 4], [2, 5, 7, -7], [8, 0, -12, 0], [0, 13, 0, -12], [-6, 0, 0, 0], [0, -3, 0, 8], [0, -25, 0, -1], [4, -1, 3, -3], [0, 19, 0, -5], [-14, 0, -4, 0], [0, -13, 0, -8], [6, -3, 3, -3], [11, -9, 2, -2], [2, 5, 7, -7], [25, 0, 7, 0], [0, 5, 0, -7], [19, -25, -6, 6], [0, 0, 6, 0], [0, -13, 0, -2], [-2, 0, 14, 0], [-13, 3, -10, 10], [-26, 25, -1, 1], [0, -9, 0, -7], [10, 1, 11, -11], [0, -5, 0, 9], [40, 0, 8, 0], [-8, 11, 3, -3], [36, 0, 6, 0], [17, 0, 3, 0], [-22, 0, 4, 0], [20, 0, 16, 0], [-8, 0, 2, 0], [0, -5, 0, -12], [0, 1, 0, 1], [-10, 5, -5, 5], [-32, 0, 8, 0], [-28, 0, -12, 0], [-10, 0, -12, 0], [0, 1, 0, 6], [12, 0, 6, 0], [-18, 0, 6, 0], [1, 11, 12, -12], [0, 13, 0, 7], [30, -29, 1, -1], [0, -29, 0, -6], [-2, 1, -1, 1], [0, 11, 0, -8], [0, 7, 0, 7], [-37, 0, 3, 0], [55, 0, 1, 0], [-1, 0, 15, 0], [-12, -3, -15, 15], [0, 13, 0, 0], [12, 0, -6, 0], [0, 25, 0, 1], [-22, 17, -5, 5], [26, 0, 12, 0], [43, -49, -6, 6], [43, -37, 6, -6], [0, -15, 0, -3], [35, 0, 9, 0], [-18, 0, -6, 0], [0, 19, 0, 12], [-32, 0, 2, 0], [-10, -7, -17, 17], [16, 1, 17, -17], [-10, 0, 0, 0], [0, -3, 0, 20], [0, -9, 0, 3], [34, -35, -1, 1], [24, -27, -3, 3], [13, 0, -11, 0], [0, -35, 0, 9], [5, 0, -9, 0], [-24, 0, -12, 0], [0, -21, 0, -9], [-4, 3, -1, 1], [-20, 1, -19, 19], [16, -11, 5, -5], [-39, 43, 4, -4], [-27, 0, -1, 0], [-28, 17, -11, 11], [19, -13, 6, -6], [0, 5, 0, -19], [0, 13, 0, 1], [37, -25, 12, -12], [24, -29, -5, 5], [0, -61, 0, 5], [-2, 0, -10, 0], [0, -53, 0, 7], [0, 29, 0, 10], [-13, 0, -21, 0], [-2, 1, -1, 1], [17, -33, -16, 16], [0, 9, 0, -9], [-29, 0, 1, 0], [0, 7, 0, -11], [0, 25, 0, 7], [-43, 0, -3, 0], [-8, 1, -7, 7], [0, 25, 0, 0], [-4, 0, 0, 0], [-24, 0, 8, 0], [0, 63, 0, 3], [45, 0, 5, 0], [48, -51, -3, 3], [2, 0, 0, 0], [28, -11, 17, -17], [6, 0, -6, 0], [10, 1, 11, -11], [-10, 29, 19, -19], [-41, 47, 6, -6], [-53, 59, 6, -6], [26, 0, 12, 0], [0, 31, 0, 1], [-28, 29, 1, -1], [0, -29, 0, 12], [-32, 25, -7, 7], [0, 27, 0, 5], [30, 0, -6, 0], [0, 1, 0, 12], [0, -57, 0, -3], [14, 0, -2, 0], [29, 0, -9, 0], [-2, 0, 8, 0], [0, -3, 0, -3], [-14, -1, -15, 15], [-21, 19, -2, 2], [0, 7, 0, 7], [-37, 27, -10, 10], [0, 23, 0, 4], [50, 0, -2, 0], [7, 11, 18, -18], [-17, -1, -18, 18], [-58, 0, 4, 0], [-3, 0, 17, 0], [0, 11, 0, -13], [0, -37, 0, 11], [9, -17, -8, 8], [-2, 0, 8, 0], [18, 0, -16, 0], [-17, 0, 19, 0], [0, -31, 0, 16], [-41, 35, -6, 6], [0, 57, 0, -7], [0, 31, 0, -6], [46, 0, 2, 0], [0, 41, 0, -8], [-2, 0, -8, 0], [34, -47, -13, 13], [0, 0, 24, 0], [-33, 0, -13, 0], [0, 45, 0, -3], [-22, 0, 0, 0], [-25, 0, -21, 0], [-12, 0, -12, 0], [43, 0, 1, 0], [8, 0, 10, 0], [8, 0, 0, 0], [-26, 25, -1, 1], [-10, 0, -8, 0], [0, -23, 0, -5], [12, 9, 21, -21], [0, -7, 0, 5], [-38, 23, -15, 15], [-56, 49, -7, 7], [-20, 0, 4, 0], [40, -47, -7, 7], [0, -3, 0, -3], [5, -21, -16, 16], [-50, 59, 9, -9], [0, 21, 21, -21], [0, 7, 0, -12], [0, -43, 0, 11], [-9, 0, -13, 0], [0, 1, 0, -24], [62, -55, 7, -7], [-53, 47, -6, 6], [0, 5, 0, -13], [-15, 0, -25, 0], [24, 0, 12, 0], [18, 0, 12, 0], [0, -13, 0, -1], [-76, 75, -1, 1], [-8, 13, 5, -5], [0, -7, 0, 25], [-28, 0, 16, 0], [0, 9, 0, -3], [0, 13, 0, -24], [-24, 0, -18, 0], [19, -25, -6, 6], [0, 49, 0, 6], [38, 0, -8, 0], [40, 0, -8, 0], [22, 0, -16, 0], [15, -17, -2, 2], [0, 19, 0, -3], [5, 0, 9, 0], [0, -3, 0, 21], [-10, 5, -5, 5], [70, -61, 9, -9], [0, 15, 0, -15], [-5, -13, -18, 18], [0, -27, 0, 9], [-31, 0, -3, 0], [0, 1, 0, 13], [35, 0, -3, 0], [-64, 0, -2, 0], [66, 0, -4, 0], [20, 0, -12, 0], [-15, 0, -25, 0], [26, -31, -5, 5], [0, 71, 0, 5], [0, -5, 0, -6], [0, -1, 0, -19], [20, -7, 13, -13], [0, 49, 0, -18], [19, -25, -6, 6], [32, -19, 13, -13], [0, 11, 0, 17], [0, -41, 0, -6], [0, -35, 0, 21], [0, -17, 0, 19], [0, 19, 0, -12], [-14, -13, -27, 27], [-66, 69, 3, -3], [2, 5, 7, -7], [-36, 0, 0, 0], [16, 0, 14, 0], [0, -67, 0, -7], [-3, 0, 11, 0], [0, -61, 0, 5], [0, 13, 0, 1], [0, -15, 0, 15], [0, -21, 0, 20], [18, 7, 25, -25], [28, -23, 5, -5], [-16, 0, -24, 0], [-5, 0, 13, 0], [44, 0, 16, 0], [0, -17, 0, 12], [-44, 0, -4, 0], [-16, 0, -14, 0], [-16, 0, 0, 0], [0, -51, 0, -7], [0, -19, 0, -7], [18, 0, 18, 0], [0, -39, 0, -10], [0, 59, 0, 11], [0, 61, 0, -15], [16, 0, 2, 0], [0, -41, 0, 19], [0, 31, 0, 9], [-14, 25, 11, -11], [0, 47, 0, 10], [6, 9, 15, -15], [53, 0, -3, 0], [-77, 0, 1, 0], [8, 0, -12, 0], [0, -75, 0, -3], [-38, 0, 4, 0], [-62, 49, -13, 13], [0, -15, -15, 15], [9, 0, -7, 0], [0, -39, 0, 21], [0, 1, 0, 24], [-41, 47, 6, -6], [-14, 1, -13, 13], [5, -9, -4, 4], [-40, 29, -11, 11], [22, -25, -3, 3], [88, -83, 5, -5], [0, 19, 0, -12], [62, -55, 7, -7], [10, 0, 20, 0], [-8, 23, 15, -15], [-39, 43, 4, -4], [-66, 69, 3, -3], [0, 79, 0, 0], [4, 0, 8, 0], [-58, 63, 5, -5], [-20, 37, 17, -17], [0, 17, 0, 11], [52, -49, 3, -3], [59, 0, -3, 0], [0, -5, 0, -23], [-24, 21, -3, 3], [-15, 7, -8, 8], [-44, 25, -19, 19], [-38, 0, -10, 0], [56, -55, 1, -1], [0, 9, 0, -22], [0, -55, 0, 11], [62, 0, -2, 0], [-23, 0, -29, 0], [26, 0, -20, 0], [76, 0, 8, 0], [0, -1, 0, -7], [13, -13, 0, 0], [66, -63, 3, -3]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4020_q_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4020_2_q_i();". // 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_4020_2_q_i(:prec:=4) chi := MakeCharacter_4020_q(); 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_4020_2_q_i();". // 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_4020_2_q_i( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4020_q(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![64, -8, 9, 1, 1]>,<11,R![36, 18, 15, -3, 1]>,<17,R![144, 108, 69, 9, 1]>],Snew); return Vf; end function;