// Make newform 3850.2.c.ba in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3850_c();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_3850_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_3850_2_c_ba();". // 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_3850_2_c_ba();". // 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 := [8, -8, 6, -6, 3, -2, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0], [0, 2, 4, 1, 0, -1], [-8, 2, -4, 3, 0, 1], [2, 4, 0, -1, 0, -1], [8, -6, 8, -7, 4, -1], [-5, 3, -3, 2, -1, 1]]; Rf_basisdens := [1, 4, 4, 2, 4, 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_3850_c();" function MakeCharacter_3850_c() N := 3850; order := 2; char_gens := [2927, 2201, 1751]; v := [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_3850_c_Hecke();" function MakeCharacter_3850_c_Hecke(Kf) N := 3850; order := 2; char_gens := [2927, 2201, 1751]; char_values := [[-1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 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, -1, 0, 0, 0], [0, 0, 1, 0, 0, 1], [0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, -1], [0, 0, 0, 0, 0, 2], [3, 1, 0, 0, 2, 0], [0, 0, 3, 0, 0, -1], [1, 3, 0, 0, 0, 0], [-4, 1, 0, 0, -1, 0], [0, 0, -1, 0, 0, 1], [3, 3, 0, 0, 2, 0], [0, 0, 4, -3, 0, 1], [0, 0, 0, 2, 0, 2], [0, 0, 5, 0, 0, 3], [0, -2, 0, 0, -2, 0], [-2, -2, 0, 0, 2, 0], [0, 0, 0, 0, 0, 4], [-10, 0, 0, 0, -2, 0], [0, 0, -4, 0, 0, 2], [-7, -1, 0, 0, -2, 0], [0, 0, -2, 0, 0, 2], [-4, 4, 0, 0, 2, 0], [0, 0, 1, -2, 0, -3], [-2, 6, 0, 0, 2, 0], [0, 0, -2, 2, 0, 0], [0, 0, -4, -1, 0, -5], [-3, -1, 0, 0, 0, 0], [0, 0, -2, 0, 0, -4], [0, 0, 4, 2, 0, -2], [-11, -3, 0, 0, 0, 0], [0, 0, -10, 0, 0, 0], [-5, -1, 0, 0, -4, 0], [7, 5, 0, 0, 4, 0], [1, 5, 0, 0, 0, 0], [0, 0, 12, 0, 0, 2], [0, 0, 4, 4, 0, 4], [0, 0, 8, 0, 0, 0], [0, 0, 2, -2, 0, 2], [4, -5, 0, 0, 1, 0], [6, -6, 0, 0, -2, 0], [-8, -2, 0, 0, 2, 0], [0, 0, 4, 3, 0, 1], [0, 0, -4, 4, 0, -2], [4, 0, 0, 0, 4, 0], [12, -2, 0, 0, 2, 0], [0, 0, 2, 2, 0, -4], [0, 0, 8, 2, 0, 6], [-16, -1, 0, 0, -5, 0], [0, 0, 18, 2, 0, 2], [19, 1, 0, 0, -2, 0], [1, 1, 0, 0, 2, 0], [-12, -10, 0, 0, 2, 0], [0, 0, -5, -2, 0, -1], [0, 0, 0, 4, 0, -4], [-8, 1, 0, 0, 1, 0], [2, -6, 0, 0, 0, 0], [0, 0, -2, -6, 0, 6], [-24, 0, 0, 0, -2, 0], [0, 0, 12, -2, 0, 6], [0, 0, -12, -1, 0, 1], [0, 0, 6, 0, 0, 2], [-16, 3, 0, 0, 1, 0], [0, 0, 21, -2, 0, 5], [0, 0, -9, -4, 0, -3], [0, 7, 0, 0, 1, 0], [0, 0, -6, -2, 0, 2], [0, 0, -8, 1, 0, 1], [8, 2, 0, 0, 4, 0], [0, 0, 13, 2, 0, 9], [1, -1, 0, 0, -2, 0], [0, 0, 6, 6, 0, 4], [0, 0, 16, -2, 0, 8], [4, 8, 0, 0, 8, 0], [0, 0, 6, 2, 0, 0], [0, 2, 0, 0, -4, 0], [0, 0, 0, -2, 0, -4], [-4, 11, 0, 0, 3, 0], [-7, -11, 0, 0, 2, 0], [8, 2, 0, 0, -6, 0], [6, 6, 0, 0, 2, 0], [1, -11, 0, 0, -8, 0], [0, 0, -1, -2, 0, -1], [-2, 0, 0, 0, 6, 0], [0, 0, 16, 2, 0, -2], [-12, -11, 0, 0, -3, 0], [0, 0, -10, -4, 0, -4], [-12, 0, 0, 0, -2, 0], [0, 0, 19, -4, 0, 3], [0, 0, -9, -6, 0, 9], [24, -4, 0, 0, -4, 0], [0, 0, 7, 6, 0, 1], [-8, 6, 0, 0, -2, 0], [20, 7, 0, 0, 5, 0], [0, 0, -28, 0, 0, -4], [6, 10, 0, 0, 2, 0], [0, -10, 0, 0, -8, 0], [0, 0, 2, 6, 0, -4], [19, 5, 0, 0, -4, 0], [0, 0, 0, 0, 0, 4], [0, 0, -16, 0, 0, -10], [0, 0, 0, -4, 0, 8], [22, 0, 0, 0, 0, 0], [10, 2, 0, 0, 4, 0], [0, 0, -9, -2, 0, -5], [0, 0, 11, -6, 0, 5], [0, 0, 8, -4, 0, 2], [-6, -4, 0, 0, -10, 0], [9, 9, 0, 0, -6, 0], [0, 0, -8, 3, 0, -1], [0, 0, -24, -4, 0, 6], [0, 0, 8, -2, 0, 16], [8, 6, 0, 0, -2, 0], [12, -2, 0, 0, -2, 0], [-6, 6, 0, 0, -6, 0], [0, 0, 11, -2, 0, -7], [0, 0, -10, 4, 0, 2], [0, 0, -7, -4, 0, -5], [18, -8, 0, 0, -2, 0], [14, 12, 0, 0, -4, 0], [0, 0, -6, 8, 0, 0], [0, 0, 2, -2, 0, -10], [0, 0, 2, 2, 0, -8], [2, -2, 0, 0, 0, 0], [-17, -7, 0, 0, 4, 0], [-12, 6, 0, 0, -4, 0], [8, 0, 0, 0, 0, 0], [0, 0, 20, 6, 0, -14], [0, 0, -42, 0, 0, 0], [12, -10, 0, 0, 2, 0], [0, 0, -44, 4, 0, 0], [-14, 4, 0, 0, -2, 0], [0, 0, -15, 0, 0, -5], [-13, -5, 0, 0, -6, 0], [1, -7, 0, 0, -2, 0], [0, 0, 14, 0, 0, -8], [0, 0, -12, 0, 0, 8], [0, 0, 12, 2, 0, -4], [16, -2, 0, 0, 4, 0], [17, -11, 0, 0, -4, 0], [-1, 13, 0, 0, 8, 0], [0, 0, 21, -2, 0, -1], [0, 0, -12, 4, 0, 4], [-8, -3, 0, 0, -11, 0], [12, 1, 0, 0, 7, 0], [0, 0, 10, 0, 0, -4], [0, 0, 8, 4, 0, -10], [0, 2, 0, 0, -6, 0], [0, 0, -31, 2, 0, 7], [0, 0, 0, 8, 0, -2], [30, 6, 0, 0, -2, 0], [0, 0, 26, 0, 0, 14], [0, 0, -32, 8, 0, 0], [0, 0, -10, -6, 0, -4], [-34, 0, 0, 0, -2, 0], [-13, 7, 0, 0, 8, 0], [0, 8, 0, 0, 10, 0], [0, 0, -6, 10, 0, -2], [-18, 4, 0, 0, 4, 0], [0, 0, -18, 0, 0, -6], [0, 0, -28, -3, 0, -5], [0, 0, 28, -16, 0, 4], [32, 0, 0, 0, 12, 0], [0, 0, -22, 4, 0, 0], [0, 0, -18, 2, 0, 0], [0, -14, 0, 0, -10, 0], [0, 0, -6, -6, 0, -6], [4, 10, 0, 0, 4, 0], [0, 0, -12, 10, 0, -8], [-2, 16, 0, 0, 10, 0], [2, -16, 0, 0, -12, 0], [11, -1, 0, 0, -8, 0], [0, 0, -22, 0, 0, 0], [-28, -7, 0, 0, -1, 0], [-10, 6, 0, 0, 2, 0], [4, -14, 0, 0, -2, 0], [18, 8, 0, 0, 4, 0], [0, 0, -4, -5, 0, 19], [18, -8, 0, 0, 0, 0], [0, 0, -19, 6, 0, -9], [-11, -11, 0, 0, -8, 0], [0, 0, 21, 4, 0, -9], [0, 0, -22, 8, 0, 4], [0, 0, 23, 6, 0, 9], [8, -1, 0, 0, 3, 0], [0, 0, 4, 10, 0, -20], [0, 0, 13, 8, 0, -11], [6, 4, 0, 0, 12, 0], [28, 0, 0, 0, -4, 0], [0, 0, -3, 10, 0, -15], [0, 0, -20, -7, 0, 5], [-12, -8, 0, 0, -8, 0], [-2, -2, 0, 0, 10, 0], [0, 0, -28, -7, 0, -3], [0, 0, -1, -10, 0, 15], [20, 0, 0, 0, 18, 0], [0, 0, 3, 12, 0, -3], [0, 0, 2, 6, 0, 10], [0, 0, -32, 7, 0, 3], [-1, -3, 0, 0, -4, 0], [24, 6, 0, 0, 10, 0], [0, 0, 14, 6, 0, -14], [-1, 3, 0, 0, 10, 0], [30, 2, 0, 0, 8, 0], [0, 0, -46, -2, 0, -10], [36, -8, 0, 0, 4, 0], [0, 0, -16, 13, 0, 5], [-10, -2, 0, 0, 10, 0], [10, -10, 0, 0, -8, 0], [0, 0, 24, 9, 0, 7], [32, 3, 0, 0, -5, 0], [0, 0, 37, -10, 0, 7], [0, 0, -19, -4, 0, -7], [32, -8, 0, 0, 0, 0], [-12, -10, 0, 0, -4, 0], [0, 0, -2, -2, 0, -4], [25, -3, 0, 0, -2, 0], [0, 0, -31, 4, 0, 9], [0, 0, 11, 0, 0, 9], [1, 15, 0, 0, 0, 0], [34, 8, 0, 0, 2, 0], [28, -15, 0, 0, -7, 0], [0, 0, -21, -6, 0, 17], [0, 0, 0, -4, 0, 0], [9, 11, 0, 0, -8, 0], [0, 0, 13, -6, 0, -7], [22, 18, 0, 0, -4, 0], [0, 0, 12, -5, 0, 19], [-14, -4, 0, 0, 10, 0], [0, 0, 27, -12, 0, 13], [-31, -5, 0, 0, -18, 0], [19, -15, 0, 0, -2, 0], [26, -2, 0, 0, -6, 0], [0, 0, 27, 0, 0, -5], [0, 0, 44, 3, 0, 3], [32, 14, 0, 0, -4, 0], [0, 0, 6, 10, 0, 2], [-32, -4, 0, 0, 0, 0], [4, -16, 0, 0, 4, 0], [0, 0, 10, -8, 0, 6], [-5, 21, 0, 0, 6, 0], [0, 0, -8, 8, 0, -16], [18, 0, 0, 0, 0, 0], [0, 0, 14, -12, 0, -8], [22, 8, 0, 0, 6, 0], [0, 0, 38, 4, 0, 10], [8, 2, 0, 0, 2, 0], [8, 4, 0, 0, 0, 0], [0, 0, 4, 12, 0, -16], [0, 0, 6, 8, 0, 12], [-29, -5, 0, 0, 2, 0], [0, 0, -31, 6, 0, -13], [34, -10, 0, 0, -2, 0], [0, 0, -24, -7, 0, 15], [-40, -10, 0, 0, -2, 0], [-8, 8, 0, 0, -6, 0], [0, 0, 10, 0, 0, 14], [0, 0, -18, 8, 0, -20], [0, 0, 42, -6, 0, -2], [0, 0, -24, -2, 0, -18], [0, 0, -12, 4, 0, -4], [-22, -6, 0, 0, -10, 0], [0, 0, -26, 12, 0, -4], [0, 0, -19, -2, 0, 9], [28, 2, 0, 0, -2, 0], [12, 8, 0, 0, 18, 0], [8, -14, 0, 0, 4, 0], [0, 0, 20, 4, 0, -20], [0, 0, -38, 2, 0, -2], [-38, 8, 0, 0, 4, 0], [0, 0, 4, -6, 0, 26], [0, 0, -17, 2, 0, -5], [1, -19, 0, 0, 0, 0], [0, 0, 6, -6, 0, 18], [0, 0, 28, -10, 0, -6], [0, 0, -8, 10, 0, 6], [33, -1, 0, 0, 4, 0], [6, 10, 0, 0, 10, 0], [-3, -5, 0, 0, -14, 0], [0, 0, 24, 8, 0, 16], [38, 10, 0, 0, -4, 0], [0, 0, 32, 9, 0, -3], [6, -10, 0, 0, -6, 0], [0, 0, 20, 0, 0, 0], [-8, -2, 0, 0, 2, 0], [0, 0, -12, 14, 0, -16], [0, 0, -14, -12, 0, -4], [12, -8, 0, 0, 4, 0], [-55, 5, 0, 0, -6, 0], [-10, 26, 0, 0, 6, 0], [0, 0, -23, 10, 0, 3], [0, 0, -14, 2, 0, 18], [3, 15, 0, 0, -4, 0], [0, 0, 30, 10, 0, 2], [-8, -18, 0, 0, -8, 0], [4, 7, 0, 0, 9, 0], [0, 0, 4, 0, 0, 6], [-17, -15, 0, 0, 6, 0], [0, 0, 2, 8, 0, -10], [0, 0, 38, -2, 0, 10], [0, 0, -12, -8, 0, 18], [-31, 13, 0, 0, 0, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3850_c_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3850_2_c_ba();". // 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_3850_2_c_ba(:prec:=6) chi := MakeCharacter_3850_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(1999) - 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_3850_2_c_ba();". // 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_3850_2_c_ba( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3850_c(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![64, 0, 68, 0, 16, 0, 1]>,<13,R![256, 0, 320, 0, 36, 0, 1]>,<17,R![64, 0, 752, 0, 60, 0, 1]>,<19,R![232, -46, -6, 1]>,<37,R![16, 0, 36, 0, 20, 0, 1]>],Snew); return Vf; end function;