// Make newform 370.2.e.f in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_370_e();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_370_e_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_370_2_e_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_370_2_e_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 := [4, -8, 18, 0, 5, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [-40, 8, -18, 25, -5, 1], [-121, 16, -36, 9, -10, 2], [72, -162, -25, -45, 9, -10], [-26, -454, -65, -117, 7, -26]]; Rf_basisdens := [1, 1, 82, 41, 82, 82]; 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_370_e();" function MakeCharacter_370_e() N := 370; order := 3; char_gens := [297, 261]; v := [3, 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_370_e_Hecke();" function MakeCharacter_370_e_Hecke(Kf) N := 370; order := 3; char_gens := [297, 261]; char_values := [[1, 0, 0, 0, 0, 0], [-1, 0, 0, 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, 0, 0, -1, 0], [0, 0, 0, 0, 0, 1], [1, 0, 0, 0, -1, 0], [-1, 1, 0, 0, 1, 1], [2, 0, 1, 0, 0, 0], [0, -3, 0, 0, 0, 1], [0, 1, 1, 3, 1, -3], [3, 0, 0, 0, -3, 1], [-4, 0, -1, 0, 0, 0], [0, 0, 2, 2, 0, 0], [-3, 0, 3, 0, 0, 0], [1, -3, 0, 1, 2, 1], [-4, 0, 0, 0, 4, -1], [-2, 0, 2, -1, 0, 0], [4, 0, -5, -2, 0, 0], [0, -3, -3, -2, -3, 2], [0, -1, -1, 2, 4, -2], [8, 2, 0, 0, -8, -2], [-4, 4, 0, 0, 4, 2], [-4, 2, 0, 0, 4, 4], [1, 0, 1, 3, 0, 0], [-6, 2, 0, 0, 6, 0], [0, -4, -4, -2, 2, 2], [0, 0, 0, -3, 9, 3], [0, 0, -6, -2, 0, 0], [-3, 0, 1, 1, 0, 0], [-2, 0, 3, 2, 0, 0], [-2, 4, 0, 0, 2, -3], [0, -2, -2, -4, -2, 4], [0, -6, -6, 2, -6, -2], [0, -4, -4, 1, -3, -1], [0, -8, -8, -2, -4, 2], [-1, 0, 9, 1, 0, 0], [0, 1, 1, -4, -6, 4], [-4, 0, -8, -8, 0, 0], [-15, -3, 0, 0, 15, 2], [0, -7, -7, -7, -6, 7], [0, -4, -4, 1, -6, -1], [-14, 2, 0, 0, 14, 0], [0, 6, 6, 3, 11, -3], [-13, 0, -6, 3, 0, 0], [15, -5, 0, 0, -15, 1], [-7, 0, 7, 6, 0, 0], [7, 0, 5, 9, 0, 0], [0, 3, 3, -2, -5, 2], [13, 0, 9, -6, 0, 0], [3, 0, 0, 5, 0, 0], [2, 0, 5, -6, 0, 0], [-4, 4, 0, 0, 4, -7], [7, -9, 0, 0, -7, 5], [-11, 0, -3, 1, 0, 0], [0, -6, -6, -2, 8, 2], [9, 2, 0, 0, -9, -1], [-15, 0, -6, -9, 0, 0], [0, 3, 3, 1, 1, -1], [-6, -5, 0, 0, 6, 0], [-1, 0, 7, -5, 0, 0], [0, -1, -1, 2, 5, -2], [-5, 9, 0, 0, 5, -2], [0, 2, 2, 4, 5, -4], [14, 2, 0, 0, -14, -1], [0, 9, 0, 0, 0, -7], [-4, 0, -4, 3, 0, 0], [0, -1, -1, -4, -21, 4], [0, 4, 4, 6, 8, -6], [0, -1, -1, 7, -14, -7], [0, -3, -3, -6, -22, 6], [-12, 0, 0, 0, 12, -2], [28, 0, 0, 2, 0, 0], [0, 0, 0, -8, -2, 8], [0, 9, 9, 1, 13, -1], [-7, 0, -5, -6, 0, 0], [9, -6, 0, 0, -9, -5], [-18, -5, 0, 0, 18, -3], [0, 13, 13, 7, 1, -7], [-18, 9, 0, 0, 18, 4], [8, 8, 0, 0, -8, -4], [0, 0, -13, -7, 0, 0], [-5, 0, -16, -5, 0, 0], [0, 2, 2, 7, 2, -7], [0, 3, 3, -10, 8, 10], [1, 0, 3, -9, 0, 0], [11, -15, 0, 0, -11, 2], [-4, 0, -8, 0, 0, 0], [-11, 9, 0, 0, 11, 4], [20, 0, -4, -1, 0, 0], [-4, -8, 0, 0, 4, -1], [-4, -2, 0, 0, 4, -12], [0, -2, -2, 2, 0, -2], [-6, -14, 0, 0, 6, 6], [-2, 0, 12, 9, 0, 0], [0, 7, 7, 12, -5, -12], [8, 0, 8, -2, 0, 0], [22, 0, -1, 4, 0, 0], [0, -16, 0, 0, 0, 4], [0, 15, 15, 5, -1, -5], [0, -9, -9, 5, -1, -5], [-19, 6, 0, 0, 19, 4], [-32, 0, 0, 0, 32, 1], [-14, 0, -14, 0, 0, 0], [24, 0, 4, 9, 0, 0], [0, -5, -5, -17, -8, 17], [0, 0, 10, -6, 0, 0], [27, 0, 2, 6, 0, 0], [0, -8, -8, -5, -9, 5], [0, 3, 3, -1, 23, 1], [16, 8, 0, 0, -16, -1], [9, 0, 11, -1, 0, 0], [15, -7, 0, 0, -15, 12], [0, -6, -6, 0, 17, 0], [0, -7, -7, -5, -31, 5], [0, 4, 4, 3, 13, -3], [0, 2, 2, 6, -16, -6], [8, 0, 20, 10, 0, 0], [0, 7, 7, -8, 13, 8], [0, 8, 8, 10, -3, -10], [-2, 0, 16, 7, 0, 0], [5, 4, 0, 0, -5, 9], [-10, -1, 0, 0, 10, -11], [-11, 12, 0, 0, 11, -7], [-23, 13, 0, 0, 23, 11], [11, 13, 0, 0, -11, -5], [17, 0, 0, 11, 0, 0], [0, 4, 4, 5, -20, -5], [0, -15, -15, 5, -7, -5], [0, -10, -10, 6, -32, -6], [12, 0, 0, 8, 0, 0], [0, -1, -1, 4, -17, -4], [-11, 0, 0, 0, 11, 5], [11, -6, 0, 0, -11, 1], [-16, 0, -11, 2, 0, 0], [8, -5, 0, 0, -8, -4], [23, 0, 11, 6, 0, 0], [0, 3, 3, -7, 30, 7], [0, 10, 10, 3, 3, -3], [-10, 0, -12, -18, 0, 0], [16, -5, 0, 0, -16, 19], [8, 0, 2, 11, 0, 0], [0, 1, 1, 5, 28, -5], [-11, -4, 0, 0, 11, -12], [-5, 18, 0, 0, 5, -13], [4, 14, 0, 0, -4, -14], [0, -9, -9, -15, -9, 15], [-18, -8, 0, 0, 18, 8], [0, 1, 1, -7, -29, 7], [0, -9, -9, -4, -13, 4], [0, 5, 5, 5, -4, -5], [21, 0, 11, 3, 0, 0], [-7, 0, 18, 1, 0, 0], [0, -10, -10, 1, -27, -1], [-18, 0, -17, -5, 0, 0], [-1, 2, 0, 0, 1, -13], [10, 2, 0, 0, -10, 13], [14, 0, 2, 4, 0, 0], [26, -4, 0, 0, -26, 8], [-1, 0, -1, -14, 0, 0], [-2, 0, 10, 6, 0, 0], [-9, 0, 0, 0, 9, 8], [0, -4, -4, 6, -16, -6], [0, 15, 15, -5, -9, 5], [0, -2, -2, -17, -12, 17], [0, 13, 13, -17, 17, 17], [-27, 10, 0, 0, 27, -7], [0, -5, -5, -16, -30, 16], [0, 0, 0, -8, -22, 8], [0, 4, 4, 3, 5, -3], [-3, 0, 9, 0, 0, 0], [0, 17, 17, 17, 2, -17]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_370_e_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_370_2_e_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_370_2_e_f(:prec:=6) chi := MakeCharacter_370_e(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 2)); 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_370_2_e_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_370_2_e_f( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_370_e(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![4, -14, 49, -4, 7, 0, 1]>,<7,R![1024, 448, 260, 36, 18, 2, 1]>],Snew); return Vf; end function;