// Make newform 1680.2.f.g in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1680_f();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1680_f_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_1680_2_f_g();". // 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_1680_2_f_g();". // 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], [0, 1, 0, 0], [-6, -2, 2, 1], [-3, -2, -1, 1]]; Rf_basisdens := [1, 1, 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_1680_f();" function MakeCharacter_1680_f() N := 1680; order := 2; char_gens := [1471, 421, 1121, 337, 241]; v := [2, 2, 1, 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_1680_f_Hecke();" function MakeCharacter_1680_f_Hecke(Kf) N := 1680; order := 2; char_gens := [1471, 421, 1121, 337, 241]; char_values := [[1, 0, 0, 0], [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, -1, 0, 0], [1, 0, 0, 0], [2, 0, 1, 0], [0, -1, 0, -1], [0, -3, -2, -3], [2, -1, 0, 1], [0, 0, -2, 0], [0, 2, -2, 2], [0, 1, 2, 1], [0, 0, 2, 0], [-2, 2, 0, -2], [-6, 0, 0, 0], [0, 2, 0, -2], [4, 1, 0, -1], [0, -2, -4, -2], [4, -2, 0, 2], [0, 0, -4, 0], [0, 2, 0, -2], [0, -4, -6, -4], [0, 0, -4, 0], [0, -1, 0, 1], [8, -4, 0, 4], [10, -2, 0, 2], [0, -3, 2, -3], [6, 0, 0, 0], [0, -3, -8, -3], [0, -4, -2, -4], [-10, 3, 0, -3], [0, 4, 4, 4], [-4, -2, 0, 2], [4, 4, 0, -4], [0, -8, -4, -8], [0, -6, 2, -6], [0, 4, 4, 4], [0, -1, 0, 1], [0, 0, 0, 0], [-8, 0, 0, 0], [12, 3, 0, -3], [6, 3, 0, -3], [0, -4, -2, -4], [0, 6, 8, 6], [0, 11, 4, 11], [2, 6, 0, -6], [0, -2, -12, -2], [0, 12, 2, 12], [4, -3, 0, 3], [0, 9, 8, 9], [-8, 7, 0, -7], [0, 6, 0, 6], [0, 4, -4, 4], [0, 11, 4, 11], [0, -12, -8, -12], [-4, -4, 0, 4], [10, 4, 0, -4], [0, -4, -6, -4], [-2, -2, 0, 2], [0, 6, 6, 6], [14, -6, 0, 6], [0, 7, -6, 7], [0, -3, -4, -3], [-18, -3, 0, 3], [0, -9, 0, -9], [0, -6, 0, 6], [0, 9, 10, 9], [0, -2, -4, -2], [4, 0, 0, 0], [2, -6, 0, 6], [0, -10, -6, -10], [0, -6, 0, -6], [18, -3, 0, 3], [0, 8, 2, 8], [0, -3, 16, -3], [-18, -2, 0, 2], [12, -4, 0, 4], [4, 4, 0, -4], [0, 13, 2, 13], [0, -9, 2, -9], [0, 1, -2, 1], [0, 18, 12, 18], [-4, 2, 0, -2], [-2, -7, 0, 7], [0, 5, 16, 5], [0, 12, -4, 12], [0, 6, -2, 6], [0, -4, -2, -4], [0, -5, -14, -5], [-6, -8, 0, 8], [-2, -2, 0, 2], [4, -12, 0, 12], [24, -3, 0, 3], [-32, 4, 0, -4], [-28, 4, 0, -4], [0, 5, 4, 5], [-20, 9, 0, -9], [4, 7, 0, -7], [-18, -6, 0, 6], [-6, 12, 0, -12], [0, 0, 6, 0], [22, -1, 0, 1], [-16, -8, 0, 8], [0, 4, 16, 4], [8, -4, 0, 4], [0, 4, -8, 4], [4, 0, 0, 0], [0, 3, 6, 3], [-8, 4, 0, -4], [2, -1, 0, 1], [0, 11, -4, 11], [0, 6, -4, 6], [0, 9, 8, 9], [-18, 4, 0, -4], [0, 4, 20, 4], [0, -6, 2, -6], [16, -3, 0, 3], [0, -20, -8, -20], [0, 15, 16, 15], [-12, 0, 0, 0], [0, -2, 20, -2], [0, 17, 4, 17], [0, 0, 4, 0], [2, 6, 0, -6], [-2, 7, 0, -7], [0, 26, 18, 26], [0, -18, -14, -18], [0, 1, 26, 1], [14, 3, 0, -3], [0, -12, 0, 12], [0, 0, 2, 0], [0, -9, 10, -9], [12, -1, 0, 1], [0, -16, -14, -16], [-8, 9, 0, -9], [-34, -6, 0, 6], [-6, -6, 0, 6], [0, -18, -4, -18], [-50, 1, 0, -1], [0, -15, -20, -15], [-26, -5, 0, 5], [0, 7, 18, 7], [0, -6, 2, -6], [0, 7, -10, 7], [28, 6, 0, -6], [0, -10, -6, -10], [0, -6, -16, -6], [40, 4, 0, -4], [0, 24, 16, 24], [18, -12, 0, 12], [0, 18, 18, 18], [0, -4, 18, -4], [-10, 12, 0, -12], [18, -6, 0, 6], [-40, 4, 0, -4], [-28, -4, 0, 4], [-8, 0, 0, 0], [0, 8, 10, 8], [8, 1, 0, -1], [34, -8, 0, 8], [0, 15, 22, 15], [-34, 2, 0, -2], [0, -4, -26, -4], [0, -14, -16, -14], [4, 6, 0, -6], [-4, 14, 0, -14], [0, -14, -16, -14], [-28, 5, 0, -5], [-16, 4, 0, -4], [0, -3, 14, -3], [-6, 7, 0, -7], [22, 4, 0, -4], [0, -16, -2, -16], [0, -6, -24, -6], [0, 5, -8, 5], [26, 6, 0, -6], [0, -12, 10, -12], [-22, 14, 0, -14], [28, -3, 0, 3], [0, 19, 14, 19], [0, -12, -22, -12], [0, 0, -12, 0], [-36, 8, 0, -8], [4, 4, 0, -4], [-2, -4, 0, 4], [2, -4, 0, 4], [0, -10, -18, -10], [6, -6, 0, 6], [6, -8, 0, 8], [0, 9, -4, 9], [-14, -7, 0, 7], [-24, -12, 0, 12], [0, -24, -12, -24], [0, 8, -10, 8], [4, -9, 0, 9], [30, 0, 0, 0], [0, 2, 10, 2], [-22, 11, 0, -11], [10, 11, 0, -11], [-26, 8, 0, -8], [-14, 1, 0, -1], [4, 19, 0, -19], [0, -20, -4, -20], [0, 0, -30, 0], [0, 12, 16, 12], [0, 18, 20, 18], [-44, 10, 0, -10], [-26, -5, 0, 5], [0, -6, -18, -6], [0, 14, 34, 14], [0, -20, -24, -20], [0, 6, -6, 6], [-14, 8, 0, -8], [-26, 10, 0, -10], [28, 6, 0, -6], [16, 1, 0, -1], [8, -10, 0, 10], [0, 0, 24, 0], [12, -4, 0, 4], [2, -4, 0, 4], [0, 20, 26, 20], [0, 16, 0, 16], [22, 8, 0, -8], [0, 0, 26, 0], [0, 19, 2, 19], [36, 2, 0, -2], [-40, -1, 0, 1], [38, -12, 0, 12], [-6, 21, 0, -21], [0, -31, -8, -31], [0, -15, -16, -15], [0, 8, 14, 8], [-10, -6, 0, 6], [0, 12, 6, 12], [16, 0, 0, 0], [0, -5, 18, -5], [0, 3, -16, 3], [4, -5, 0, 5], [0, 18, 20, 18], [0, -8, 8, -8], [0, -7, 12, -7], [-16, -4, 0, 4], [0, 14, -14, 14], [0, -6, -14, -6], [0, -9, 4, -9], [46, 8, 0, -8], [-30, -3, 0, 3], [8, -10, 0, 10], [0, 24, 26, 24], [-28, 14, 0, -14], [-44, -9, 0, 9], [0, -28, -14, -28], [-58, 0, 0, 0], [18, 12, 0, -12], [0, 20, 10, 20], [0, 12, 24, 12], [-2, 4, 0, -4], [0, -1, -8, -1], [6, -20, 0, 20], [0, -12, 14, -12], [-2, -11, 0, 11], [0, 15, 22, 15], [4, -6, 0, 6], [0, 14, 18, 14], [0, -6, 24, -6], [0, 3, 34, 3], [-38, 4, 0, -4], [0, 6, 18, 6], [0, 1, 18, 1], [10, -8, 0, 8], [16, 6, 0, -6], [0, 34, 4, 34], [0, -30, -8, -30], [16, -6, 0, 6], [0, -9, -34, -9], [24, 8, 0, -8], [0, -36, -12, -36], [0, -9, 4, -9], [0, 38, 26, 38], [14, -12, 0, 12], [34, -7, 0, 7], [68, 2, 0, -2], [-4, 11, 0, -11], [-32, 3, 0, -3], [36, -12, 0, 12], [0, -24, 4, -24], [0, 21, -4, 21], [0, -25, -4, -25], [-46, 12, 0, -12], [0, -2, 4, -2], [0, 12, 2, 12], [34, 4, 0, -4], [0, -5, -42, -5], [24, -9, 0, 9], [0, -14, 0, -14], [-20, -2, 0, 2], [6, 16, 0, -16], [-2, 16, 0, -16], [0, 18, -18, 18], [-2, -5, 0, 5], [52, -8, 0, 8], [0, -3, 4, -3], [0, 3, -26, 3], [0, 4, 32, 4], [-16, 7, 0, -7]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1680_f_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1680_2_f_g();". // 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_1680_2_f_g(:prec:=4) chi := MakeCharacter_1680_f(); 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_1680_2_f_g();". // 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_1680_2_f_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1680_f(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![4, 0, 7, 0, 1]>,<17,R![-6, -3, 1]>,<41,R![6, 1]>],Snew); return Vf; end function;