// Make newform 800.2.n.j in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_800_n();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_800_n_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_800_2_n_j();". // 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_800_2_n_j();". // 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]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]]; 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_800_n();" function MakeCharacter_800_n() N := 800; order := 4; char_gens := [351, 101, 577]; v := [2, 4, 3]; // 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_800_n_Hecke();" function MakeCharacter_800_n_Hecke(Kf) N := 800; order := 4; char_gens := [351, 101, 577]; char_values := [[-1, 0], [1, 0], [0, 1]]; 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], [2, 2], [0, 0], [2, -2], [0, 0], [1, -1], [5, 5], [-4, 0], [2, 2], [0, -4], [0, 4], [-1, -1], [0, 0], [-6, -6], [-2, 2], [7, -7], [-4, 0], [-4, 0], [-10, 10], [0, -12], [3, -3], [-16, 0], [-2, -2], [0, 0], [3, 3], [-6, 0], [-6, -6], [6, -6], [0, 10], [9, -9], [10, -10], [0, -8], [-1, -1], [12, 0], [0, -18], [0, -12], [9, 9], [2, 2], [-2, 2], [-13, 13], [-12, 0], [-10, 0], [0, 20], [5, -5], [-5, -5], [-24, 0], [0, -16], [-10, -10], [10, -10], [0, -20], [5, -5], [-8, 0], [-16, 0], [0, 24], [-7, -7], [6, 6], [0, -10], [0, -20], [9, 9], [8, 0], [6, 6], [5, -5], [-10, 10], [0, 28], [-15, 15], [-11, -11], [0, 0], [23, 23], [18, -18], [0, 20], [-9, 9], [16, 0], [-22, 22], [-21, 21], [-28, 0], [22, 22], [0, 18], [-13, -13], [34, 0], [0, 2], [12, 0], [20, 0], [0, -4], [19, -19], [16, 0], [22, 22], [0, -26], [-15, -15], [-14, 0], [-22, -22], [2, -2], [24, 0], [-6, 6], [0, 16], [-4, 0], [10, 10], [0, 36], [6, 0], [14, 14], [-30, 0], [-6, 6], [15, 15], [-6, -6], [0, -2], [0, 16], [-15, -15], [-14, 14], [-1, 1], [8, 0], [-8, 0], [-18, 18], [9, -9], [29, 29], [28, 0], [0, -4], [-48, 0], [-10, -10], [10, -10], [-1, 1], [-20, 0], [12, 0], [-5, 5], [3, 3], [-22, -22], [0, 32], [20, 0], [0, -12], [-8, 0], [18, -18], [-21, 21], [-44, 0], [30, 30], [0, -44], [1, 1], [-22, 0], [0, 40], [1, -1], [30, -30], [29, 29], [0, 24], [0, 24], [-28, 0], [-30, -30], [2, -2], [0, 6], [24, 0], [-15, 15], [7, 7], [28, 0], [34, 34], [-15, -15], [-40, 0], [10, 10], [10, -10], [6, -6], [0, 4], [24, 0], [0, -6], [-27, -27], [46, 0], [-10, 10], [9, -9], [-10, 10], [0, -8], [-21, -21], [-34, -34], [0, -4], [21, 21]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_800_n_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_800_2_n_j();". // 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_800_2_n_j(:prec:=2) chi := MakeCharacter_800_n(); 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_800_2_n_j();". // 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_800_2_n_j( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_800_n(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![8, -4, 1]>,<7,R![8, -4, 1]>,<11,R![0, 1]>,<13,R![2, -2, 1]>],Snew); return Vf; end function;