// Make newform 825.2.a.n in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_825_a();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_825_2_a_n();". // 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_825_2_a_n();". // 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, -3, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0], [0, 1, 0], [-2, -1, 1]]; Rf_basisdens := [1, 1, 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_825_a();" function MakeCharacter_825_a() N := 825; order := 1; char_gens := [551, 727, 376]; v := [1, 1, 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; function MakeCharacter_825_a_Hecke(Kf) return MakeCharacter_825_a(); 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 := [[1, 0, 1], [-1, 0, 0], [0, 0, 0], [3, -1, -1], [-1, 0, 0], [1, 3, -1], [1, 1, 3], [-2, 4, 2], [4, 0, 0], [-2, -2, -2], [2, 2, 0], [2, -2, -2], [-2, -2, -2], [3, -1, -1], [2, 2, -2], [-4, 4, 2], [-4, 4, -2], [2, -4, 0], [4, 0, 4], [-4, 0, -2], [7, -3, -3], [-2, 0, 2], [-1, 1, 1], [0, 2, 0], [-4, 4, 0], [-14, 2, 2], [10, -2, 2], [3, 5, -3], [-2, 0, -8], [-10, -2, -4], [13, -3, -3], [0, -4, 0], [8, -8, -6], [2, -4, -6], [2, -2, 2], [-6, 4, -2], [-6, -2, 6], [2, -10, -6], [1, 3, -5], [5, -7, -5], [4, 0, -4], [-6, 8, -2], [8, -8, -4], [-1, 1, 5], [5, -7, 3], [4, 0, 8], [14, 4, 6], [-2, -6, 10], [-7, -5, -1], [2, 4, 12], [-9, 3, -3], [0, -4, 8], [10, -8, 4], [8, -8, 2], [-2, 6, 0], [3, -3, 1], [-18, 0, 4], [10, -8, 2], [-3, 3, 3], [10, 2, 14], [7, 7, 7], [-1, 7, -7], [-9, -9, -1], [0, 4, -6], [-2, 10, 10], [6, 6, 4], [2, -6, 8], [-5, 13, 1], [3, -15, -3], [-2, 8, -8], [4, -12, -10], [-4, -4, -12], [6, -6, -14], [-5, -3, 5], [-4, -8, -4], [10, -14, 2], [6, -12, 0], [-16, -8, 0], [-16, -6, -16], [-10, -8, 4], [20, 0, -4], [-2, -8, 14], [8, 4, 0], [-24, 0, -8], [-22, -4, 6], [-18, 6, 10], [-16, 6, 8], [-3, 7, -1], [-14, 6, -2], [6, 10, 2], [4, 0, -4], [16, 4, 4], [-4, 16, -4], [0, 8, 16], [-8, -8, -4], [21, 3, -9], [-30, 0, 4], [8, 10, 0], [9, 9, -7], [-6, -4, 4], [9, -7, -15], [-3, -3, 11], [-3, 11, 15], [10, -18, 6], [-10, 12, 2], [2, 6, 14], [8, -4, 8], [-11, 9, 7], [24, 0, -8], [2, 12, -4], [-3, -19, 5], [7, -11, -7], [-26, 14, 8], [-6, 2, 8], [-28, 8, 0], [-12, -10, -4], [22, -6, 6], [14, 6, -2], [-10, -2, 0], [0, 4, -4], [14, -4, -4], [7, -11, -3], [-5, 11, -7], [6, -10, 6], [4, -16, 0], [6, 10, -18], [-10, 12, 14], [-8, 16, -4], [-24, 12, 12], [-13, -11, 13], [-14, -8, -10], [19, -11, 5], [-12, -8, -28], [-34, 2, 6], [6, -14, -6], [30, 0, -4], [-18, -2, 8], [-5, 15, 15], [-18, 6, 0], [-2, 18, -10], [14, -16, -2], [14, -10, -6], [-18, 26, 14], [-17, -3, 1], [18, -28, -10], [-32, 4, 10], [3, -11, 21], [9, -3, 15], [-14, -6, -8], [34, 10, 2], [27, -7, -11], [6, 4, -12], [12, -16, -8], [-7, -17, -9], [-14, -10, 6], [-20, 4, 12], [-22, 4, -2], [-22, 4, 4], [21, 11, 7], [-2, 6, 10], [-10, -10, -26], [-1, -9, -11], [37, 5, 5], [-40, -4, -16], [16, -24, -18], [2, 18, 26], [6, 14, 8], [-29, 1, -11]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_825_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_825_2_a_n();". // 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_825_2_a_n(:prec:=3) chi := MakeCharacter_825_a(); 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_825_2_a_n();". // 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_825_2_a_n( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_825_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![5, -1, -3, 1]>,<7,R![-4, 16, -8, 1]>],Snew); return Vf; end function;