// Make newform 825.2.a.k 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_k();". // 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_k();". // 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], [-2, 0, 1], [2, 2, -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 := [[0, 1, 0], [-1, 0, 0], [0, 0, 0], [0, -1, 1], [1, 0, 0], [0, 1, 1], [2, -3, -1], [2, 0, 2], [0, 2, -2], [-4, 2, 0], [4, -2, -2], [2, 0, 0], [-4, -2, 0], [0, -1, -3], [4, -2, -2], [2, -2, 2], [4, 2, -2], [-2, -2, 2], [0, 2, 2], [4, -2, -2], [4, 3, -1], [6, -4, -2], [-2, 3, 3], [-2, -4, 0], [-6, -2, -2], [-8, -4, 2], [4, -4, 0], [-2, -1, -1], [-2, 4, -4], [6, 0, 0], [12, -3, -1], [0, 2, 2], [-2, 4, 4], [14, -2, 0], [-4, -4, -2], [-2, 8, 2], [-2, 6, -2], [0, 0, 4], [-2, -3, 5], [6, -3, -1], [0, 8, -4], [-6, 4, 0], [-4, 0, 4], [0, 3, -5], [6, -7, -5], [4, 0, 4], [2, 4, -2], [0, 2, -2], [-14, 1, 1], [14, -4, -4], [10, -1, -3], [8, -2, -6], [10, -2, -6], [24, -2, -2], [-2, -2, 2], [10, -1, 7], [6, 4, 4], [2, -4, 2], [0, 9, 1], [-4, -2, 0], [-24, 3, 1], [6, 3, 1], [20, -1, 1], [16, -10, -6], [-2, -4, 4], [-14, -4, -4], [4, 2, 6], [-20, 3, -1], [-2, -1, -1], [14, 2, -2], [-2, 2, 6], [12, -2, -2], [8, 14, -6], [-8, 7, -5], [-20, 0, 0], [12, -12, -8], [2, 4, 0], [14, -2, -2], [2, 0, 0], [-2, 8, 4], [8, -4, 0], [18, 0, -4], [4, 0, 12], [2, 6, -2], [-18, 6, 4], [8, -8, -4], [-2, 0, -12], [0, -11, 1], [4, 4, -10], [8, 2, 6], [-12, -2, 2], [-16, 2, -2], [28, 6, -2], [24, 4, 0], [0, 8, -4], [-2, 5, -3], [-2, 16, 0], [-14, 8, 0], [-20, -3, 3], [-18, 0, 0], [-4, 13, -5], [10, -7, -1], [-18, 17, 5], [-20, -4, 2], [10, -14, 0], [2, -10, -10], [-28, 6, 2], [-26, 5, 3], [-16, 8, 0], [-10, 2, 2], [12, -3, -1], [-32, -1, -5], [6, -4, -4], [-4, -14, -2], [-12, -4, 8], [-2, -4, 8], [-8, 0, -4], [4, -12, 8], [-6, 10, -2], [-28, -4, 4], [-10, 4, -4], [8, 3, 3], [-14, -9, 1], [0, 10, -6], [-4, 8, 0], [-16, 2, -4], [10, 8, -4], [-4, -4, 8], [0, -12, 4], [28, 3, 3], [-14, -10, -4], [6, 3, -9], [-4, 12, 8], [-10, 2, 2], [-12, 0, 6], [-10, 2, -6], [-38, -2, 2], [-16, -13, 1], [-14, 20, 4], [-24, 2, 0], [-2, 6, 8], [-28, -6, 8], [-4, 20, 0], [-10, -1, -9], [2, -8, -4], [-8, 2, 6], [-16, -13, 7], [-14, 17, 3], [12, 18, -2], [4, -8, 12], [0, -9, -5], [-10, 8, 4], [20, -16, -8], [-30, 5, -7], [4, -2, 18], [12, -4, -8], [14, -22, 0], [-10, -8, 4], [12, -7, 1], [8, 12, 10], [8, 6, -2], [22, 3, -3], [-12, 5, -9], [-4, 12, -12], [6, -16, -8], [4, 8, 4], [-12, -2, -18], [16, -17, -5]]; 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_k();". // 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_k(: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_k();". // 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_k( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_825_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![1, -5, -1, 1]>,<7,R![-16, -16, 0, 1]>],Snew); return Vf; end function;