// Make newform 825.2.n.g in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_825_n();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_825_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_825_2_n_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_825_2_n_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 := [1, 3, 5, 3, 4, -3, 5, -3, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [-7, -12, -7, -4, 0, -3, 2, -1], [9, 6, 19, -16, 20, -7, 0, 1], [-1, -10, 13, -16, 12, -9, 4, -1], [-5, -8, -13, -4, 8, -17, 10, -3], [-1, 6, 1, 16, -20, 23, -12, 3], [-7, -12, -11, -28, 32, -35, 18, -5]]; Rf_basisdens := [1, 1, 8, 8, 8, 8, 8, 8]; 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_n();" function MakeCharacter_825_n() N := 825; order := 5; char_gens := [551, 727, 376]; v := [5, 5, 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_825_n_Hecke();" function MakeCharacter_825_n_Hecke(Kf) N := 825; order := 5; char_gens := [551, 727, 376]; char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 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 := [[-1, -1, -1, 1, -1, 1, 1, -1], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, -2, 0, 1, 1, 0], [0, -1, 0, -1, -1, 3, 1, 1], [-1, 1, 1, 1, -2, -1, -1, -2], [0, 0, -2, 0, 1, 2, 2, 0], [0, 1, -1, 0, 0, 0, -1, -2], [-2, 4, 4, 1, 0, -3, 0, -1], [3, -6, 0, 3, 3, 1, 1, 0], [4, -4, -5, -4, -1, 4, 5, -1], [-1, -3, 0, -4, -1, 6, 6, 0], [0, -1, 0, -4, 4, 0, 1, -4], [4, -5, -5, -3, 0, 5, 0, 3], [0, -1, 4, 2, -2, 0, 1, 0], [1, -3, 0, -1, 7, 3, 0, 7], [2, -5, 0, 4, 2, 2, 2, 0], [5, 0, 1, 0, 2, -1, -7, 5], [-3, 0, 0, 7, 0, -3, 0, -7], [-5, 0, -5, 0, -9, 5, 3, -5], [4, -5, 0, -1, 4, 0, 0, 0], [-1, -2, 2, 1, 4, 2, -2, 4], [5, 0, 3, 0, 5, -3, -1, 5], [-5, 4, 4, 3, 0, -8, 0, -3], [-7, -5, -4, 7, -5, 5, 4, -5], [0, -3, -1, 0, 12, 3, 1, 12], [2, -4, 0, 4, 2, -5, -5, 0], [0, 3, -5, 4, -4, 0, -3, 2], [-1, -1, -1, 7, 0, 1, 0, -7], [0, 5, 5, -13, 13, 0, -5, 5], [1, 0, 1, 0, -6, -1, -2, 1], [-9, 1, 1, 4, 0, -6, 0, -4], [1, 0, 0, 0, 2, 0, -11, 1], [-3, -11, 0, -8, -3, 13, 13, 0], [-6, 0, 4, 0, -7, -4, 3, -6], [0, -4, -8, 13, -13, 0, 4, -7], [0, 3, -1, 9, -9, 0, -3, -4], [5, 6, 0, -5, -4, -6, 0, -4], [-6, 10, 6, 6, 6, -10, -6, 6], [0, -4, -3, 14, -14, 0, 4, -7], [0, -5, 1, -4, 4, 0, 5, -2], [-6, 0, 5, 0, -15, -5, 1, -6], [7, -4, 0, 1, 7, -3, -3, 0], [12, 0, 8, 0, 9, -8, -12, 12], [0, -9, -9, 3, 0, 9, 0, -3], [7, 5, 5, -12, 0, 8, 0, 12], [-6, 9, 0, 6, -4, -9, 0, -4], [0, 0, -4, 11, -11, 0, 0, -7], [2, 7, 0, -6, 2, -6, -6, 0], [7, -3, 0, -7, 12, 3, 0, 12], [-2, -9, -11, 2, -13, 9, 11, -13], [0, 11, -2, 4, -4, 0, -11, -4], [-13, 0, 0, 6, 0, 0, 0, -6], [9, 4, 2, -9, 9, -4, -2, 9], [-6, -4, 0, -4, -6, 1, 1, 0], [-16, 7, 7, 7, 0, -11, 0, -7], [-1, 0, -5, 0, -6, 5, -6, -1], [-18, 9, 0, 0, -18, -4, -4, 0], [16, -8, -1, -16, 15, 8, 1, 15], [4, 0, -2, 0, -9, 2, -12, 4], [0, -6, -7, -2, 2, 0, 6, -2], [6, 10, 0, 5, 6, -13, -13, 0], [-17, 7, 7, -2, 0, -13, 0, 2], [0, -1, -6, -10, 10, 0, 1, 3], [-1, 0, 2, 0, 8, -2, 6, -1], [10, -2, 4, -10, 10, 2, -4, 10], [0, 5, 5, 8, 0, -5, 0, -8], [8, 6, 0, -4, 8, 3, 3, 0], [17, 0, -6, 0, -5, 6, 7, 17], [0, -3, -3, 3, -3, 0, 3, -9], [5, -10, -10, -9, 0, 20, 0, 9], [-2, 19, 0, 12, -2, -19, -19, 0], [-4, 5, 0, -12, -4, 7, 7, 0], [-12, 9, 9, -3, 0, -4, 0, 3], [13, 0, -5, 0, 19, 5, 0, 13], [15, -9, 6, -15, 13, 9, -6, 13], [-2, 9, 0, -5, -2, 5, 5, 0], [-8, 4, 4, -12, 0, 1, 0, 12], [-1, 0, -4, 0, -6, 4, -3, -1], [2, -4, 4, -2, 2, 4, -4, 2], [22, -7, -7, -16, 0, -5, 0, 16], [0, 9, 10, -8, 8, 0, -9, 5], [-9, -4, -8, 9, 4, 4, 8, 4], [-9, -4, 0, 10, -9, -7, -7, 0], [19, 4, 4, -13, 0, -5, 0, 13], [0, -10, -10, -1, 1, 0, 10, -4], [-15, 14, -7, 15, -9, -14, 7, -9], [-7, 0, -15, 0, -19, 15, 3, -7], [-12, 2, 2, -5, 0, 20, 0, 5], [-23, 16, 16, -3, 0, 2, 0, 3], [-16, 0, 4, 0, -21, -4, 19, -16], [12, 0, 3, 0, 6, -3, -5, 12], [0, -10, 0, 9, -9, 0, 10, 6], [-7, 3, 0, 18, -7, -3, -3, 0], [-21, -2, 0, 17, -21, 1, 1, 0], [0, -8, 12, 0, 0, 0, 8, -11], [0, -2, 25, -9, 9, 0, 2, 16], [10, -1, 0, -19, 10, -2, -2, 0], [-3, 0, -7, 0, 2, 7, 5, -3], [-8, -3, 3, 8, 9, 3, -3, 9], [0, 4, -10, 9, -9, 0, -4, -23], [14, 3, 0, 8, 14, -11, -11, 0], [5, 3, -2, -5, 17, -3, 2, 17], [0, -1, -1, 6, -6, 0, 1, -16], [18, 1, 1, -4, 0, -14, 0, 4], [12, 0, -1, 0, 18, 1, -17, 12], [-12, -13, 0, -18, -12, 15, 15, 0], [20, -10, -10, 11, 0, 6, 0, -11], [5, 0, -6, 0, 27, 6, 0, 5], [8, -23, 0, -7, 8, 12, 12, 0], [0, -19, -9, 0, 8, 19, 9, 8], [0, 10, 19, -5, 5, 0, -10, 17], [-6, 11, 11, 17, 0, -12, 0, -17], [0, -6, -8, 4, -4, 0, 6, -31], [-18, 25, 0, -4, -18, 0, 0, 0], [0, 3, 14, -7, 7, 0, -3, 14], [-17, 0, -10, 0, -21, 10, -8, -17], [-13, 16, -5, 13, -9, -16, 5, -9], [-7, 4, 0, 8, -7, 0, 0, 0], [3, -12, -12, 4, 0, 8, 0, -4], [-15, 0, 0, 34, 0, -12, 0, -34], [3, 26, 25, -3, 7, -26, -25, 7], [27, 0, 7, 0, 11, -7, -30, 27], [22, -26, -26, 5, 0, 16, 0, -5], [-24, 5, 9, 24, -9, -5, -9, -9], [0, 2, 11, -8, 8, 0, -2, -12], [-5, 0, 4, 0, -14, -4, 22, -5], [-19, 9, 0, -9, -19, 7, 7, 0], [28, 0, 0, 4, 0, -10, 0, -4], [7, -20, 0, -1, 7, 19, 19, 0], [9, -19, -2, -9, -17, 19, 2, -17], [8, 0, -7, 0, -9, 7, 5, 8], [0, -8, -13, 30, -30, 0, 8, -8], [12, -16, -10, -12, 2, 16, 10, 2], [16, 12, 17, -16, 27, -12, -17, 27], [-9, -2, -2, -4, 0, 12, 0, 4], [0, -7, 3, -26, 26, 0, 7, 9], [12, 0, 1, 0, 25, -1, -3, 12], [0, 0, 5, 0, 3, -5, 3, 0], [18, 0, -3, 0, -18, 3, 7, 18], [0, 6, 3, -18, 18, 0, -6, 27], [-9, 11, 0, 7, -9, -6, -6, 0], [35, -5, -5, -35, 23, 5, 5, 23], [-41, -4, -5, 41, -33, 4, 5, -33], [3, -6, 0, -6, 3, 7, 7, 0], [0, 20, 0, -14, 14, 0, -20, 29], [10, 0, -8, 0, 27, 8, 1, 10], [14, -3, -3, -14, 0, 13, 0, 14], [-9, -2, -2, -4, 0, -11, 0, 4], [17, 0, 2, 0, 23, -2, -9, 17], [0, 1, -21, 9, -9, 0, -1, -16], [-6, -15, -15, 27, 0, 0, 0, -27], [0, -4, -22, 17, -17, 0, 4, -6], [-14, 5, 0, -2, -14, -2, -2, 0], [-16, 0, 2, 0, 3, -2, -8, -16], [10, -10, -14, -10, 7, 10, 14, 7], [12, 0, -9, 0, -2, 9, -1, 12], [-20, 0, 5, 0, 9, -5, -19, -20], [0, 29, 29, 0, 4, -29, -29, 4], [-4, 0, -4, 0, -17, 4, 10, -4], [10, 11, 11, 1, 0, -3, 0, -1], [-21, -13, 0, -3, -21, 14, 14, 0], [-5, 10, 10, 17, 0, -3, 0, -17], [0, -24, -6, -4, 4, 0, 24, -6], [-5, -27, -8, 5, -12, 27, 8, -12], [-6, 17, 0, -6, -6, 1, 1, 0], [13, -19, -19, -12, 0, 8, 0, 12], [12, 10, 0, -10, 12, 6, 6, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_825_n_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_825_2_n_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_825_2_n_g(:prec:=8) chi := MakeCharacter_825_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_825_2_n_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_825_2_n_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_825_n(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![1, 7, 21, 21, 24, 13, 9, 4, 1]>,<13,R![121, -77, -19, 39, 74, -33, 29, -4, 1]>],Snew); return Vf; end function;