// Make newform 1925.2.b.n in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1925_b();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1925_b_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_1925_2_b_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_1925_2_b_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 := [2, -4, 4, 2, 2, -2, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0], [38, 2, -1, -4, 8, -1], [29, 10, -5, -20, 17, -5], [-13, 32, 30, 5, -10, 7], [27, -70, -11, -21, 19, -11], [26, -64, -37, -10, 20, -14]]; Rf_basisdens := [1, 23, 23, 23, 23, 23]; 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_1925_b();" function MakeCharacter_1925_b() N := 1925; order := 2; char_gens := [1002, 276, 1751]; v := [1, 2, 2]; // 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_1925_b_Hecke();" function MakeCharacter_1925_b_Hecke(Kf) N := 1925; order := 2; char_gens := [1002, 276, 1751]; char_values := [[-1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 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 := [[0, 0, 0, 1, 0, -1], [0, 0, 0, -1, 1, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0], [-1, 0, 0, 0, 0, 0], [0, 0, 0, 1, -1, -2], [0, 0, 0, -1, 3, 0], [-2, 2, 0, 0, 0, 0], [0, 0, 0, -3, -1, 1], [4, -2, 2, 0, 0, 0], [-4, 1, -2, 0, 0, 0], [0, 0, 0, 5, 1, -3], [0, -3, 0, 0, 0, 0], [0, 0, 0, -1, 1, -5], [0, 0, 0, 7, -1, -2], [0, 0, 0, -3, -3, 1], [-6, -1, -4, 0, 0, 0], [4, -5, 2, 0, 0, 0], [0, 0, 0, 1, -1, -5], [-10, 2, -6, 0, 0, 0], [0, 0, 0, -1, 7, 0], [-1, -5, 5, 0, 0, 0], [0, 0, 0, -2, -4, -4], [-8, 4, -4, 0, 0, 0], [0, 0, 0, 2, -6, 6], [2, 5, -2, 0, 0, 0], [0, 0, 0, -1, 1, -4], [0, 0, 0, 6, -6, 2], [0, -4, 6, 0, 0, 0], [0, 0, 0, -6, 0, 2], [0, 0, 0, 14, -2, 0], [-4, 0, -4, 0, 0, 0], [0, 0, 0, 9, -7, 5], [2, 0, -6, 0, 0, 0], [-8, 0, -10, 0, 0, 0], [7, -7, 1, 0, 0, 0], [0, 0, 0, 8, 4, -2], [0, 0, 0, 1, -5, 3], [0, 0, 0, -2, 4, 2], [0, 0, 0, 1, 9, -2], [14, 2, 6, 0, 0, 0], [4, -2, -8, 0, 0, 0], [-4, 2, 8, 0, 0, 0], [0, 0, 0, -1, -9, 7], [0, 0, 0, -7, 9, -3], [2, 1, 6, 0, 0, 0], [6, -8, 2, 0, 0, 0], [0, 0, 0, -13, 1, -4], [0, 0, 0, 12, -8, 8], [6, 14, 0, 0, 0, 0], [0, 0, 0, -3, 5, -1], [10, 12, -6, 0, 0, 0], [2, -5, -2, 0, 0, 0], [2, 7, -8, 0, 0, 0], [0, 0, 0, -10, -2, 12], [0, 0, 0, -2, -10, 0], [10, -4, 2, 0, 0, 0], [-2, -4, 2, 0, 0, 0], [0, 0, 0, 7, 7, -7], [-10, 8, -8, 0, 0, 0], [0, 0, 0, 2, -6, 6], [0, 0, 0, 11, 7, 2], [0, 0, 0, -8, -2, -8], [-2, -5, 10, 0, 0, 0], [0, 0, 0, 16, 2, -10], [0, 0, 0, -2, 0, 8], [10, -4, 2, 0, 0, 0], [0, 0, 0, -7, -3, -11], [0, 0, 0, -3, 15, -11], [-14, 3, -12, 0, 0, 0], [0, 0, 0, 8, 10, 6], [17, -7, -1, 0, 0, 0], [0, 0, 0, 13, 9, -4], [0, 0, 0, 13, 13, 9], [6, -6, -14, 0, 0, 0], [0, 0, 0, -5, -11, 10], [6, -6, 12, 0, 0, 0], [0, 0, 0, 6, 10, -4], [14, -4, 8, 0, 0, 0], [4, 7, 2, 0, 0, 0], [4, -9, 8, 0, 0, 0], [9, -9, -3, 0, 0, 0], [-19, 1, -5, 0, 0, 0], [0, 0, 0, -12, 18, -12], [4, 2, -4, 0, 0, 0], [0, 0, 0, 23, -15, 3], [-1, 1, -9, 0, 0, 0], [0, 0, 0, -3, 1, -3], [12, -7, -8, 0, 0, 0], [0, 0, 0, -5, -7, -11], [0, 0, 0, -1, -7, 4], [16, -12, 10, 0, 0, 0], [0, 0, 0, -5, 5, 3], [3, -9, -11, 0, 0, 0], [-20, -8, 4, 0, 0, 0], [0, 0, 0, 10, -16, 8], [28, -4, 2, 0, 0, 0], [4, 6, -8, 0, 0, 0], [0, 0, 0, 12, 14, 8], [0, 12, 2, 0, 0, 0], [0, 0, 0, 8, 4, 14], [0, 0, 0, 9, 1, -1], [0, 0, 0, -8, 8, -6], [12, 4, 10, 0, 0, 0], [26, 12, -2, 0, 0, 0], [0, 0, 0, 20, 8, -4], [0, 0, 0, -7, 3, 2], [0, 0, 0, -7, 7, 6], [8, -8, -4, 0, 0, 0], [6, 11, -6, 0, 0, 0], [0, 0, 0, -10, 12, -6], [0, 0, 0, -15, -15, -9], [0, 0, 0, -3, -15, -7], [-10, -1, -14, 0, 0, 0], [-8, -8, -4, 0, 0, 0], [11, 13, -9, 0, 0, 0], [0, 0, 0, -11, -11, 0], [0, 0, 0, -9, -5, 20], [0, 0, 0, 26, -8, 6], [-5, 1, -11, 0, 0, 0], [-10, 10, 10, 0, 0, 0], [0, 0, 0, -25, 3, -5], [0, 0, 0, -19, 15, -4], [0, 0, 0, 1, -21, -1], [-10, 3, -6, 0, 0, 0], [6, 10, 12, 0, 0, 0], [6, 4, -8, 0, 0, 0], [-16, -11, 8, 0, 0, 0], [0, 0, 0, 3, 15, -4], [0, 0, 0, 13, -7, 4], [-36, -4, 0, 0, 0, 0], [0, 0, 0, -26, -2, -2], [-6, 16, -18, 0, 0, 0], [0, 0, 0, 8, -22, 0], [14, -13, 20, 0, 0, 0], [-2, -17, -10, 0, 0, 0], [0, 0, 0, 6, 2, 22], [0, 0, 0, 8, 4, 2], [0, 0, 0, 8, -12, -10], [12, 14, -14, 0, 0, 0], [34, 0, 12, 0, 0, 0], [6, -2, 0, 0, 0, 0], [0, 0, 0, 3, -3, 1], [0, 0, 0, 3, -15, 7], [-2, 4, 0, 0, 0, 0], [8, 29, -12, 0, 0, 0], [0, 0, 0, 21, 7, -6], [0, 0, 0, -17, -5, 10], [8, -5, 6, 0, 0, 0], [0, 0, 0, 3, -11, 7], [0, 0, 0, 3, -1, 13], [-20, 10, 16, 0, 0, 0], [0, 0, 0, -13, 1, 21], [0, 0, 0, 22, -26, 8], [0, 0, 0, 7, -7, 19], [-26, 2, -6, 0, 0, 0], [-26, 0, 14, 0, 0, 0], [-16, -6, 0, 0, 0, 0], [0, 0, 0, 3, -11, -6], [-24, -1, -2, 0, 0, 0], [0, 0, 0, 23, -19, -1], [0, 0, 0, 37, 1, -1], [0, 0, 0, 17, -5, 17], [4, -9, 8, 0, 0, 0], [0, 0, 0, 42, -4, 6], [0, 0, 0, -19, 9, -10], [32, -10, 8, 0, 0, 0], [0, 0, 0, -43, 5, 0], [18, 6, -12, 0, 0, 0], [0, 0, 0, -38, 2, 6], [-22, -8, -10, 0, 0, 0], [-6, 22, -10, 0, 0, 0], [-33, 5, -11, 0, 0, 0], [0, 0, 0, -5, -5, -3], [20, -1, -8, 0, 0, 0], [8, 32, -14, 0, 0, 0], [-19, -15, 15, 0, 0, 0], [-19, 3, 17, 0, 0, 0], [0, 0, 0, -2, 0, -16], [24, 1, 12, 0, 0, 0], [0, 0, 0, -9, -15, -1], [8, 4, 2, 0, 0, 0], [0, 0, 0, -24, -10, -8], [0, 0, 0, -37, -7, 0], [0, 0, 0, 5, -9, -13], [-28, 10, 8, 0, 0, 0], [0, 0, 0, -31, 13, -9], [0, 0, 0, 5, 5, 20], [-6, -8, -8, 0, 0, 0], [-12, -6, 6, 0, 0, 0], [0, 0, 0, -2, -20, -2], [0, 0, 0, -28, 20, -8], [-4, 10, -20, 0, 0, 0], [-8, -8, -16, 0, 0, 0], [0, 0, 0, 3, -3, -13], [0, 0, 0, 26, 6, 16], [-32, 6, -18, 0, 0, 0], [0, 0, 0, -45, 15, -1], [0, 0, 0, -5, 13, -6], [0, 0, 0, -50, 6, -4], [38, -2, 12, 0, 0, 0], [6, 16, 20, 0, 0, 0], [0, 0, 0, 2, -12, -14], [-30, 5, -28, 0, 0, 0], [20, 19, 2, 0, 0, 0], [0, 0, 0, 18, -20, 12], [-20, -15, 18, 0, 0, 0], [0, 0, 0, -10, 2, -24], [8, -20, -2, 0, 0, 0], [22, -19, 28, 0, 0, 0], [0, 0, 0, 9, 13, 11], [26, 6, -12, 0, 0, 0], [0, 0, 0, -15, -9, 21], [0, 0, 0, -25, -9, -6], [4, 8, -10, 0, 0, 0], [-18, 10, 2, 0, 0, 0], [0, 0, 0, 15, -7, -13], [10, -23, -2, 0, 0, 0], [0, 0, 0, 17, 11, 19], [0, 0, 0, 2, -12, 2], [42, 6, 8, 0, 0, 0], [-16, -16, 18, 0, 0, 0], [13, 11, -7, 0, 0, 0], [0, 0, 0, 41, 3, -9], [0, 0, 0, 26, 16, -14], [-2, 10, 16, 0, 0, 0], [0, 0, 0, 22, -8, 14], [-20, 20, 4, 0, 0, 0], [0, 0, 0, -34, -2, 0], [-13, -11, 1, 0, 0, 0], [0, 0, 0, 3, 3, -13], [-4, 8, -16, 0, 0, 0], [-4, 12, 8, 0, 0, 0], [-4, 10, -22, 0, 0, 0], [0, 0, 0, -21, 15, -14], [0, 0, 0, -8, -30, -18], [-8, -24, -6, 0, 0, 0], [0, 0, 0, -5, 27, -33], [-22, -22, -10, 0, 0, 0], [-6, 17, -14, 0, 0, 0], [0, 0, 0, 25, 19, 7], [44, 6, 0, 0, 0, 0], [0, 0, 0, -31, -3, -4], [3, -1, 31, 0, 0, 0], [0, 0, 0, 13, 19, 8], [-30, -20, -4, 0, 0, 0], [0, 0, 0, -3, 1, -10], [44, 1, 16, 0, 0, 0], [-48, 0, -4, 0, 0, 0], [0, 0, 0, 14, -22, 24], [0, 0, 0, -15, 9, -19], [-16, -19, 18, 0, 0, 0], [0, 0, 0, -29, -15, -1], [10, -10, -22, 0, 0, 0], [0, 0, 0, 23, 19, -6], [8, 20, 4, 0, 0, 0], [-21, 19, -17, 0, 0, 0], [0, 0, 0, 12, -22, 26], [0, 0, 0, -20, 38, -14], [0, 0, 0, -13, 1, -20], [0, 0, 0, -5, -3, 15], [0, 0, 0, -20, 4, 28], [4, -33, 16, 0, 0, 0], [0, 0, 0, -15, 5, 34], [0, 0, 0, -14, 2, 8], [-18, -5, -26, 0, 0, 0], [29, 5, 9, 0, 0, 0], [4, 0, -2, 0, 0, 0], [0, 0, 0, 11, 17, 19], [0, 0, 0, 3, 23, -21], [-52, 10, 10, 0, 0, 0], [0, 0, 0, 7, -27, -3], [0, 0, 0, 4, -6, -28], [-12, 12, -36, 0, 0, 0], [0, 0, 0, -39, 1, 16], [0, 0, 0, 29, -5, -18], [0, 0, 0, 19, 25, -15], [24, 22, -10, 0, 0, 0], [38, -18, -8, 0, 0, 0], [-8, -2, 28, 0, 0, 0], [0, 0, 0, 26, -24, -10], [4, -10, 28, 0, 0, 0], [0, 0, 0, -32, -6, 6], [-36, -11, -6, 0, 0, 0], [0, 0, 0, 12, 14, 10], [-48, 12, -20, 0, 0, 0], [0, 0, 0, 1, -15, 17], [0, 0, 0, -29, 3, 27], [-6, -3, -6, 0, 0, 0], [14, -25, 32, 0, 0, 0], [-59, 1, -11, 0, 0, 0], [0, 0, 0, 5, -1, 30], [0, 0, 0, 7, -5, -7], [-2, 8, 22, 0, 0, 0], [0, 0, 0, 29, 13, -19], [32, 27, -16, 0, 0, 0], [-12, -31, 14, 0, 0, 0], [0, 0, 0, 0, 4, -36], [34, 4, 22, 0, 0, 0], [0, 0, 0, -20, 0, -2], [0, 0, 0, -3, -17, 6], [0, 0, 0, -5, -37, -17], [13, -9, 19, 0, 0, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1925_b_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1925_2_b_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_1925_2_b_n(:prec:=6) chi := MakeCharacter_1925_b(); 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_1925_2_b_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_1925_2_b_n( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1925_b(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![25, 0, 31, 0, 11, 0, 1]>,<3,R![4, 0, 12, 0, 8, 0, 1]>,<19,R![-8, -4, 6, 1]>],Snew); return Vf; end function;