// Make newform 1925.2.a.r in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1925_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_1925_2_a_r();". // 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_a_r();". // 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, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0], [-1, 2]]; Rf_basisdens := [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_1925_a();" function MakeCharacter_1925_a() N := 1925; order := 1; char_gens := [1002, 276, 1751]; 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_1925_a_Hecke(Kf) return MakeCharacter_1925_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], [-1, 1], [0, 0], [-1, 0], [-1, 0], [-1, -1], [1, 1], [2, 2], [2, -2], [4, 2], [-5, 1], [4, 2], [-9, -1], [-8, 0], [-5, 1], [-4, 2], [1, -1], [-5, -1], [-10, -2], [-6, -2], [3, -1], [0, -4], [-2, 6], [2, 0], [-4, -6], [-7, 1], [-1, -3], [4, 0], [0, -2], [-2, 0], [-12, 4], [4, 8], [-12, -2], [-6, 2], [14, 0], [0, 4], [-2, -4], [-10, 6], [4, -4], [15, -1], [0, -4], [-12, -6], [-12, 4], [6, -8], [2, 0], [9, -5], [4, -8], [-1, 5], [12, -8], [0, -2], [4, -6], [8, -8], [7, -9], [11, -3], [6, 0], [0, 0], [0, 6], [-6, -2], [2, 8], [-8, 2], [-12, -8], [5, 9], [0, 4], [-15, 3], [-6, -4], [-14, 0], [4, -8], [16, -2], [-12, -4], [-5, -1], [-2, -8], [-16, -4], [3, 9], [-6, 0], [6, 14], [-11, 7], [-2, 8], [18, 8], [14, 4], [-11, 9], [9, 7], [0, 10], [12, 0], [4, 2], [6, 2], [16, 4], [-24, -2], [2, 12], [-1, -5], [10, -2], [5, -13], [18, -2], [14, 10], [0, 0], [6, -2], [-10, -6], [36, 2], [8, 10], [-44, 0], [-6, 12], [-28, 0], [30, 4], [-26, 6], [12, -2], [-24, 4], [24, -2], [-7, -9], [39, 3], [-10, -6], [-17, -9], [-4, -4], [16, -14], [-20, 6], [-9, 9], [-24, 0], [-20, 2], [9, -9], [-7, -13], [-34, 4], [0, -8], [-14, -12], [8, 6], [3, 3], [-12, 8], [-15, 15], [-20, 2], [-6, -4], [-9, -19], [13, -17], [-11, 1], [16, 4], [16, -8], [34, -2], [2, -8], [-7, -11], [25, -3], [16, 10], [30, 6], [6, 4], [30, -4], [-8, -12], [18, 12], [32, -8], [4, 4], [10, 12], [15, 13], [15, 7], [-13, -1], [7, -15], [12, -4], [-28, 6], [16, -6], [0, 4], [-18, 10], [-18, -2], [38, 2], [24, 8], [-12, -18], [-5, 7], [-17, 11], [32, -12], [-14, -4], [-4, 8], [-9, 9], [-14, -4], [-33, 5], [14, 18], [-17, 7], [-30, -8], [-20, -2], [4, 16], [-10, 8], [0, 4], [52, 2], [-3, -1], [36, 6], [-4, -4], [12, -18], [2, 6], [-15, 1], [28, -4], [2, -14], [-30, -8], [13, -3], [-4, 4], [-16, -2], [-14, 16], [-39, 7], [-8, -6], [-12, -4], [-40, 6], [4, 0], [42, 2], [18, 12], [32, -8], [-34, 0], [12, -2], [36, 10], [7, -13], [30, -14], [38, 0], [-32, 16], [-24, 18], [25, 13], [-41, 1], [2, 16], [1, -13], [16, -12], [10, 16], [1, -17], [32, -2], [-4, -6], [-24, 0], [-1, 1], [2, -18], [18, -8], [4, -8], [-19, -7], [22, 10], [-10, -4], [12, 10], [-46, -10], [36, 10], [-24, -16], [-32, -4], [36, 10], [16, -10], [8, 16], [22, -14], [-28, 4], [12, -6], [48, -12], [-44, 4], [-26, -12], [1, 7], [-10, 10], [-4, 14], [14, 12], [-2, -10], [-17, -19], [10, 2], [-18, -2], [-31, -5], [32, 2], [21, -19], [36, -4], [1, 3], [21, 11], [-12, -8], [-52, -4], [18, 16], [-3, 9], [-42, -6], [-10, 12], [37, 1], [56, -4], [-4, -26], [-46, -6], [30, -4], [-15, -7], [-4, 16], [-32, 12], [-21, 23], [-69, 3], [-22, 12], [-1, -7], [-12, -26], [24, 18], [-32, 8], [36, -10], [2, 20], [18, 26], [-8, 6], [0, -16], [-45, -9], [7, 5], [12, -8], [26, 4], [-10, 12], [-46, -6], [-8, 8], [38, -14], [-18, -14], [-11, 5], [26, 18], [46, -14], [-20, 22], [38, -4], [-59, -1], [-13, -5], [-24, -6], [1, 23], [8, 30], [-10, 6], [-34, -4], [31, -21], [-1, 13], [-12, 14], [6, -18], [-14, -14], [-25, 27], [28, 22], [-52, 16], [2, 26], [18, 2], [-60, -2], [-40, -12], [6, 24], [-4, 4], [6, -20], [-52, 12], [-26, -16], [-12, -10], [8, -28], [4, 0], [-51, -15], [35, 5], [4, -28], [-32, 6], [-16, -6], [20, 8], [10, -4], [-73, 3], [12, -12], [-2, 16], [10, -16], [-18, 30], [-31, -17], [-16, 20], [28, -18], [42, 20], [-12, -30], [-72, 8], [6, 6], [-12, 4], [-3, -21], [12, 30], [49, -15], [-22, -24], [42, -2], [-54, -16], [-12, -10], [9, 1], [-72, 0], [-34, -12], [24, 4], [20, 2], [2, -14], [-28, -4], [-4, -2], [6, -18], [-40, -6], [-58, -8], [-14, -10], [-2, 24], [-7, -15], [-71, 3], [-20, 0], [26, 20], [-12, -4], [-18, -4], [31, -25], [28, 12], [-12, 4], [-95, -1], [4, 30], [63, -1], [56, 8], [6, 28], [14, -2], [-27, 27], [-4, 28], [2, 44], [24, 32], [-16, 14], [-9, 1], [76, 0], [-71, 9], [23, 7], [-15, -7], [-10, 8], [32, 2], [60, -8], [14, -8], [-6, -38], [11, 17], [56, 0], [42, 8], [36, -8], [-61, 1], [20, 2], [-32, 2], [26, 26], [-79, 7], [-12, -20], [24, 14], [-78, 6], [-66, -12], [-42, -10], [4, -22], [-53, -1], [-42, 8], [0, 16], [-86, 4], [-39, 1], [-20, -28], [66, 0], [2, -16], [-41, 9], [13, -13], [31, -21], [40, 26], [-16, -12], [16, 0], [12, 18], [-58, 0], [-56, 8], [37, -17], [30, 24], [52, -12], [-86, 4], [21, -23], [18, -26], [-20, -16], [-18, 16], [-24, 18], [-10, -34], [20, -26], [-63, -1], [30, 18]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1925_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1925_2_a_r();". // 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_a_r(:prec:=2) chi := MakeCharacter_1925_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(2999) - 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_a_r();". // 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_a_r( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1925_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![-5, 0, 1]>,<3,R![-4, 2, 1]>,<13,R![-4, 2, 1]>],Snew); return Vf; end function;