// Make newform 4225.2.a.bl in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4225_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_4225_2_a_bl();". // 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_4225_2_a_bl();". // 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, 4, -3, -2, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-2, -1, 1, 0], [1, -3, -1, 1]]; Rf_basisdens := [1, 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_4225_a();" function MakeCharacter_4225_a() N := 4225; order := 1; char_gens := [677, 3551]; v := [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_4225_a_Hecke(Kf) return MakeCharacter_4225_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, 0], [0, 0, 0, 1], [0, 0, 0, 0], [3, 0, 1, -1], [0, -2, -1, 2], [0, 0, 0, 0], [0, 0, -2, 1], [-4, 0, -1, 0], [4, -2, 0, -1], [1, 2, 2, 0], [-2, 0, 2, 0], [-1, -2, -1, 3], [-2, 0, 1, 0], [0, 0, -2, 1], [4, -4, -2, 0], [2, 2, 0, 0], [-4, 2, -1, 0], [7, -2, -2, 2], [7, 0, -1, 1], [2, 0, 3, -6], [0, -2, 4, -2], [0, -6, 0, 2], [-4, 0, 2, 2], [0, 8, -1, -2], [1, 2, -1, -3], [1, -6, -4, 2], [12, 2, -2, 0], [8, 0, -2, 1], [-6, -2, -6, 4], [6, -4, -2, 5], [2, 2, 0, -3], [0, -2, 2, 2], [1, -4, 5, -7], [3, 4, -2, 4], [4, -4, -1, -2], [-2, -2, 0, -4], [-4, -2, -6, 6], [-1, 6, -7, 5], [3, -2, 1, -5], [4, 6, 8, -3], [7, 2, -8, 2], [10, -2, 2, -6], [5, 4, -2, -6], [13, 2, -5, 1], [-7, 2, -3, 1], [5, 10, -2, -2], [-5, 0, 0, -2], [7, -2, 3, 5], [7, -2, 3, 3], [12, 0, 4, -4], [8, -2, 6, 2], [12, -4, -2, -4], [-8, -10, -9, 8], [7, 2, 4, -4], [14, -6, 2, -1], [0, 0, 8, -3], [13, -8, 0, 2], [-10, 12, 5, -6], [10, 6, 12, -9], [8, -4, 8, 0], [2, 4, -2, 5], [11, 4, -3, 1], [6, 2, 2, -4], [4, -2, 2, -2], [-8, 0, -6, -6], [2, -4, 0, -2], [-12, 12, 3, 2], [2, -2, 8, -8], [2, 6, -6, -3], [4, -6, 9, 2], [5, 0, 9, -7], [-12, 12, -2, -8], [-4, 0, 6, 5], [8, -4, 0, 7], [-16, 0, -5, 8], [-9, 0, -1, 1], [-6, 2, -8, 2], [9, -12, -3, 7], [16, -6, -3, 0], [2, -12, 7, 2], [21, -2, 4, 2], [-12, 4, 0, 0], [-10, 2, 5, 6], [4, -10, 0, -3], [-5, 2, 0, 6], [14, -2, 12, -2], [4, 0, 13, -2], [7, 0, -7, -9], [-6, 4, -5, 0], [-6, 6, -10, -2], [-6, 12, 0, -6], [6, -6, -5, 12], [-11, 12, -1, 1], [17, -10, 2, -4], [-26, 4, 2, 4], [-2, 8, 12, -11], [22, -12, -7, -2], [-22, -4, -12, 8], [8, -16, 8, 5], [0, 6, -6, 4], [6, 6, 8, -10], [1, -8, -9, 15], [0, 10, 6, -7], [-1, 14, 2, -12], [12, -4, 6, -12], [-8, 14, 0, -6], [-17, -4, -13, 9], [-4, 6, 4, -6], [4, -14, 8, 2], [-9, -18, -6, 10], [2, -12, -2, -3], [-13, 2, -7, 7], [3, 4, 23, -5], [-4, -4, 6, -4], [12, 4, 13, -2], [-11, 12, 4, 4], [13, -2, -1, 1], [22, 4, 4, -7], [-18, -4, -12, 5], [7, -6, 4, -2], [-10, -4, -5, 8], [-16, -10, 2, 5], [4, 22, 2, -6], [-5, -2, 9, -11], [4, 4, -1, -2], [-6, -4, 0, 16], [10, 4, 5, -14], [21, -6, -4, 6], [10, -14, -12, -2], [-6, 0, -12, 6], [-28, -6, 3, -4], [-9, 18, 11, -5], [-15, -4, -10, 14], [-24, -6, -4, 11], [-14, -2, 11, 4], [26, 2, -9, 2], [-3, 20, -1, -11], [3, 20, 13, -15], [8, -8, 2, 9], [9, -12, -8, -6], [-8, -8, -10, 8], [2, 2, -1, -12], [-2, 6, -4, 7], [-6, -18, -6, 16], [23, -16, 0, -2], [32, -6, 7, -4], [4, -2, 12, 0], [-4, 12, -18, 6], [4, 14, 10, -18], [14, 2, -6, -12], [-13, 4, -1, 9], [5, 0, 12, 8], [12, -12, 10, -16], [28, -6, -2, -3], [10, 2, 16, 1], [32, 8, 6, -8], [-13, 16, -8, 12], [6, 6, 13, -10], [28, -10, -2, -2], [8, 2, -14, 4], [-1, 6, -27, 5], [-26, 8, 6, -1], [20, 4, -2, 2], [-3, -6, 20, -12], [5, 10, 13, -13], [-10, 6, -10, 18], [-5, 8, -20, 10], [6, 12, -4, -3], [-18, -12, -4, 4], [-12, 6, 18, -6], [-2, 22, 8, -4], [6, -26, 7, 4], [-25, 12, -2, -14], [-23, 0, -1, -7], [-4, -8, 0, -8], [9, -8, -12, 2], [2, 4, 9, -18], [-18, 2, 14, -8], [22, 10, 10, -15], [9, 14, 6, 0], [4, 16, 6, -12], [-12, -4, 4, -8], [-12, 4, 2, -2], [16, 2, 12, 8], [1, -6, 13, 11], [-5, 10, 16, 0], [20, -12, 14, -6], [0, 8, 2, 10], [-10, -8, -5, -6], [-8, 20, 5, 2], [-8, 14, -4, -17], [37, 2, -13, 5], [4, -10, -8, -2], [-26, 8, -5, 0], [-6, 4, -6, -5], [-14, 16, -16, -9], [-8, 14, 2, -12], [-16, 10, -12, 17], [36, -14, -4, -12], [16, -18, -10, 8], [-42, 26, 15, 6], [-15, 14, -6, -12], [-9, 8, -19, 27], [14, -14, 2, -6], [8, 8, -15, 4], [30, 2, -2, -11], [-20, -2, 4, 4], [-6, -26, -8, 17], [-36, 14, 3, 10], [-1, 26, 8, -12], [-10, 8, 18, -9], [2, -2, -20, -10], [-16, 26, -4, 5], [7, -12, 1, -11], [24, 16, 3, -8], [28, 20, 24, -24], [8, -8, -26, 0], [47, -10, -8, -2], [15, -16, -5, 11], [-32, -2, 12, -2], [-33, 12, 2, -16], [-4, -12, 18, -12], [-8, 26, 6, -24], [-37, -8, 3, 7], [14, -4, -16, -11], [-14, 20, 0, -16], [-34, -12, 12, -7], [39, -2, 6, -6], [-22, -4, -14, -1], [-20, -6, -12, 0], [-14, -8, 16, -5], [-15, 4, 28, -4], [-38, 28, 7, 2], [14, 30, -4, -2], [8, -10, 2, -8], [-46, 2, -2, 0], [4, -24, -15, 6], [13, 22, 7, -13], [41, -12, -8, 4], [7, 0, 10, -2], [25, -12, 11, -7], [25, -6, -14, 4], [-18, 14, 0, 5], [-34, -14, -29, 22], [23, -14, -9, 17], [0, 16, 24, -8], [9, -10, 13, -7], [-26, 16, -3, 14], [10, -8, 15, -28], [12, -2, -8, 15], [-9, -18, -5, 15], [2, 14, -1, 4], [-34, 10, 2, 16], [15, 16, 6, -22], [0, 2, -2, -2], [12, -20, 13, -8], [-1, -24, 6, 0], [3, 8, 19, -13], [30, -14, 0, -4], [-19, 10, -1, -5], [4, 14, 6, 0], [-34, -8, -28, 27], [28, -24, 4, 12], [2, -22, 8, 1], [-25, 18, 5, 5], [-5, 0, 24, 4], [-20, -14, 21, -6], [4, 0, -8, -12], [37, 4, -7, 3], [2, -18, 10, -13], [-6, 0, 18, 8], [34, -22, -18, -8], [19, -12, -17, 19], [9, -14, 18, 14], [0, -2, -2, 3], [3, -10, 19, 15], [11, 14, 27, -25], [-4, 4, 12, 8], [0, 14, -9, 0], [19, -14, -28, 0], [8, 4, -24, 13], [6, -4, -11, 22], [18, 0, 16, -30], [-12, 16, -27, 16], [6, -18, -14, 12], [12, -20, -16, 8], [0, -14, -18, 2], [-28, 10, 8, -2], [38, -8, 5, 12], [-21, -10, -14, 12], [-17, 2, -36, 18], [30, -4, 6, -15], [-5, 22, 1, 7], [-28, 4, -27, 10], [56, 4, 0, 3], [6, 0, -16, 4], [20, 2, 18, -14], [-16, 2, 2, 21], [5, 4, -8, -8], [-23, 12, 13, -1], [18, 6, 26, -17], [10, 12, -24, -2], [-25, 14, -10, 10], [36, -26, -6, 8], [21, -26, 4, -12], [-13, -4, 5, 19], [-2, -16, -16, 14], [14, -24, -6, 28], [34, -4, -3, 8], [0, -22, -10, 6], [24, -10, 22, -1], [-22, -26, -1, 10], [-10, 40, 6, -24], [-32, 0, -24, 17], [-13, 16, 21, -5], [-15, 34, 4, -12], [-14, 0, -9, 28], [-14, -10, 8, -4], [-17, -8, -11, -1], [-21, 10, -2, -28], [-36, -16, -34, 24], [22, -8, 4, 12], [7, 16, -28, 0], [-33, 12, 9, -1], [10, -20, 20, -6], [3, 18, 10, -10], [-16, 0, -18, 4], [29, -10, 19, -13], [-46, 2, -12, 9], [26, 2, 20, -15], [-14, -2, -35, 24], [0, -10, -10, -14], [-11, 8, -10, -4], [-11, 8, -31, 9], [-8, 4, 11, -26], [-18, 18, -22, 30], [4, -2, 3, 22], [-39, 2, -1, -1], [-10, 2, 29, -4], [-20, 22, -10, 0], [12, -22, 0, 2], [-52, 16, 2, -1], [-8, 24, -16, 8], [-7, -12, 24, -6], [23, -14, -17, -5], [-4, -4, -28, 20], [-26, 24, -12, -4], [45, 10, -15, 1], [4, 18, 11, -6], [22, 6, -12, -5], [-4, -10, 12, -4], [7, -14, 15, 17], [-6, 26, 15, 0], [28, 4, -6, -19], [-9, -10, -22, 8], [2, -22, 24, 8], [-12, 14, -5, -20], [0, 38, -15, -10], [-12, -16, 14, 10], [-14, -26, -26, 14], [17, 2, 7, 17], [13, -12, 12, 4], [26, -10, -8, -17], [12, -8, 19, -16], [-4, 22, -10, 11], [-34, 0, 12, 9], [-5, -6, 9, 1], [23, -10, -19, 17], [-14, 24, 10, -20], [-27, 40, 8, 2], [-15, 28, 22, -20], [-40, 16, -18, -14], [-10, 2, -6, 18], [39, -20, -10, -14], [-18, -14, 16, -23], [16, -4, -22, 12], [7, -2, 18, -12], [-3, -12, -9, -15], [-35, -12, 0, -6], [12, 34, 10, -11], [-32, 20, 4, 0], [-27, 6, -11, 33], [-2, -14, 34, -10], [48, -10, -8, -2], [-42, 16, -3, -20], [-33, 24, 5, -9], [20, -10, 5, -2], [0, 10, -2, 10], [36, 24, 14, -16], [-14, -10, -22, 29], [-12, 20, 3, -6], [-11, 16, -27, -9], [-6, -10, 4, 12], [-16, 16, 10, -11], [-18, -18, -17, 34], [-6, 2, 6, 17], [-2, 34, 5, 0], [22, -14, 20, 26], [28, 18, -10, -10], [38, -20, 7, -8], [30, -10, -19, -14], [18, 8, 12, -29], [-13, 6, 13, -33], [36, 2, -8, -6], [10, -22, -1, 2], [-7, 18, 4, -4], [-25, -8, 17, -19], [42, -22, 1, 4], [-2, -30, -18, 12], [10, -20, -31, 8], [-2, 18, -4, -16], [4, 2, 44, -21], [-26, 22, -24, -19], [23, 6, 0, -16], [0, 6, -12, 1], [-34, -18, -28, 14], [-38, -4, -13, 32], [6, 10, 16, -26], [31, 22, 27, -9], [14, -18, 20, -13], [51, 2, -8, 16], [-28, 10, 12, -30], [3, 0, -5, -3], [-44, 0, -28, 36], [-23, 28, -3, 3], [-23, 46, -1, -15], [34, 18, -8, 14], [-48, 16, 20, 4], [-14, 0, -31, 6], [-9, 18, -24, -6]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4225_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4225_2_a_bl();". // 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_4225_2_a_bl(:prec:=4) chi := MakeCharacter_4225_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_4225_2_a_bl();". // 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_4225_2_a_bl( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4225_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![1, 4, -3, -2, 1]>,<3,R![1, 10, -6, -2, 1]>,<7,R![-11, -22, 30, -10, 1]>,<11,R![33, 0, -30, 0, 1]>],Snew); return Vf; end function;