// Make newform 3850.2.a.bt in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3850_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_3850_2_a_bt();". // 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_3850_2_a_bt();". // 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, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0], [-3, 1, 1], [-3, -1, 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_3850_a();" function MakeCharacter_3850_a() N := 3850; order := 1; char_gens := [2927, 2201, 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_3850_a_Hecke(Kf) return MakeCharacter_3850_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 := [[-1, 0, 0], [-1, 0, -1], [0, 0, 0], [1, 0, 0], [1, 0, 0], [0, -1, 1], [0, 0, 2], [-3, 2, -1], [-3, 0, 1], [-1, 0, -3], [-4, 1, 1], [-1, 0, 1], [3, -2, 3], [-4, 3, -1], [0, 2, 2], [-5, 0, -3], [0, -2, 2], [-2, -2, -2], [0, 0, 4], [-10, 2, 0], [4, 0, -2], [7, -2, 1], [2, 0, -2], [4, 2, -4], [1, -2, -3], [-2, -2, 6], [2, -2, 0], [-4, -1, -5], [3, 0, 1], [2, 0, 4], [4, 2, -2], [-11, 0, -3], [-10, 0, 0], [5, -4, 1], [-7, 4, -5], [1, 0, 5], [12, 0, 2], [-4, -4, -4], [8, 0, 0], [-2, 2, -2], [-4, 1, 5], [6, 2, -6], [-8, -2, -2], [-4, -3, -1], [-4, 4, -2], [-4, 4, 0], [12, -2, -2], [-2, -2, 4], [8, 2, 6], [16, -5, 1], [-18, -2, -2], [-19, -2, -1], [1, -2, 1], [-12, -2, -10], [-5, -2, -1], [0, -4, 4], [8, 1, -1], [2, 0, -6], [-2, -6, 6], [-24, 2, 0], [-12, 2, -6], [12, 1, -1], [6, 0, 2], [-16, -1, 3], [-21, 2, -5], [-9, -4, -3], [0, -1, 7], [-6, -2, 2], [-8, 1, 1], [-8, 4, -2], [-13, -2, -9], [-1, -2, 1], [6, 6, 4], [-16, 2, -8], [-4, 8, -8], [-6, -2, 0], [0, -4, -2], [0, -2, -4], [-4, -3, 11], [7, 2, 11], [-8, -6, -2], [6, -2, 6], [1, 8, -11], [1, 2, 1], [2, 6, 0], [-16, -2, 2], [12, -3, 11], [-10, -4, -4], [-12, 2, 0], [-19, 4, -3], [-9, -6, 9], [-24, -4, 4], [7, 6, 1], [-8, 2, 6], [-20, 5, -7], [28, 0, 4], [-6, 2, -10], [0, 8, -10], [-2, -6, 4], [19, 4, 5], [0, 0, 4], [-16, 0, -10], [0, 4, -8], [-22, 0, 0], [10, -4, 2], [-9, -2, -5], [11, -6, 5], [-8, 4, -2], [6, -10, 4], [9, 6, 9], [-8, 3, -1], [24, 4, -6], [8, -2, 16], [-8, -2, -6], [12, 2, -2], [-6, 6, 6], [-11, 2, 7], [-10, 4, 2], [7, 4, 5], [-18, -2, 8], [14, 4, 12], [6, -8, 0], [2, -2, -10], [-2, -2, 8], [2, 0, -2], [-17, -4, -7], [12, -4, -6], [-8, 0, 0], [20, 6, -14], [42, 0, 0], [-12, 2, 10], [44, -4, 0], [-14, 2, 4], [-15, 0, -5], [-13, 6, -5], [-1, -2, 7], [-14, 0, 8], [-12, 0, 8], [12, 2, -4], [-16, 4, 2], [17, 4, -11], [-1, -8, 13], [-21, 2, 1], [-12, 4, 4], [8, -11, 3], [-12, 7, -1], [-10, 0, 4], [8, 4, -10], [0, -6, -2], [31, -2, -7], [0, 8, -2], [30, 2, 6], [-26, 0, -14], [-32, 8, 0], [-10, -6, -4], [-34, 2, 0], [13, 8, -7], [0, 10, -8], [-6, 10, -2], [-18, -4, 4], [-18, 0, -6], [28, 3, 5], [28, -16, 4], [32, -12, 0], [-22, 4, 0], [18, -2, 0], [0, 10, -14], [-6, -6, -6], [-4, 4, -10], [12, -10, 8], [2, 10, -16], [2, 12, -16], [11, 8, -1], [22, 0, 0], [28, -1, 7], [10, 2, -6], [4, 2, -14], [18, -4, 8], [4, 5, -19], [-18, 0, 8], [-19, 6, -9], [-11, 8, -11], [-21, -4, 9], [-22, 8, 4], [-23, -6, -9], [-8, 3, 1], [4, 10, -20], [-13, -8, 11], [-6, 12, -4], [28, 4, 0], [3, -10, 15], [20, 7, -5], [-12, 8, -8], [-2, -10, -2], [-28, -7, -3], [1, 10, -15], [20, -18, 0], [-3, -12, 3], [2, 6, 10], [32, -7, -3], [1, -4, 3], [24, -10, 6], [14, 6, -14], [1, 10, -3], [-30, 8, -2], [-46, -2, -10], [-36, 4, 8], [16, -13, -5], [10, 10, 2], [10, 8, -10], [24, 9, 7], [32, 5, 3], [-37, 10, -7], [-19, -4, -7], [-32, 0, 8], [-12, 4, -10], [-2, -2, -4], [25, 2, -3], [-31, 4, 9], [-11, 0, -9], [1, 0, 15], [-34, 2, -8], [-28, -7, 15], [21, 6, -17], [0, -4, 0], [-9, -8, -11], [-13, 6, 7], [-22, -4, -18], [12, -5, 19], [-14, -10, -4], [-27, 12, -13], [31, -18, 5], [19, 2, -15], [26, 6, -2], [-27, 0, 5], [44, 3, 3], [-32, -4, -14], [-6, -10, -2], [32, 0, 4], [4, -4, -16], [-10, 8, -6], [-5, -6, 21], [8, -8, 16], [-18, 0, 0], [-14, 12, 8], [-22, 6, -8], [38, 4, 10], [8, -2, 2], [-8, 0, -4], [-4, -12, 16], [6, 8, 12], [-29, -2, -5], [-31, 6, -13], [-34, -2, 10], [24, 7, -15], [40, -2, 10], [-8, 6, 8], [10, 0, 14], [-18, 8, -20], [42, -6, -2], [24, 2, 18], [-12, 4, -4], [22, -10, 6], [26, -12, 4], [-19, -2, 9], [-28, -2, -2], [-12, 18, -8], [8, -4, -14], [-20, -4, 20], [38, -2, 2], [-38, -4, 8], [4, -6, 26], [17, -2, 5], [-1, 0, 19], [6, -6, 18], [-28, 10, 6], [-8, 10, 6], [-33, 4, 1], [6, -10, 10], [-3, 14, -5], [-24, -8, -16], [38, 4, 10], [32, 9, -3], [6, 6, -10], [20, 0, 0], [-8, -2, -2], [12, -14, 16], [-14, -12, -4], [-12, 4, 8], [55, -6, -5], [-10, -6, 26], [-23, 10, 3], [14, -2, -18], [3, 4, 15], [-30, -10, -2], [8, -8, 18], [4, -9, 7], [-4, 0, -6], [17, 6, 15], [2, 8, -10], [-38, 2, -10], [-12, -8, 18], [31, 0, -13], [-22, -8, 14], [4, -16, 8], [14, 14, 6], [18, -8, -2], [-4, -17, 9], [-26, -12, 26], [-4, 16, -6], [68, 3, 11], [18, 10, -18], [42, 4, 12], [22, -16, 2], [-18, 10, 0], [35, -10, 11], [-14, -12, 2], [29, -14, 3], [35, -6, 3], [-28, 4, -18], [9, 14, 7], [38, -6, 18], [-22, -2, -6], [-19, -4, -11], [4, -13, 13], [-24, -10, -4], [-36, -11, 9], [-1, 2, 1], [-20, -2, 2], [-36, 0, -6], [-27, 12, 7], [-41, 0, -15], [26, -8, 10], [10, 0, 22], [18, -14, 8], [-5, 4, -25], [-12, 6, -20], [-22, 2, -18], [16, 10, -16], [-12, 13, -19], [13, 12, 3], [-2, 4, 12], [-40, 6, 0], [4, 8, 20], [5, 4, -5], [40, -2, -6], [24, -9, -11], [16, 0, -4], [26, -8, 26], [-46, 2, -14], [-9, 0, 7], [-28, -8, -18], [-24, 4, 6], [-20, 0, -20], [25, 12, 3], [28, -6, 8], [40, -1, 19], [-17, -6, 25], [-4, 7, 17], [37, -2, -1], [18, 0, 8], [-33, -2, -17], [19, -12, 11], [-32, 22, 2], [25, 2, -5], [-8, -4, 6], [-16, -15, -5], [-14, -4, 14], [0, 12, -22], [-12, 12, -4], [10, -12, -14], [6, 8, 22], [-7, 0, 7], [66, 0, 2], [-7, -4, -9], [-14, -6, -16], [19, -14, 17], [-2, 16, -24], [35, -10, 19], [14, 2, 26], [-24, -1, 5], [-56, 16, -6], [-18, -16, -2], [-2, 6, -2], [41, 10, 19], [40, 0, 8], [24, 16, -8], [-34, 6, -14], [-20, 17, 5], [22, 14, -20], [-36, -1, -11], [18, 0, 8], [8, -3, 5], [-19, -12, -7], [-4, 4, 8], [0, -3, 27], [-26, -6, -16], [52, 0, 2], [-32, 7, -1], [23, 4, -15], [74, 4, 8], [-8, -16, 2], [-12, 16, -12], [7, 10, 15], [-20, 8, -2], [58, -2, 16], [9, 12, -25], [-52, -6, -28], [-17, 8, -25], [32, 6, 2], [-38, 12, 0], [38, 2, 6], [56, 0, 4], [22, -10, -8], [10, -2, 10], [34, -6, -10], [-35, -8, 17], [-74, -4, -6], [-57, 14, -5], [-16, -9, -13], [-34, 14, -10], [28, -11, -13], [41, -12, -7], [15, 6, 5], [13, 10, -19], [6, 0, 20], [64, -2, -6], [56, -14, 0], [4, -20, 12], [-48, 6, -10]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3850_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3850_2_a_bt();". // 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_3850_2_a_bt(:prec:=3) chi := MakeCharacter_3850_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_3850_2_a_bt();". // 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_3850_2_a_bt( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3850_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-8, -6, 2, 1]>,<13,R![-16, -16, 2, 1]>,<17,R![8, -28, 2, 1]>,<19,R![-232, -46, 6, 1]>],Snew); return Vf; end function;