// Make newform 7938.2.a.bh in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_7938_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_7938_2_a_bh();". // 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_7938_2_a_bh();". // 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 := [-8, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]]; 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_7938_a();" function MakeCharacter_7938_a() N := 7938; order := 1; char_gens := [6077, 3727]; 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_7938_a_Hecke(Kf) return MakeCharacter_7938_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, 0], [-1, -1], [0, 0], [2, -1], [-2, 0], [2, -1], [-5, 0], [-5, 1], [-2, -2], [-2, 0], [2, 0], [-8, 1], [2, -3], [0, 0], [-2, -2], [0, 3], [7, -3], [8, -3], [3, -3], [-2, -3], [5, -3], [-4, -4], [-8, -2], [-2, 3], [-5, 1], [10, 0], [-4, -1], [14, 0], [-1, -1], [-7, 3], [5, -1], [14, -1], [7, -6], [10, -2], [-1, 3], [1, -3], [2, -6], [-8, 4], [6, 0], [10, -2], [-11, 3], [-9, -3], [-7, 0], [-6, 0], [10, 0], [-16, 0], [4, 0], [19, -2], [13, -3], [-7, 2], [-7, 5], [-8, -3], [9, 0], [-10, 5], [11, -1], [5, -1], [4, -6], [8, -6], [13, 1], [-17, -3], [7, 1], [13, 0], [12, -6], [10, 3], [-6, 0], [2, 6], [2, -3], [0, -3], [22, 0], [10, 1], [5, 5], [-8, 6], [-10, 0], [2, -3], [26, 2], [-16, 8], [22, 0], [-5, -2], [-8, 9], [-11, 7], [14, 6], [-20, 4], [10, 3], [-8, 0], [-30, -3], [33, 0], [5, -12], [-19, 5], [-13, -3], [4, -11], [-26, -2], [5, 9], [-22, -1], [8, 3], [12, 6], [2, 8], [8, 5], [25, 3], [14, -6], [8, 3], [16, 4], [-3, 0], [6, 3], [8, -9], [-20, 3], [-1, -4], [-8, -2], [24, 0], [-14, -3], [-2, -6], [-16, -6], [2, 5], [-5, -12], [-7, -9], [-13, 14], [4, -9], [8, -4], [-20, -2], [-14, -2], [-17, -3], [-13, 3], [6, -12], [-14, 13], [13, 3], [-18, -6], [44, 0], [4, -2], [16, 6], [31, 3], [2, -3], [20, -10], [-7, -3], [26, 6], [-22, -4], [10, 0], [23, 5], [4, 0], [25, -5], [10, -5], [-32, -3], [-24, 6], [8, -6], [12, 0], [34, -6], [8, -4], [25, 3], [-14, 16], [-44, 3], [-7, 5], [2, -18], [0, 6], [-10, -9], [8, -10], [-16, -3], [-41, 1], [-37, -3], [-24, -6], [10, -12], [-9, -3], [24, 3], [18, 3], [41, -3], [-5, 1], [30, 3], [46, -2], [8, 12], [-5, 3], [26, 0], [5, -13], [50, -1], [-11, 9], [13, 13], [-19, 18], [40, 0], [-20, -2], [-10, -3], [-20, -2], [22, 6], [-41, 3], [11, 9], [-16, 11], [-58, 0], [-38, 1], [15, 9], [-2, -8], [2, -6], [49, 3], [-37, 6], [20, -4], [-20, 3], [22, 7], [20, 3], [-39, -3], [-2, -5], [12, -9], [17, 6], [-40, 6], [-22, -7], [-8, -2], [-6, 12], [-38, 12], [-17, -15], [4, -9], [-27, -12], [-1, -1], [10, -18], [4, 1], [-26, -11], [-5, -18], [35, 6], [-14, 22], [44, -6], [-3, 3], [-36, 6], [22, -9], [29, -3], [28, -11], [8, -22], [-28, 2], [-4, -6], [10, -12], [-47, -8], [-16, -6], [16, -5], [-16, 6], [-24, 18], [37, -5], [-56, 0], [12, 0], [-16, -6], [-5, -15], [29, 9], [-30, -3], [-11, 15], [-50, 4], [-20, 6], [-10, 20], [-16, -1], [-6, -6], [56, -1], [-44, 0], [28, -12], [26, -18], [-50, 1], [0, -18], [-2, 0], [21, 12], [-16, -12], [15, 3], [20, -12], [-8, -5], [-27, 15], [-32, 6], [10, -8], [20, 5], [2, -12], [-11, -9], [-9, 3], [-44, 9], [-16, 18], [-44, 1], [10, 12], [7, 3], [-60, 6], [4, 9], [-42, 12], [8, 11], [20, 9], [0, 6], [-14, 0], [-4, -12], [16, -9], [-16, 12], [22, -3], [-8, 0], [4, 16], [-58, -6], [-58, -3], [15, 6], [48, 0], [-16, 18], [-34, 8], [-53, 3], [-23, 6], [-19, -1], [5, -3], [-52, 2], [-2, -18], [-42, -3], [-4, 2], [29, -10], [13, 10], [52, -5], [-10, 0], [57, -9], [-50, -6], [5, 17], [28, 4], [4, 0], [-56, 9], [-16, -16], [-13, 15], [-4, -4], [38, 6], [-25, 3], [32, -16], [7, 9], [52, -8], [38, 6], [6, -24], [-2, -14], [-18, 12], [2, 6], [-27, -15], [16, -21], [5, 8], [51, -15], [46, -3], [-22, 8], [4, -9], [35, -18], [42, -18], [-25, 9], [-19, 5], [4, -3], [-16, -9], [-8, 3], [-3, 21], [30, -12], [-22, 0], [12, -12], [4, -24], [19, -14], [-22, 21], [-21, 9], [38, 0], [-22, 17], [-26, -15], [-44, 12], [38, -12], [31, -5], [-2, -14], [-67, -3], [12, -18], [-42, -15], [49, -3], [-16, 21], [38, -10], [-86, 4], [13, 15], [-43, 12], [-40, -10], [4, 0], [-4, 24], [-54, 12], [-46, -4], [-73, 5], [-59, 13], [-32, -8], [-40, 0], [54, -3], [-20, 16], [-12, -3], [16, -3], [53, -18], [38, -10], [8, 0], [17, -12], [-52, 8], [-8, 3], [-23, -11], [-8, -8], [-14, 24], [14, 6], [21, -6], [-39, -9], [40, -6], [34, 7], [16, -6], [1, -5], [17, 20], [47, 9], [23, -4], [-41, 6], [56, 2], [56, 12], [-5, -27], [2, -30], [16, 10], [26, -15], [-9, -9], [34, -17], [13, 6], [12, -30], [47, -21], [-56, 6], [2, 23], [2, 3], [-42, 0], [-41, -3], [-7, -16], [-16, 0], [-88, 2], [-19, 29], [-14, 30], [26, -6], [-11, 16], [-29, 0], [12, -12], [4, 3], [-18, 12], [-36, -9], [-10, -9], [-7, -3], [-35, 7], [8, 20], [-38, 18], [-78, 3], [32, -4], [-58, 14], [-17, 27], [-47, 1], [-11, 22], [16, 18], [18, -18], [72, -3], [-3, 9], [25, -21], [46, 10], [4, -3], [-16, 17], [14, 21], [66, -12]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_7938_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_7938_2_a_bh();". // 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_7938_2_a_bh(:prec:=2) chi := MakeCharacter_7938_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(3023) - 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_7938_2_a_bh();". // 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_7938_2_a_bh( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_7938_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-6, 3, 1]>,<11,R![-6, -3, 1]>,<13,R![2, 1]>,<17,R![-6, -3, 1]>,<23,R![12, 9, 1]>],Snew); return Vf; end function;