// Make newform 7938.2.a.bm in Magma, downloaded from the LMFDB on 19 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_bm();". // 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_bm();". // 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 := [-6, 0, 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, 0], [0, 2], [2, 0], [-5, -1], [1, 0], [2, -2], [-6, 0], [2, -4], [0, -4], [2, -2], [0, -4], [6, -2], [-2, 0], [-9, -1], [-8, 2], [-5, -2], [2, -2], [3, 2], [2, 0], [-12, 2], [2, 2], [-5, 5], [-14, 0], [12, 0], [-2, -6], [11, 2], [-3, 0], [-11, -1], [2, 4], [-7, -1], [6, 0], [-5, 0], [1, -3], [-10, 4], [4, 6], [-8, -2], [-6, 6], [3, -3], [-9, -2], [-13, 2], [-2, 6], [-2, -2], [8, -2], [16, 2], [-3, 1], [11, 5], [7, 0], [3, 4], [4, 2], [15, -1], [18, 4], [-21, 2], [-11, 3], [-12, -2], [-4, 6], [19, 0], [23, -1], [15, 5], [-13, 5], [16, -6], [-10, -6], [-6, 6], [-10, -6], [6, 12], [0, 8], [-16, -2], [-6, 0], [-1, -4], [4, -4], [-2, 2], [-2, -10], [2, 2], [20, -2], [-24, 6], [15, -2], [4, -4], [27, -1], [18, -2], [-12, 8], [-2, -4], [-12, -4], [10, -2], [-1, 8], [7, 10], [13, 3], [-11, -6], [10, 0], [-10, -8], [-7, 12], [-6, 4], [4, 12], [0, 10], [-12, -2], [-24, -6], [7, -3], [6, 10], [12, 8], [8, -2], [-13, 11], [30, 0], [24, -4], [-4, -8], [-1, -7], [0, -6], [24, -4], [-2, 6], [-6, 6], [0, 6], [-14, -12], [-13, 7], [23, -2], [-17, -10], [10, -12], [-36, 6], [0, 4], [10, -6], [-7, -1], [11, 8], [0, -6], [-42, 4], [-29, 9], [-22, -12], [8, -8], [0, -4], [-16, -10], [15, -1], [-6, -8], [-36, 0], [21, 8], [-6, 12], [2, 0], [-10, -18], [7, -11], [-18, -12], [9, 11], [26, 4], [2, 0], [-20, -8], [16, 12], [12, -10], [-16, -6], [-40, -2], [7, 13], [-30, 2], [10, 0], [-7, 2], [-2, -10], [0, -8], [-10, 4], [-24, -4], [-12, -6], [-3, 20], [-11, 6], [0, -14], [-26, 8], [1, 1], [-28, 10], [-26, -12], [5, -8], [-27, -11], [18, -8], [6, 16], [8, 4], [-35, 7], [-34, 4], [3, -17], [-22, -10], [-13, -3], [-27, -6], [-27, 12], [12, 20], [-20, 0], [-6, 10], [4, -2], [34, -4], [1, -21], [-33, 4], [2, 16], [-14, -2], [22, 10], [-19, 0], [24, 14], [30, -4], [31, -3], [-21, 0], [36, 6], [8, 12], [18, -12], [28, -6], [23, 7], [18, 6], [-4, -2], [-13, -10], [-8, -4], [-6, 18], [-36, -2], [-30, 10], [-6, 6], [1, -13], [0, 10], [-27, 7], [9, 5], [34, 10], [24, -8], [6, 8], [13, -5], [17, -6], [-8, 2], [-36, -8], [-35, -1], [-48, -2], [10, 0], [43, 10], [32, -4], [16, 12], [-32, 6], [52, 6], [-22, 18], [-9, 24], [-56, -4], [30, -16], [-28, -6], [-4, -8], [-1, -4], [-12, 4], [12, 6], [-36, -2], [-19, -21], [-29, -2], [-18, 8], [-29, 5], [-18, 10], [44, 4], [6, -12], [36, -6], [10, 12], [-68, 2], [-2, 8], [-20, 6], [6, 8], [14, 2], [-48, 10], [-30, 2], [53, 3], [32, -6], [15, 6], [-32, 0], [12, -20], [45, 2], [-4, 12], [-64, -2], [-20, -14], [2, -12], [-3, 9], [7, -23], [16, -22], [16, -16], [-16, -6], [-16, -10], [-55, -1], [-60, -2], [50, 0], [-30, 8], [4, 4], [40, 4], [12, -18], [12, -6], [8, 10], [36, -14], [32, -8], [2, -10], [12, -8], [-20, -24], [30, 10], [10, -20], [-15, -23], [12, 0], [-24, 4], [-18, -18], [19, -5], [-9, 17], [-17, 0], [21, 20], [40, 8], [-14, -2], [66, -2], [0, 6], [69, -5], [9, 30], [-14, 0], [14, 8], [57, 11], [34, -6], [27, -17], [-66, 0], [-6, -28], [-28, 20], [36, 18], [37, 18], [24, -2], [-34, -12], [-25, -12], [56, -4], [7, 7], [-20, 24], [-34, 6], [18, 6], [-34, 4], [2, 8], [2, -32], [-19, -6], [-20, -4], [3, -5], [-13, 14], [-26, -8], [-42, 4], [-38, -8], [-53, 8], [-24, 2], [-23, -10], [-7, 6], [-8, -12], [-40, 10], [50, 16], [31, -4], [-50, -10], [50, -10], [-20, 22], [-12, 22], [35, -11], [-14, -16], [13, 29], [-30, -2], [22, 24], [46, 14], [44, -6], [-46, -18], [3, -6], [0, -18], [27, 22], [-48, 4], [78, 0], [-23, 25], [-52, 16], [-18, 14], [48, 6], [-35, 17], [-67, 4], [0, -28], [-12, -10], [64, 10], [-10, 32], [22, -14], [-15, 25], [67, -4], [16, -8], [16, -12], [38, -22], [20, 8], [40, 0], [26, 24], [-49, 6], [48, -10], [-48, 16], [19, -22], [-32, -2], [-58, -8], [-15, 6], [-16, -2], [26, 4], [-14, 12], [-3, 15], [-53, -12], [8, -2], [-10, -14], [-32, -26], [33, 11], [5, -10], [1, 6], [33, -12], [-31, -1], [4, 4], [8, 28], [11, 11], [34, 24], [-64, 8], [-58, -12], [-33, -19], [-74, 4], [-35, 13], [-76, 0], [15, -12], [44, 16], [2, -22], [18, 32], [70, -2], [7, 11], [15, 16], [-56, -20], [-8, -20], [-21, -29], [-26, 14], [-26, -12], [-63, -14], [-11, -17], [-10, 4], [80, -4], [-26, 12], [16, 18], [-6, -8], [41, 4], [53, -13], [-80, -4], [-46, 8], [-42, -8], [-32, -2], [-26, 4], [31, -9], [1, -30], [-21, 23], [28, 8], [-8, -38], [-16, -10], [-21, 0], [-63, -15], [-34, -24], [-28, -26], [0, 30], [14, -8], [34, 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_bm();". // 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_bm(: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_bm();". // 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_bm( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_7938_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-5, -2, 1]>,<11,R![2, 1]>,<13,R![-24, 0, 1]>,<17,R![-2, 1]>,<23,R![-1, 1]>],Snew); return Vf; end function;