// Make newform 6762.2.a.ci in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_6762_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_6762_2_a_ci();". // 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_6762_2_a_ci();". // 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, 2, -3, -2, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [1, -2, -2, 1], [-3, -4, 2, 0], [-2, 4, 2, -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_6762_a();" function MakeCharacter_6762_a() N := 6762; order := 1; char_gens := [2255, 3727, 3823]; 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_6762_a_Hecke(Kf) return MakeCharacter_6762_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, 0, 0, 0], [-2, -1, 0, 0], [0, 0, 0, 0], [1, 0, 1, 0], [-3, 1, 0, 1], [1, -1, 0, 1], [-1, 1, -1, -2], [1, 0, 0, 0], [2, -2, -2, -2], [1, 1, -1, -2], [1, -1, 0, 1], [2, 1, 2, 0], [1, 1, 0, 3], [-3, -1, -1, 2], [1, 1, -2, -1], [-4, 2, 2, 2], [-4, 1, 2, 0], [5, 5, 0, -1], [-6, -1, 1, -1], [-2, 4, 1, 1], [-8, -3, -1, 1], [1, -1, -1, 0], [-5, -3, 2, -3], [-1, 7, 0, -1], [-3, -1, 0, -3], [0, 0, -2, -2], [5, -4, 1, 4], [-3, 7, 0, -1], [-12, 1, 1, 1], [-8, -3, -3, 1], [6, -4, 2, 4], [-12, 2, 0, -2], [-8, 2, 0, 0], [3, 3, 2, -1], [-2, -3, -1, 5], [-4, -4, 3, 3], [-2, -5, -1, -1], [-11, -7, 1, 0], [1, -7, -2, -1], [-2, 4, 2, -4], [-8, -3, -2, 0], [-6, -3, 1, -3], [2, 6, -4, -2], [4, 1, -3, 1], [10, 0, -4, -2], [-4, -5, 1, -1], [-3, -9, 1, 0], [1, 13, 1, -2], [0, 9, -4, -2], [-8, -8, 2, 0], [-16, 1, 1, -3], [1, -5, -4, -3], [-13, -5, -1, -2], [-4, 5, 2, 2], [-20, 8, 0, 0], [-11, 5, 2, 7], [3, 0, -2, 1], [-8, 6, 4, 6], [2, -6, 2, -2], [3, 10, 2, -1], [-8, 0, -1, -5], [2, 8, 0, 6], [-13, 6, -2, -7], [3, 5, -4, -1], [4, -15, 1, 1], [12, -4, 0, 0], [4, -1, 3, 3], [-10, 5, -3, -7], [-3, 1, 2, -5], [-2, 2, -3, 5], [-2, -5, -1, -1], [2, -8, 0, 2], [-1, -19, 0, -3], [7, 12, -1, 0], [4, -6, -2, 6], [-7, 1, -2, 5], [13, 7, 2, -1], [-8, 2, -4, 2], [-12, -9, 4, 0], [-5, -5, -1, 6], [15, 3, 2, 5], [-2, 16, -2, 0], [-3, 9, 0, 5], [11, 0, 2, -3], [6, -3, -3, -7], [-24, 6, 2, 6], [-12, 0, 2, 0], [-1, -3, -4, -1], [-4, 2, 2, 10], [-1, 0, 2, -1], [6, -2, -2, 8], [0, -10, 2, -2], [2, -11, -3, -7], [14, -5, 3, 3], [10, -6, 4, 2], [-9, 1, -4, -1], [3, -7, 2, 1], [-11, -3, -1, 0], [-4, -4, 2, -8], [8, 4, 0, 8], [-7, 3, 0, 7], [-9, -10, 8, 1], [-20, 0, -2, 4], [5, 1, -4, -5], [6, -14, 1, -7], [4, -4, 6, -2], [-4, -3, 4, -8], [-8, -4, -4, 4], [-2, 20, 3, -1], [-11, 9, 7, 10], [15, -3, 0, 5], [-18, 4, 0, -4], [27, -11, 3, 6], [2, 1, 5, -11], [-12, -4, 2, -4], [7, -3, -1, -2], [-5, -12, 4, -5], [-4, -6, 4, 6], [15, 7, 4, 1], [8, -17, 2, 8], [10, 0, -4, -8], [14, 1, 0, -4], [8, 15, 5, -1], [-20, 12, -2, -2], [13, 17, 2, 3], [11, 9, -4, 7], [11, -2, -2, 1], [2, -6, 2, 0], [-14, 1, 6, 6], [2, -13, -5, -5], [-12, -3, -1, 5], [6, 2, -4, 2], [13, -9, 8, 7], [-20, -4, -1, 1], [9, -3, 2, -9], [2, -8, -5, -11], [-5, -1, -3, 4], [10, 0, -7, 3], [-12, -12, 4, 0], [-4, 12, -4, 0], [6, -13, -3, -5], [2, -25, 1, -1], [11, -8, -5, -8], [3, 19, -8, -5], [18, -6, 0, -6], [11, 11, 0, -1], [6, 0, 1, -7], [6, -20, 0, 2], [-12, -1, 7, -5], [-6, -5, 9, 7], [27, -9, 2, 13], [8, 9, -5, 1], [15, 19, -5, 0], [7, 2, 1, 14], [2, -15, -3, 5], [-8, 9, -1, -7], [6, 7, 0, -2], [7, 15, -6, 1], [-30, 13, -4, -4], [6, 9, 1, 1], [2, 7, 5, -1], [4, 2, 2, -10], [25, -10, 0, -9], [16, -4, -2, 8], [20, 18, 6, -6], [-2, -9, 9, 7], [9, -7, -4, -3], [-36, -14, -4, 2], [-11, -5, -10, -9], [9, -24, -7, -8], [-19, 5, -2, 7], [0, 17, 7, 9], [-8, 9, 1, -15], [-7, 4, -2, -9], [-11, -3, 8, 9], [26, -14, -4, -6], [0, 16, -2, -12], [16, -8, 0, 12], [-24, 12, 3, 7], [10, 9, 7, 1], [33, -5, 3, 4], [-10, 1, -5, -7], [42, -10, 1, 5], [-18, 2, 0, 6], [-6, -12, 3, -3], [-10, -4, 4, -8], [-7, -3, -1, -12], [-10, -10, -10, -2], [-5, 11, 7, 8], [16, -24, -3, -9], [-20, -3, 7, 1], [-3, 2, -1, -2], [-31, -13, 6, 3], [15, 7, -4, 1], [5, 5, 0, 1], [12, 7, -9, -1], [-5, -29, 2, 3], [-7, 7, -8, -11], [-3, -8, -4, -3], [-11, 3, -4, 1], [15, 2, -4, 7], [-23, 1, 0, 1], [-1, -3, -8, -13], [23, 1, 5, 12], [-11, 15, -2, 3], [2, 2, 16, 10], [-12, -3, 7, 1], [-20, 8, -6, 0], [-2, -6, -2, 12], [-4, -30, -2, -2], [-3, -37, -4, 1], [14, 8, -6, -8], [13, -7, 5, -10], [-33, -11, -3, 6], [31, 23, 2, -3], [-8, 22, -6, 2], [12, 14, 7, 9], [-16, 1, -5, -7], [52, -14, 4, 10], [12, 9, 1, -7], [-4, -10, 2, 6], [-10, 20, -4, -12], [-10, 9, 1, 13], [-18, -24, -2, 4], [12, -5, -1, 7], [1, -1, -10, -9], [10, 7, 9, -1], [6, 6, -10, -4], [-14, -2, 8, 14], [10, -17, -3, -5], [7, 21, 3, 2], [30, -3, 5, 1], [-32, -14, 6, -6], [-27, -6, -10, -7], [28, -14, 8, 8], [-15, -5, 0, -15], [17, -21, 0, -7], [-24, -4, 4, -12], [-7, 11, -1, -2], [3, 9, 6, -5], [24, 16, 0, -4], [3, 9, -9, 0], [-6, -4, 8, 0], [12, 12, 5, 3], [-11, 4, 4, 13], [13, 18, 0, 1], [7, -13, 7, -6], [19, 1, 2, -9], [28, -11, 7, 17], [-21, -17, -10, 9], [45, -1, 2, -1], [-36, 4, -4, 8], [17, -23, 2, 3], [-23, 7, 0, 9], [-30, 20, 6, 8], [21, -13, 12, 19], [-17, 12, -8, -11], [11, 9, 6, 1], [-16, 7, -2, 10], [6, -8, 14, 4], [33, 21, 0, 3], [31, 9, -2, -7], [6, 6, -7, -5], [-14, -6, -5, -5], [11, 15, -7, 0], [-11, 11, -12, -17], [9, 9, 12, 9], [-31, 5, -4, -1], [-2, 18, 6, -2], [-19, 13, 2, 19], [-1, -4, 7, -8], [7, -23, 6, 13], [14, 10, 16, 10], [-14, 21, 2, 8], [-7, 2, 4, 9], [16, 6, 2, 10], [-2, -32, -4, -4], [22, 0, 4, 0], [23, -21, -9, -14], [-49, -4, -4, 7], [6, 12, 2, 8], [30, 8, 2, 0], [2, -18, -1, -11], [-14, -28, 4, -6], [-4, 17, 9, 13], [24, 24, 6, -8], [19, 5, -4, -7], [-13, -26, -4, 11], [10, 15, 2, 8], [1, -15, -2, 5], [9, -26, 0, 7], [-14, 32, 2, 0], [39, -4, -6, -1], [20, 0, -6, 8], [2, -1, 0, -12], [-30, 12, 6, 20], [-47, -15, -8, 1], [-14, -22, 2, -4], [-30, -4, 10, 8], [11, -15, 2, -7], [5, -11, -2, -15], [-12, 2, 2, 6], [-4, 27, -11, -5], [15, 33, -10, -1], [-4, 6, 2, -2], [38, -4, 10, 8], [12, 15, -2, 8], [-4, 8, -4, -12], [-10, 22, 2, 6], [-19, -8, 8, 1], [-15, -23, -6, -3], [-4, 10, 8, 2], [4, 7, -7, -1], [-18, -28, -2, 0], [-45, 11, -4, -13], [10, 8, -12, -6], [-10, -11, -5, 5], [-11, 5, 16, 17], [-58, 8, 2, 8], [27, -10, -2, 11], [2, -29, 9, 3], [-15, 7, 12, 5], [-8, -14, -10, 6], [26, -1, -11, -13], [25, 9, 2, -9], [37, -5, 10, 5], [-50, -10, -6, 4], [-16, -5, -3, -9], [-39, -27, 2, -1], [4, -12, -6, -8], [28, 38, 4, 2], [19, 12, 2, 5], [36, -18, 6, 6], [-21, 17, 2, 15], [14, -34, -2, -4], [-15, 15, 0, 1], [-33, -3, 6, 13], [-28, -23, -6, 10], [50, -2, 0, 6], [6, 30, 2, 10], [-4, 20, 10, 16], [-29, -7, 2, -7], [4, 14, -6, 6], [17, -31, 6, 5], [-14, 34, 0, -6], [-29, -27, -10, 9], [-10, 22, 0, -2], [26, 2, 4, -6], [-36, 10, -7, -11], [0, -2, -2, -10], [20, 8, 2, -8], [-56, -19, 9, -7], [-30, -12, -6, 0], [-31, -17, -8, -1], [-34, 24, 5, 5], [2, 4, 6, 20], [3, 18, -4, -7], [-8, -32, 4, -4], [8, -20, -4, 8], [21, 1, -6, -1], [-2, 37, 0, -2], [16, -6, -10, -18], [-3, -17, 2, -3], [-16, 16, -8, -8], [18, -16, 4, 16], [-3, -1, 14, 7], [44, -22, -2, 6], [44, 4, 2, -4], [46, -3, -7, 1], [-56, 4, 2, 2], [12, 22, 6, 2], [45, 23, 0, -5], [-26, -10, 6, 4], [-10, 24, -4, 4], [-14, -18, -6, -8], [6, 22, -12, 2], [19, 13, -8, 15], [15, 1, -14, -3], [-26, -25, -12, -6], [-10, -6, -13, -19], [-30, -23, 11, -3], [12, 3, 3, -1], [-36, 16, -2, 8], [1, -18, -2, 5], [-3, -2, -6, -17], [-24, -19, -7, -7], [-7, -19, -4, -11], [-29, 9, 2, -9], [-34, 12, 6, -4], [-12, 45, -3, -7], [-19, -5, 6, 19], [8, -30, 10, 14], [20, 24, 6, 6], [-18, 29, 9, 5], [14, -39, 3, 5], [54, 6, 8, 2], [-3, 27, -2, -1], [23, -8, 11, 12], [-28, -12, 10, 8], [-19, -17, -12, -3], [12, -28, -12, -4], [-8, 21, 11, 5], [23, 19, 0, 7], [-7, -31, 12, 5], [13, 3, -11, -20], [-5, -7, 4, -15], [-8, 24, 12, 8], [-21, 0, -8, -7], [34, -6, 14, 16], [-40, -24, 3, 1], [32, 12, 6, -4], [17, -5, 10, -7], [19, 6, 1, 2], [32, 15, -13, 3], [-27, -5, 16, 9], [4, 4, -24, -16], [5, -1, -11, -4], [-23, 5, -8, -7], [52, -10, 2, -2], [21, -3, -10, -1], [-34, -5, 4, -4], [0, -24, 0, 0], [40, 20, -16, 0], [-92, 3, 0, 4], [-17, 3, 6, 9], [-9, 34, -3, -14], [2, 32, -10, 4], [8, 28, -14, -2], [-45, -13, -1, 8]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_6762_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_6762_2_a_ci();". // 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_6762_2_a_ci(:prec:=4) chi := MakeCharacter_6762_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_6762_2_a_ci();". // 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_6762_2_a_ci( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_6762_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![2, 4, 1]>,<11,R![16, 48, -20, -4, 1]>,<13,R![-112, -16, 36, 12, 1]>,<17,R![-16, 48, -12, -4, 1]>],Snew); return Vf; end function;