// Make newform 6030.2.a.bt in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_6030_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_6030_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_6030_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, 1, -7, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-4, -7, 0, 1], [4, -5, -2, 1]]; Rf_basisdens := [1, 1, 2, 2]; 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_6030_a();" function MakeCharacter_6030_a() N := 6030; order := 1; char_gens := [4691, 1207, 3151]; 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_6030_a_Hecke(Kf) return MakeCharacter_6030_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], [0, 0, 0, 0], [1, 0, 0, 0], [-1, -1, 0, -1], [-2, 1, 1, 1], [-3, 0, 0, 1], [-1, 2, -1, -2], [0, 0, -2, 2], [0, -2, 2, 2], [-5, 2, 0, -1], [1, -2, 0, -1], [1, 3, -1, -2], [-5, 2, -3, 0], [-3, -2, 1, 0], [-3, -2, -2, -3], [-3, 0, -2, 3], [-5, 4, 1, -2], [-1, -1, 3, 4], [1, 0, 0, 0], [-7, -1, -1, 4], [3, -2, 3, 2], [-4, 0, 2, -2], [0, -1, 4, 2], [-4, 1, 1, 1], [0, 1, 3, -1], [1, -3, -2, -3], [-11, -2, 0, -1], [-2, -2, 1, -5], [1, -3, 0, 3], [-4, 5, 0, -4], [1, -8, 2, 1], [2, 0, 2, 4], [-8, -3, -2, -2], [-4, 6, -3, -3], [4, 4, -2, -4], [-7, 3, -5, 0], [-2, -4, 1, -3], [-2, 3, -1, -1], [-2, 0, -4, -6], [3, 1, 7, 4], [0, -6, -1, 3], [1, -8, 4, 5], [1, -6, 0, -1], [3, 2, -5, 2], [2, -2, 2, -6], [11, 1, 4, 3], [3, 4, 5, 2], [-12, -2, 2, -2], [-10, 4, -8, -2], [-3, 7, -5, -2], [-8, 3, -5, -1], [-8, 12, -2, -2], [8, 5, 4, 4], [8, 1, 2, 0], [-1, -6, -5, 2], [-9, -2, 2, 7], [10, 0, 2, 2], [6, 4, 2, 4], [3, -7, -2, -1], [8, -6, -4, 6], [4, -5, 4, -2], [7, -5, 4, 1], [-12, 1, -6, 0], [5, 6, 0, -5], [14, -5, 4, 6], [-22, 0, -6, -2], [-14, 7, -10, -2], [4, -6, -1, 5], [16, -4, 6, 2], [-4, -6, -2, 4], [2, 10, -7, -11], [-9, 1, -3, 0], [-2, 8, -1, -7], [2, -2, -8, -12], [20, 6, -3, -3], [-11, -1, -7, -6], [6, 6, 4, -12], [13, 9, -4, 3], [-9, 0, 3, -2], [-1, 4, 9, 0], [0, 2, -2, 6], [-9, 4, 0, 3], [-3, 1, 1, -10], [0, 6, -1, 1], [-1, 3, -4, -5], [-8, 4, -2, -6], [-4, -5, 2, 14], [2, -4, 6, 6], [16, -6, 6, 4], [6, -8, -2, 0], [-8, -3, -2, 0], [1, -3, -2, -1], [-7, 5, 0, 1], [11, -10, -5, 4], [-20, -1, -3, 3], [11, 1, 13, -4], [5, 4, 6, 7], [-7, 2, 3, 8], [20, -3, 13, 3], [-2, 4, 11, 7], [-12, -8, 4, 12], [18, -4, 4, 0], [-13, -8, 5, 2], [12, 5, -6, -6], [-11, 8, 9, -4], [-4, 0, -2, -8], [10, 4, 2, 12], [2, 7, 5, 7], [21, -8, 10, 9], [18, 1, -6, -8], [-8, -2, 8, 8], [30, 4, 1, -3], [-1, 6, -7, 0], [10, 6, -4, 10], [-5, 0, 4, 5], [-9, 4, -3, -8], [-6, 6, 1, 7], [-3, 3, 3, -12], [11, -2, 8, 3], [-1, 0, -1, 8], [-8, 4, -2, 0], [-10, 6, -5, -9], [-4, -2, 4, 10], [-22, 4, -4, -2], [9, -6, 9, -4], [19, -13, 6, 7], [-1, 4, -8, -5], [-12, -8, 2, -10], [14, -12, -1, 9], [3, 4, 0, -5], [44, -2, 3, 3], [-7, -16, 2, -7], [-2, 0, 4, -2], [1, 4, 0, -3], [-22, -3, -7, -5], [13, -6, 1, -10], [3, -5, 5, 6], [-7, 14, -5, -2], [-17, -7, 5, 6], [-15, 8, -3, -10], [-18, 3, -8, -4], [3, -8, -6, 13], [-23, 2, 2, -3], [-6, 3, 3, -1], [4, -20, 4, 6], [0, 0, -12, -12], [14, -8, 6, 2], [-4, -1, -11, -11], [35, -8, 13, 2], [-2, -6, 8, -14], [-14, 4, 8, -8], [-2, -1, -4, 6], [-14, 8, 6, 8], [-13, 10, -16, 1], [-10, -5, -8, 0], [-21, 3, 1, 12], [6, 10, -2, -12], [12, -4, 8, 2], [6, 3, -7, 11], [2, 4, -4, -8], [-22, -1, -16, 4], [-10, -10, 2, 22], [-11, 0, -4, -1], [-14, 2, 2, 8], [-3, 2, 7, 16], [-14, -4, -10, 12], [-14, 4, -6, 8], [-7, -1, 12, -5], [30, 5, 1, -5], [7, 10, 2, 1], [-3, -10, -5, 2], [19, -12, -4, 11], [18, 4, 6, -4], [-2, -1, 4, 6], [3, 8, -10, -17], [-7, 6, -1, 12], [10, -11, 2, -2], [16, 4, 0, -6], [17, 17, -6, -13], [-27, 6, -12, -3], [-3, -2, 6, 13], [4, 18, -2, -6], [12, -4, 0, -2], [-2, 6, -2, 2], [3, 9, -1, -2], [28, -4, 12, -6], [25, 0, 12, 1], [4, 20, -6, 2], [2, 14, -8, -4], [-20, -2, 6, 6], [4, -12, 2, 0], [-6, -1, 0, 20], [-8, 13, -5, -7], [23, -9, 0, 5], [27, -12, 3, 0], [4, 2, -8, -18], [2, -7, 7, -11], [-19, 16, -4, -3], [32, -10, 19, 9], [-5, 2, 12, -3], [1, -10, -8, 9], [-17, 15, -10, 5], [-19, -12, 8, 9], [6, 14, 4, 12], [-14, 10, -7, -5], [18, 8, -10, 10], [-23, 17, -19, -2], [6, 15, -5, 3], [-4, 3, -11, 1], [18, -3, 3, -1], [-10, 2, 0, -8], [39, -11, -2, 7], [-42, 12, -17, -7], [13, 2, 17, 16], [37, -2, 0, -5], [5, -8, -1, -8], [0, 14, -10, -2], [6, -1, 14, 0], [-25, 9, -5, -10], [37, -3, 6, 1], [0, 12, -4, -10], [-2, 8, -7, -5], [46, -12, -2, 6], [-37, 0, -16, -3], [12, -10, 6, -6], [-11, 4, 12, 1], [18, 8, -8, 4], [-25, 12, 8, -3], [-3, 12, 6, 1], [-6, 10, -17, -15], [26, 2, 10, 14], [6, -3, 8, 8], [27, -9, -5, -2], [13, 14, -9, 0], [-28, -13, 2, 4], [10, -20, 12, 2], [4, -11, 9, 5], [23, -11, 18, 3], [20, 2, 14, 10], [10, 4, -15, -17], [-18, 3, 3, -13], [30, 13, -3, 5], [-11, -7, -21, -16], [7, 15, 4, 5], [24, -9, 14, 2], [-27, -16, 6, -19], [9, 16, 2, -15], [-28, 3, 5, -5], [8, -12, 12, 12], [1, -5, 9, 6], [13, 3, 17, -20], [-40, 1, 7, 3], [9, 15, 13, 10], [-6, 5, -4, -6], [18, -10, 6, 6], [12, -5, -10, 4], [-29, 3, -17, -12], [-20, 3, 6, 0], [-41, 3, -5, -8], [21, -20, -7, 2], [14, -2, -14, 12], [-44, -3, 6, -4], [-31, -3, -3, -8], [-22, -2, -18, -22], [-26, 6, 8, -4], [-18, 14, 4, -10], [-15, -5, -5, -2], [-20, 24, -18, -16], [1, -14, 1, 4], [-27, -10, 4, -11], [33, -5, 18, 1], [14, -2, 8, 2], [2, -11, -16, 10], [-4, -16, 14, 2], [-31, 6, -13, 4], [-11, -21, 10, 3], [9, 10, -1, 2], [-12, 10, -18, 4], [24, -7, -10, 6], [-6, -2, -19, -9], [1, -12, 8, -17], [9, -11, -13, 4], [23, -13, 9, 8], [27, 1, 13, 6], [-3, -26, 9, 24], [-21, 7, 10, 13], [-8, 16, -2, 0], [-11, -9, 10, 9], [-25, 4, -8, 5], [-15, -4, -15, -6], [-14, -8, -14, -6], [16, -34, 14, 6], [23, -6, -1, 6], [13, -6, 13, -4], [1, 2, 18, -5], [20, 12, -4, 10], [14, -14, 12, 2], [-27, -2, -18, 15], [7, -6, -17, -12], [3, 6, -21, -20], [8, 11, -5, 11], [-9, 15, 2, -1], [-49, -1, -4, 3], [-36, -5, -5, 5], [18, 4, 16, -14], [-2, -18, 7, -5], [-18, 14, 3, -11], [10, 4, 10, -18], [-22, 0, 1, 7], [21, -20, -14, -5], [5, -17, -4, 5], [-19, 14, 8, -11], [12, -17, 21, 1], [-18, -11, 7, 3], [-35, -18, 8, -9], [-20, 6, -2, 0], [-73, 10, 1, -2], [30, -4, 8, 2], [-28, -10, 10, -4], [-6, -12, -20, -8], [-16, -6, -22, 18], [-6, 2, 18, 6], [24, 12, -10, -16], [11, -11, -17, -2], [5, 0, -1, -20], [16, -11, -5, 11], [24, -6, 26, -6], [-20, 19, -19, -17], [3, 23, 6, -7], [-7, 4, 14, -17], [30, -10, -12, 8], [-39, 11, 2, -3], [-7, 2, 22, 29], [38, -4, 8, 14], [-55, 6, -13, -2], [38, -1, 9, 1], [-4, -16, -8, 22], [-18, 4, 4, -4], [13, -16, -7, 18], [17, 6, 16, 11], [-26, 8, 2, -2], [-23, 6, -15, -10], [3, -23, -14, -17], [-17, 1, 0, 3], [-21, -5, -5, -4], [4, -10, -15, -15], [-8, -16, 6, 16], [-53, 8, -35, -14], [-32, 8, -4, 4], [56, -18, 14, 8], [14, 10, -4, -10], [32, -17, 22, 6], [-18, -22, -4, -12], [-24, 8, 4, 12], [56, 0, 4, 6], [-32, -1, -8, -4], [-3, -11, 8, 13], [-14, -23, -4, 0], [-38, -15, 1, 7], [-9, 19, -18, -11], [-7, -11, -2, 1], [26, -25, 17, 15], [68, -6, 6, 2], [11, 18, 3, -12], [4, 7, 20, -10], [-10, -13, 9, -5], [-11, -3, -18, -15], [-24, 4, 16, -4], [-20, 20, 2, -16], [20, 1, 22, 16], [-37, -20, 3, -24], [8, -2, -10, -2], [-6, 4, 15, -21], [4, -20, 4, 8], [-5, 20, -16, -13], [-28, 6, 0, -12], [30, -16, 10, -4], [27, -2, 11, -6], [38, -2, -4, 0], [-20, -2, -6, 20], [-31, 5, -31, 0], [-56, 9, -3, 5], [-44, 16, -6, -2], [-40, 11, 11, 7], [48, -15, -4, 10], [-9, 15, -20, -21], [-34, -22, 8, 30], [-71, -5, 3, -6], [-24, 26, -22, -14], [-3, -1, 15, 8], [-24, 24, -26, -24], [9, -14, -6, -17], [-3, -8, -19, 8], [24, -24, -8, 4], [-28, 10, -24, -8], [48, -2, 4, -14], [-17, 25, -3, -16], [-4, -14, -7, 15], [40, -7, 18, 16], [-6, -12, 12, -8], [-20, 0, 8, -18], [13, -18, 9, 18], [5, 5, 5, 2], [-8, -5, -13, -33], [-17, -5, -5, 24], [-34, 8, 10, 4], [-18, 4, 0, 12], [30, -18, -7, 5], [-50, 11, -29, -5], [-10, -21, 10, 30], [-44, 26, -24, -6], [-39, 35, 6, -15], [-34, -3, -9, -13], [-46, 8, -4, 6], [70, 1, 12, 10], [0, -10, -12, 2], [8, -4, 16, 0], [-21, 12, 6, 19], [5, 8, -7, -10], [-6, -12, -22, -20], [-23, -19, 19, 26], [1, 9, 1, -20], [-38, 40, -15, -17], [-20, 25, 6, 8], [18, 9, -9, 17], [-48, 8, -4, 10], [-38, -13, 9, -3], [30, -6, 29, 25], [-37, 2, 7, -8], [20, 8, -14, 6]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_6030_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_6030_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_6030_2_a_bt(:prec:=4) chi := MakeCharacter_6030_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_6030_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_6030_2_a_bt( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_6030_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![-88, -75, -9, 5, 1]>,<11,R![8, -85, 3, 9, 1]>,<13,R![4, 58, 46, 12, 1]>,<17,R![52, 98, -44, 0, 1]>,<23,R![-32, -184, -44, 6, 1]>,<29,R![-412, 34, 90, 18, 1]>],Snew); return Vf; end function;