// Make newform 3120.2.a.bh in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3120_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_3120_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_3120_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, -14, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0], [0, 1, 0], [-9, -2, 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_3120_a();" function MakeCharacter_3120_a() N := 3120; order := 1; char_gens := [1951, 2341, 2081, 2497, 2641]; v := [1, 1, 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_3120_a_Hecke(Kf) return MakeCharacter_3120_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 := [[0, 0, 0], [-1, 0, 0], [-1, 0, 0], [-2, 1, 0], [-1, 0, -1], [1, 0, 0], [0, 1, 0], [-1, -1, 1], [-2, 1, 0], [-1, -1, -1], [-4, 2, 0], [1, 2, -1], [1, 2, -1], [-4, 0, 0], [0, -2, 0], [-5, 2, 1], [0, 0, 0], [7, 0, 1], [-2, -2, -2], [1, -2, 1], [9, 1, 1], [7, 0, -1], [-2, 2, 2], [5, -4, -1], [8, 1, 0], [3, 1, -1], [0, 0, 0], [3, 0, -1], [1, 3, 1], [3, -1, -1], [2, 0, 2], [9, 1, -1], [0, -2, -2], [15, 2, -1], [-9, 2, 1], [2, 2, -2], [-8, 2, -2], [3, 4, -1], [4, 0, 0], [-4, -2, 2], [7, -1, 1], [9, 2, -1], [8, 4, 0], [4, -1, 0], [-2, -2, 0], [16, -4, 0], [12, 0, 0], [1, -5, -1], [-4, 6, 0], [-1, -1, -1], [6, 5, -2], [-3, 2, -3], [2, 4, 0], [9, -1, 3], [-1, 3, 3], [1, 3, -1], [-5, 1, -1], [0, -2, 4], [4, 2, 2], [6, 4, 0], [-4, -4, 0], [-18, 2, 0], [-7, 6, 1], [0, 0, 0], [-4, -2, -2], [0, 0, -2], [5, -5, -1], [14, -6, 0], [5, 2, -3], [-15, -1, 1], [0, -6, -2], [-2, 2, -2], [4, 0, -4], [2, 2, 0], [-3, 5, 3], [-14, 0, -2], [-21, 1, -1], [-7, 2, -1], [4, 6, 2], [-4, 2, -2], [-3, 1, 3], [-1, 1, 3], [14, -6, -2], [-2, 2, 0], [17, -4, 1], [-11, -2, -3], [-17, -4, 1], [20, 1, 4], [-17, -4, -3], [-12, 3, 2], [11, -8, -1], [-17, -2, -1], [-16, -3, 2], [-13, 1, 1], [17, 1, -1], [5, -5, 3], [-15, 4, -1], [-4, -6, -2], [4, -4, 0], [5, 1, 1], [4, 0, 0], [-6, -8, 0], [5, 2, 5], [18, 2, 4], [21, 2, -3], [26, 3, -2], [-28, 2, 0], [24, -4, 2], [-4, 4, 4], [-21, 2, -3], [-4, 0, 4], [7, 6, -3], [4, -2, 2], [17, -3, -1], [16, -4, 0], [0, -4, 6], [-15, 2, 1], [-10, 7, -4], [22, 0, 0], [-9, -3, -3], [-9, 1, 3], [6, -4, -4], [-23, -4, 3], [10, 6, -2], [-17, 11, 1], [7, 5, -5], [9, -5, 1], [14, -2, 2], [0, -6, -4], [-23, -2, -1], [-19, 1, -5], [-18, 4, 2], [-11, -4, -3], [-16, -4, 2], [0, 6, -2], [-20, -4, 2], [-22, 2, -4], [28, 0, 0], [-27, 2, 3], [18, 0, 0], [-7, -1, 3], [17, -2, 3], [16, -8, 0], [-24, -6, 4], [30, 0, 0], [29, -6, -3], [-1, -10, 5], [0, 3, 4], [33, -4, -3], [-2, -12, 2], [10, -12, -4], [28, -2, -2], [20, -4, 0], [8, 9, 2], [-6, 4, -2], [-34, -2, -6], [15, 8, -1], [-7, -8, -5], [26, 0, 0], [19, -10, -3], [2, -2, 6], [0, 9, 0], [-5, -7, -3], [21, -5, -1], [4, 4, -2], [-20, 8, 0], [-11, 8, 5], [26, -10, 0], [-18, 4, -4], [-21, 6, 1], [2, -6, 6], [-25, -5, -1], [2, 6, -2], [-26, -1, 2], [1, 8, 1], [10, 2, 4], [-5, -9, -3], [7, 2, 1], [-18, -2, -6], [0, 6, -2], [-18, 5, 0], [-1, 5, 5], [-18, -4, 0], [38, 0, 0], [18, 4, -2], [-7, 3, 1], [-8, -8, -6], [11, 10, 3], [34, 2, 4], [47, 2, -1], [-4, 0, -6], [0, -2, 4], [-5, 8, -1], [9, -2, 3], [30, 6, -2], [10, -5, -2], [-16, 2, 2], [-24, 2, -2], [4, -8, 6], [4, 3, 2], [10, -4, 0], [-13, 8, 3], [25, 8, -5], [-23, 2, 3], [-55, 0, 1], [-17, 4, 1], [-8, -2, 4], [6, 6, -2], [-33, -6, 5], [5, 0, 9], [22, 6, 0], [7, -5, -1], [6, 6, -6], [-4, 14, 0], [10, -14, 2], [-40, 0, -2], [6, -18, 2], [20, 6, -2], [30, 4, 2], [10, 0, 0], [-29, 6, 1], [46, 0, -2], [28, -6, 2], [-25, 1, 1], [1, -2, 1], [13, 10, 3], [-10, -3, -10], [-16, 14, 4], [4, 6, 8], [-53, 0, 3], [6, -14, -4], [-19, -4, -7], [-10, -6, 2], [-6, 6, -4], [6, -12, 2], [-2, -12, -6], [-22, -6, 4], [38, 4, -4], [15, 11, -7], [26, -18, -2], [18, -2, -2], [-7, 8, -3], [50, -6, -2], [27, 5, 7], [-20, -4, 6], [-16, 4, -8], [-31, -5, -9], [9, 10, -3], [-35, -5, -1], [21, -1, 3], [-32, 2, 6], [-42, 8, 0], [0, 2, 8], [27, -12, 1], [5, 6, 3], [7, 2, 3], [26, -4, -4], [-45, 0, -1], [-20, -10, 2], [16, 9, 8], [-16, -6, 4], [25, 8, 5], [13, -5, 5], [-20, 0, -2], [-6, 4, 4], [25, 6, 1], [13, 4, 11], [-35, 4, -1], [23, 10, -1], [-6, -12, 4], [12, -14, 2], [1, -8, -3], [22, -9, 2], [20, -12, 4], [18, 0, 0], [8, 9, -6], [-2, -6, 2], [-13, 5, -1], [38, 6, 8], [35, 3, -3], [20, 3, -6], [-36, 0, -4], [-3, 15, -5], [-9, -5, -9], [2, -2, -6], [22, -2, 2], [-2, -14, 0], [6, -6, 8], [-10, 12, 6], [0, 4, -2], [13, 11, -3], [-24, -12, 4], [-28, -4, 6], [26, -6, -2], [-6, 4, 4], [21, 5, 9], [28, -20, -4], [35, 4, -3], [-41, 1, -3], [3, -16, -1], [-4, 14, -2], [30, 6, -8], [11, 4, 3], [-1, -6, -1], [8, 8, -4], [16, -13, -4], [37, -12, 1], [23, 14, -3], [-10, -14, -2], [36, 8, -2], [-10, 1, 0], [-42, -8, -4], [4, -10, 6], [16, -8, 4], [34, 10, 2], [-49, 4, 5], [-19, -12, 5], [9, -14, 1], [29, -7, -3], [28, 4, 2], [30, 6, 6], [33, -7, 1], [33, -11, 5], [-40, -5, -2], [38, 8, 0], [-33, 2, -7], [-11, 5, 3], [1, -10, 1], [-4, 15, -6], [64, 2, 6], [-19, 9, -7], [7, -4, 1], [-48, -8, 0], [22, -2, 2], [-63, 11, 3], [-36, 6, 0], [5, 13, 1], [10, 2, -8], [-16, -2, -6], [-42, 4, 6], [-3, -16, 7], [20, -1, 0], [3, -18, 5], [4, 8, 4], [20, -6, 6], [-41, -3, -3], [-13, 16, 5], [-31, -2, -7], [-10, 10, -2], [-13, -12, -3], [5, -19, -5], [24, -21, -4], [41, -10, 3], [-20, -2, -8], [55, 10, -3], [-39, -3, -7], [17, 14, -7], [-24, 20, 0], [37, -3, -3], [16, -6, 8], [-18, 4, 8], [-68, 0, 2], [49, -9, -1], [-22, 2, -2], [46, -4, 2], [26, -10, -4], [2, 2, 4], [-20, -9, 2], [31, 4, 5], [33, 1, 7], [18, 2, -6], [-8, 0, -4], [-5, 15, -5], [-15, 12, 9], [-24, 20, 2], [47, -2, 3], [-46, 0, 2], [10, -21, -2], [-46, 6, -4], [28, 6, -2], [-40, 10, 2], [-8, -2, 6], [1, 3, -1], [-6, -8, -4], [3, -7, -7], [62, 6, -2], [30, -8, 6], [16, -8, -6], [-70, -2, -6], [-40, -9, -2], [-12, -4, 2], [6, -16, 4], [-37, 0, -5], [24, 0, 12], [13, 10, 5], [-32, -6, -6], [-28, -6, -16], [-30, -4, 0], [9, 12, 5], [2, 0, -8], [-23, -7, -3], [-16, 11, 4], [22, 7, -4], [16, -4, 2], [-17, -12, 5], [-15, 4, -7], [28, -6, 2], [6, -8, -8], [-19, 8, 9], [-13, 10, -1], [0, 2, 2], [25, -6, -1], [1, -6, 9], [-41, -18, 7], [-42, -4, -12], [15, -1, 7], [11, -6, 11], [-18, -12, -2], [-16, -4, 10], [46, -9, -4], [23, -17, -9], [-17, 0, 9], [-6, -18, 10], [-11, 23, -1], [25, -15, -7], [-24, -16, -2], [-37, 4, -9], [-29, -12, -3], [-23, -11, -1], [6, 0, -2]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3120_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3120_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_3120_2_a_bh(:prec:=3) chi := MakeCharacter_3120_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_3120_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_3120_2_a_bh( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3120_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![-32, -6, 5, 1]>,<11,R![-128, -40, 3, 1]>,<17,R![-8, -14, -1, 1]>,<19,R![-176, -52, 4, 1]>,<31,R![-256, -24, 10, 1]>],Snew); return Vf; end function;