// Make newform 4851.2.a.bp in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4851_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_4851_2_a_bp();". // 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_4851_2_a_bp();". // 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, -6, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0], [0, 1, 0], [-4, 0, 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_4851_a();" function MakeCharacter_4851_a() N := 4851; order := 1; char_gens := [4313, 199, 442]; 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_4851_a_Hecke(Kf) return MakeCharacter_4851_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, -1, 0], [0, 0, 0], [0, 1, -1], [0, 0, 0], [-1, 0, 0], [0, -1, 1], [0, -2, 0], [-4, 1, 1], [2, 2, 0], [-4, 1, -1], [2, 0, -2], [0, 3, 1], [2, 2, 2], [2, -2, 0], [-8, 1, 1], [0, 0, 2], [-8, 1, -3], [-6, 0, 0], [4, 1, 1], [-4, 4, 0], [-8, 1, -1], [4, 4, 0], [-6, 0, 2], [6, 0, 4], [-8, 0, -2], [-4, -6, 4], [0, -2, -2], [4, -1, -1], [-4, 2, -4], [2, 4, 0], [6, -4, -2], [4, -4, -4], [0, -2, -4], [-8, -8, 4], [-12, 1, -1], [-6, -4, 2], [10, 6, -2], [-4, -7, 1], [-2, 2, 4], [-10, -2, -2], [12, 0, 0], [-8, 4, -2], [2, -4, 2], [4, 0, 6], [-2, 4, 0], [2, -2, 4], [4, 2, 2], [4, 10, -2], [-8, 2, -2], [0, 0, 2], [-10, 4, 4], [-8, -7, 1], [0, -1, -3], [0, -5, -1], [-4, 5, 3], [8, -1, -1], [-10, -4, 4], [0, 1, 1], [-10, 2, 2], [4, -7, 3], [-4, -3, 5], [10, 2, 6], [-4, -4, -4], [8, -8, 0], [4, 6, 0], [-4, -12, 2], [8, 8, -4], [4, -8, -2], [-4, 0, -8], [8, 7, 1], [-4, -5, -7], [-4, 0, -4], [4, -2, 2], [12, -2, 0], [4, 3, -5], [0, -4, 4], [-10, 10, -2], [-4, 6, 4], [-16, -4, 2], [6, -12, -4], [-4, -9, 7], [-20, -3, -1], [8, -5, 3], [-10, -6, 2], [0, 5, -3], [0, 8, -4], [-6, -14, 6], [0, 0, 2], [2, 10, -2], [0, 13, -3], [-20, 5, 5], [14, 0, -6], [24, -4, -4], [-20, -5, 3], [-4, 3, -5], [-2, -6, -4], [-18, 0, 0], [8, 1, -5], [-8, -9, 3], [2, -10, 2], [20, 0, 0], [8, -3, -1], [-2, -6, 8], [14, 4, -4], [0, -4, -8], [0, 6, -4], [-8, 3, -1], [-2, -14, -2], [-4, -12, 0], [4, -3, -1], [-4, 5, -7], [-12, 2, -4], [-22, 4, 0], [-18, -8, -2], [4, 0, 4], [-12, -6, -4], [10, 6, 8], [-4, 3, -9], [-18, 14, 2], [-4, -9, 7], [-4, 0, -10], [-20, 4, -14], [18, -12, 0], [14, -4, 10], [16, 6, -6], [10, 0, -8], [-4, -15, 3], [0, -9, -1], [18, 6, -4], [18, -12, -4], [30, 2, 0], [16, 7, -1], [-20, 9, -3], [-8, 7, -3], [14, -2, 2], [-16, -13, 1], [-36, -3, -1], [-36, -1, -1], [0, -1, -11], [4, -7, 3], [-24, -1, -5], [-12, 11, 9], [28, 5, 1], [-16, -1, 3], [6, -10, 2], [-8, -3, 5], [14, 8, -4], [-22, 12, 4], [-4, 4, 4], [10, 2, 0], [-34, -2, -2], [12, -7, 7], [0, 7, -5], [30, -4, -2], [20, 16, -8], [8, -14, 10], [-26, 6, 0], [16, 11, -11], [-10, -4, 4], [-26, 2, 2], [-4, 10, -6], [16, 1, -1], [8, 2, 2], [8, 15, -5], [-16, -8, -2], [12, 12, 0], [-4, 17, 5], [-6, 4, 12], [-6, -16, 8], [0, 9, 7], [-16, 11, -1], [6, 20, 0], [16, 0, -8], [36, 2, 0], [12, 10, 6], [34, -8, 0], [38, 6, 4], [-2, 12, -8], [4, -9, 11], [0, 17, -13], [-20, 1, -11], [10, -2, 0], [26, 4, 0], [-6, -6, 2], [-24, -12, -4], [8, 13, 3], [-26, 2, -6], [2, 16, -6], [-20, 10, 0], [2, 0, 6], [6, -8, 0], [-36, -7, 1], [4, -19, -3], [-4, -17, -7], [16, -1, -5], [28, -1, 5], [0, -22, 4], [18, 4, 0], [20, 4, 2], [30, -10, 8], [2, 4, 4], [8, -5, 3], [-28, -20, 10], [-4, 9, -5], [12, -4, 4], [6, 4, -4], [28, -4, -8], [20, 1, -7], [26, 4, 0], [6, 2, 0], [10, -4, -12], [-4, 7, -7], [-24, 3, 3], [-56, 0, -4], [16, 2, 10], [-6, 4, -16], [8, -12, -4], [4, -10, -12], [-34, 4, -10], [-16, 22, 0], [0, 10, 8], [32, 7, -1], [0, 12, -14], [-4, -21, -1], [-8, 8, -12], [-10, 0, -16], [-8, -13, 13], [40, 2, -6], [-16, -21, 3], [-32, 1, 5], [38, -4, 4], [4, 19, -5], [-20, -18, 2], [-12, 1, -5], [-30, 2, 4], [-8, 18, -6], [-16, -14, 8], [16, -7, 11], [-30, -4, 6], [-4, 5, -7], [22, -14, 0], [-20, 0, -8], [10, 8, -10], [-14, -4, 8], [14, 2, 14], [20, 16, 4], [-4, -16, -4], [-16, 21, 1], [0, 0, 12], [52, 7, 3], [-18, -8, 8], [52, 4, 2], [4, -10, 10], [-30, 4, 8], [10, 8, -12], [12, 8, 8], [4, -21, 5], [-8, -1, -21], [-12, -9, 17], [12, 5, -1], [0, 22, -10], [44, -8, 8], [-54, -8, 0], [-28, -23, 7], [-20, 23, 5], [46, -8, 6], [30, -12, -8], [-4, 13, -13], [0, -4, 8], [-22, -8, 8], [-18, -2, -14], [-12, -3, -3], [-36, 8, -6], [-10, -12, 6], [28, 21, -9], [46, 0, -6], [-28, -20, 12], [-14, 12, -24], [44, 2, 8], [-30, 6, 0], [-30, -2, 4], [16, -8, 0], [22, 4, 14], [-32, -3, 7], [16, 1, 5], [46, -10, 0], [8, 13, -1], [-10, -8, -20], [14, -12, 14], [-12, 22, 4], [-12, -2, 4], [20, 24, 0], [-8, -7, -1], [14, -26, -4], [38, -2, 6], [36, -16, -6], [-24, 2, -14], [14, -4, -12], [-22, -26, 0], [32, -7, -11], [-20, -13, 13], [32, -5, 21], [14, 26, -12], [46, 10, 0], [-36, 5, 5], [-20, 9, -9], [-6, -6, 12], [52, -4, 2], [-18, 6, -8], [-16, 16, -18], [-32, 26, 10], [-30, 14, -10], [-8, 5, 3], [4, 16, -16], [-8, 8, -8], [-2, 12, 12], [-40, -12, 8], [0, 32, 0], [-48, -4, 2], [-6, 4, 8], [-4, -3, 13], [-36, 1, -9], [42, -4, -8], [12, -4, 0], [4, -5, 5], [-18, -26, 6], [-28, 16, 0], [22, -6, 8], [-28, 1, -19], [-4, 9, -1], [34, -6, 22], [10, 14, 6], [4, 21, 1], [6, 4, -6], [14, -26, 12], [48, 9, -3], [-8, -3, -5], [-24, 4, -18], [8, -18, 4], [32, 20, 4], [18, -4, 16], [12, 6, -8], [-16, -2, 4], [-36, -1, -21], [-48, -14, -4], [12, 11, 11], [-50, 4, -16], [24, -16, 4], [-4, 4, 8], [-2, 8, -16], [0, 28, 8], [-16, 31, 1], [-6, 6, -2], [44, -11, -7], [-12, -14, 12], [16, 2, -4], [-12, -15, -11], [60, 8, -8], [12, 13, 7], [-62, -6, 8], [-30, 2, -10], [2, 22, -10], [-44, -4, 8], [36, -1, -1], [4, -4, 12], [-28, -19, 11], [-24, 6, -4], [16, 10, 18], [-62, 8, 0], [2, 16, -14], [50, -8, -6], [-8, -35, 5], [-52, 3, 9], [16, 4, 12], [0, 13, 3], [0, 35, -1], [16, -27, -11], [-36, -11, -9], [10, 20, -12], [62, 4, 4], [-64, -13, 9], [0, -12, 18], [24, 14, 14], [-8, -3, 11], [4, -8, 16], [-16, 13, 13], [0, 9, 9], [22, 12, -16], [6, -16, -2], [40, 7, 7], [24, -1, 17], [-6, 4, -16], [-30, -14, 12], [-6, 14, -16], [-48, 6, -2], [-48, 10, -4], [20, 5, 17], [24, -9, 17], [40, 0, 4], [40, -15, -5], [-56, 15, 9], [-4, 6, 0], [6, -12, 6], [-22, -8, 8], [-20, -24, 2], [16, -13, 27], [12, -27, 15], [-10, -24, 16], [4, 18, 2], [-40, 0, -12], [50, -20, 0], [6, 8, -4], [-8, -8, 20], [30, 2, -16], [30, -4, 8], [-4, 13, 7], [-20, 27, 15], [22, -4, 14], [-2, -8, -4], [6, 12, 22], [4, 6, -20], [20, 15, 5], [-22, 4, 2], [-10, 20, 2], [-2, 16, 8], [-20, 5, -1], [-34, 4, 10], [-16, 21, 3], [32, -30, -2], [24, 28, -20]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4851_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4851_2_a_bp();". // 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_4851_2_a_bp(:prec:=3) chi := MakeCharacter_4851_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_4851_2_a_bp();". // 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_4851_2_a_bp( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4851_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![1, -6, 0, 1]>,<5,R![2, -15, 0, 1]>,<13,R![-2, -15, 0, 1]>,<17,R![8, -24, 0, 1]>],Snew); return Vf; end function;