// Make newform 5070.2.a.bg in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_5070_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_5070_2_a_bg();". // 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_5070_2_a_bg();". // 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 := [-3, 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_5070_a();" function MakeCharacter_5070_a() N := 5070; order := 1; char_gens := [1691, 4057, 1861]; 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_5070_a_Hecke(Kf) return MakeCharacter_5070_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], [-1, 0], [1, 0], [2, 0], [3, -2], [0, 0], [4, 0], [4, -2], [-2, 1], [2, 1], [0, 1], [4, -3], [-2, 0], [-5, 4], [7, 2], [-6, -4], [-5, 2], [0, 6], [8, -2], [-2, 6], [-2, 0], [-7, 4], [2, -4], [-4, -2], [4, 2], [4, 4], [-2, -8], [6, 8], [-2, 8], [6, 3], [-2, -4], [-10, 5], [-1, -2], [-6, 4], [-17, -2], [0, -6], [-5, 0], [4, 11], [-1, -10], [-4, 4], [18, -1], [-4, -4], [18, -2], [-6, -10], [6, 6], [12, 4], [6, 10], [24, 2], [-6, 6], [-6, -8], [6, 7], [-6, 6], [-16, 1], [-14, 1], [14, 5], [-6, -7], [12, 0], [-4, -3], [-3, -4], [2, -4], [3, -4], [18, -8], [-16, 2], [18, -6], [-28, 0], [18, 2], [4, 12], [8, 10], [12, 6], [18, -2], [-2, 0], [-12, -4], [26, 6], [5, 12], [-14, 8], [15, -6], [22, -1], [0, 7], [-32, 0], [4, 0], [12, -6], [-6, -6], [-18, 6], [-2, 10], [4, -8], [6, -6], [-12, -16], [-28, -2], [-3, -2], [-14, -4], [2, -8], [2, -2], [-8, -18], [0, -10], [10, 2], [0, 18], [9, 6], [0, 10], [-9, -12], [-20, 4], [16, -4], [-10, -16], [22, 10], [-12, 10], [-14, -18], [-10, 0], [-8, -6], [-7, 22], [0, 6], [1, 12], [12, 18], [20, -11], [-19, 2], [0, -14], [-28, 2], [-14, -8], [-24, -2], [-4, -10], [-20, -14], [-2, -1], [-26, -10], [-32, 0], [22, -6], [20, -12], [-4, -18], [2, 1], [16, -4], [0, -20], [4, -16], [-44, 4], [-2, 10], [37, -2], [-21, -4], [-18, 0], [-12, -4], [-4, -9], [0, -16], [0, 1], [24, 8], [-22, 12], [20, -12], [-25, -6], [-8, 18], [0, 10], [-20, -2], [2, -24], [-4, 23], [-2, 13], [-4, 18], [39, 2], [-16, -7], [14, 0], [-23, 4], [2, -7], [7, -8], [0, -14], [-20, -20], [-20, -8], [10, 20], [6, 12], [6, -2], [14, 1], [-38, 2], [8, 2], [-23, -6], [47, 6], [-39, 8], [-6, -24], [12, 3], [-18, 4], [-12, 24], [2, 10], [-46, -6], [28, -6], [21, 0], [-38, 2], [18, 0], [-10, -20], [0, 0], [-10, -16], [-2, 2], [-6, 5], [-17, -20], [-27, 14], [39, -2], [-10, -13], [-43, 4], [-4, -13], [-28, -1], [12, -24], [8, 34], [48, 10], [-14, 26], [27, -2], [-16, 24], [12, -8], [16, -17], [-23, -24], [31, -18], [38, -1], [-14, -12], [-29, -4], [-28, 4], [22, 20], [36, 8], [34, 10], [-4, -6], [4, -30], [-22, 20], [8, -22], [-34, 6], [18, -11], [-28, -2], [-16, 4], [12, -30], [8, -15], [10, 4], [-20, 6], [8, 4], [24, 2], [4, 20], [-8, 10], [-24, -10], [20, -20], [2, -12], [6, 12], [2, -27], [12, 32], [16, 18], [27, 10], [7, 20], [38, -20], [20, 6], [6, 6], [9, 0], [24, 0], [20, 7], [-8, 30], [-12, 30], [16, 16], [56, 0], [16, -10], [6, 30], [4, -30], [-14, 24], [14, -16], [-20, -16], [8, 4], [54, -2], [-34, -17], [-24, -25], [-4, 16], [-73, -2], [37, -4], [-34, -6], [20, 12], [-4, -24], [-40, 6], [54, 0], [-10, -8], [-8, -2], [56, 6], [2, 22], [-14, -8], [29, -26], [38, -20], [-14, -28], [-12, -14], [-52, -1], [-10, 26], [8, 8], [-24, 27], [-28, -28], [53, 12], [8, -12], [52, 0], [18, 8], [48, -14], [0, -15], [38, 13], [-8, -10], [-40, -25], [6, 23], [4, 20], [-52, 3], [-4, 20], [10, 6], [26, -14], [4, 37], [-4, -16], [42, -19], [10, 4], [-27, 2], [47, 6], [-34, 12], [-18, 19], [-11, -32], [44, 22], [6, -29], [28, -35], [50, -4], [66, 10], [-19, -16], [-4, 0], [30, 10], [-16, -20], [42, -16], [-18, 6], [-60, 8], [-14, 20], [-78, 7], [-51, -14], [-6, 36], [13, 4], [8, 16], [-2, -20], [39, -22], [32, -26], [26, 18], [-54, -8], [-22, 14], [-8, 36], [18, -41], [30, 34], [6, -48], [33, -24], [-10, 44], [-28, 1], [2, -7], [6, -10], [4, -24], [14, -34], [35, -12], [2, 6], [-80, 0], [-32, -26], [18, -20], [3, -42], [-8, 45], [2, 18], [-28, -13], [60, -16], [13, -2], [-1, -8], [24, 20], [10, 1], [-8, -42], [-40, 5], [-12, 14], [30, 16], [38, 0], [-28, 16], [-46, 16], [10, 24], [16, 10], [18, 3], [18, 38], [23, -22], [66, 5], [-17, 6], [44, 4], [-2, 36], [6, 33], [12, -36], [-56, -4], [68, 10], [-20, 40], [2, -40], [69, -16], [-24, -26], [22, 18], [-44, -20], [-76, 0], [-13, -4], [2, 4], [-24, 12], [-62, -8], [-48, -20], [-38, 10], [30, 6], [7, 2], [6, 32], [48, 16], [9, 34], [28, 30], [20, 20], [-44, 21], [17, 24], [-4, -1], [-36, 38], [-44, -24], [-48, 6], [-85, 6], [13, -32], [44, 28], [26, 0], [-76, 11], [-52, -8], [-74, 4], [-27, -26], [62, 18], [38, 7], [28, -42], [6, -24], [-54, 12], [19, 40], [40, 32], [48, -14], [-80, 1], [-40, -20], [12, -26], [96, 0], [-58, 16], [56, -22], [20, -36], [-6, -29], [-52, 16], [14, -16], [-57, 2], [40, 14], [0, -8], [-40, 9], [0, 16], [10, -41], [52, 12], [-48, -22], [16, 8], [-72, -20], [-26, 16], [42, -22]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_5070_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_5070_2_a_bg();". // 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_5070_2_a_bg(:prec:=2) chi := MakeCharacter_5070_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_5070_2_a_bg();". // 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_5070_2_a_bg( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_5070_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![-2, 1]>,<11,R![-3, -6, 1]>,<17,R![-4, 1]>,<31,R![-3, 0, 1]>],Snew); return Vf; end function;