// Make newform 4410.2.a.bx in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4410_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_4410_2_a_bx();". // 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_4410_2_a_bx();". // 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, 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_4410_a();" function MakeCharacter_4410_a() N := 4410; order := 1; char_gens := [3431, 2647, 1081]; 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_4410_a_Hecke(Kf) return MakeCharacter_4410_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], [-4, -2], [-2, 1], [-2, 2], [-2, -4], [-4, 1], [-6, 1], [-2, 1], [-2, -4], [0, 5], [-4, -1], [2, 6], [-2, -2], [-4, -4], [-4, -7], [2, -6], [-10, 2], [8, -4], [8, 6], [-6, 6], [-6, 6], [-4, 4], [0, 4], [8, -2], [6, -6], [10, 4], [0, 10], [-2, -6], [4, 6], [-10, -8], [0, -9], [6, 2], [-10, 0], [0, -5], [8, 1], [6, -10], [2, 9], [-12, -6], [-2, -8], [6, -10], [6, 4], [-14, 3], [-4, 14], [8, -2], [8, -12], [8, 2], [18, -4], [-14, 4], [-12, 5], [20, 4], [6, 7], [-10, 0], [-2, -12], [-6, -3], [-30, 1], [4, 18], [-2, -12], [-14, 8], [-4, 16], [4, 18], [-2, 4], [-2, -12], [-4, -14], [-18, 6], [12, -4], [12, -6], [-2, 7], [-30, 2], [-8, -6], [26, 3], [-4, -6], [0, -21], [-8, -17], [-14, -4], [-12, -12], [0, 19], [12, -20], [6, -2], [-2, 10], [-30, 2], [-10, 3], [16, 12], [-36, 2], [-14, -10], [14, 8], [36, 0], [36, -2], [28, -2], [-12, -2], [26, 5], [0, 22], [24, -1], [4, 28], [18, -2], [-2, 16], [-10, 24], [-20, -9], [-18, -8], [4, 14], [-12, -8], [-16, -8], [14, -16], [4, 10], [10, -21], [-10, 16], [28, -13], [0, -22], [18, -1], [-8, 22], [2, 2], [-10, -18], [-4, -16], [10, 12], [32, 13], [14, -22], [-30, -1], [28, 4], [-6, -12], [-10, -18], [32, 4], [18, 12], [12, 3], [-10, -22], [-8, 18], [12, 18], [-10, -12], [0, 2], [40, -4], [-22, 14], [-14, -25], [-14, -16], [20, 5], [-2, 12], [20, 12], [-22, -10], [-8, 6], [2, -16], [36, 5], [16, 4], [4, -14], [16, 0], [-4, -20], [46, -4], [-42, 7], [-14, 12], [12, -12], [-26, 11], [-38, 10], [-12, -23], [-36, 7], [-4, 37], [-14, -28], [6, 18], [-2, 36], [2, -12], [-56, 0], [-16, 16], [48, 2], [24, -12], [-4, 4], [-18, 8], [4, 13], [2, 10], [14, 4], [16, -20], [-42, -2], [-10, -15], [-20, -2], [14, -8], [10, 18], [42, -15], [-26, -8], [32, 4], [32, 7], [12, -2], [32, -12], [-12, -18], [-14, 6], [30, -7], [-2, -23], [-6, 36], [2, 8], [2, -1], [30, -24], [42, -8], [8, -18], [-26, 18], [28, -22], [-36, 6], [10, -12], [-36, 8], [-38, 7], [-24, 4], [14, -13], [-18, -5], [-16, -33], [12, 9], [58, 3], [-8, 38], [4, 1], [36, -24], [2, 32], [-2, -5], [20, 18], [-4, 32], [-42, -8], [-6, 28], [-22, 16], [-36, 16], [-12, -6], [-52, -8], [-12, -29], [-20, -10], [10, 12], [-14, 0], [-14, 2], [-30, 2], [6, 9], [-12, -2], [12, 22], [4, 26], [22, 22], [2, -35], [-46, 16], [-8, -36], [-22, 25], [10, 9], [10, 14], [-48, -16], [-16, -12], [4, 4], [-4, 5], [-16, -7], [-2, 36], [-34, -21], [40, 20], [-16, -30], [-38, -22], [-12, -12], [-18, 40], [-50, -3], [-24, 30], [-60, -10], [46, 18], [-12, -2], [20, -28], [-6, 1], [-30, 24], [28, 16], [-16, 9], [-22, -24], [-30, 9], [6, -18], [30, 32], [-26, 4], [30, 10], [40, -8], [-20, 0], [-24, 0], [-24, 30], [-42, -17], [46, 8], [-44, 15], [38, -2], [-12, 19], [10, -24], [-32, 14], [32, -25], [18, -44], [-48, -8], [-6, -40], [-48, 14], [20, 10], [-34, -18], [-24, 0], [50, -10], [12, -13], [-40, 28], [-20, -17], [-12, 16], [18, -12], [38, 22], [-2, -26], [-2, 6], [-62, -11], [-42, -12], [-20, -17], [20, 36], [34, -4], [2, -22], [-18, 11], [20, -16], [26, -31], [6, 22], [60, 4], [60, -12], [-18, -20], [-30, 2], [-64, -16], [12, -4], [-44, -12], [-34, 12], [-32, -38], [-8, 20], [2, 6], [2, -17], [4, 23], [-36, -35], [-12, 0], [-12, -33], [-12, -36], [-36, -3], [44, -4], [2, 26], [34, 38], [0, -22], [6, 4], [-62, -8], [-44, 0], [-4, 4], [-8, -14], [12, -3], [-24, -4], [-24, 48], [20, 32], [-14, 18], [-14, -40], [2, -6], [10, 3], [60, -8], [-20, -20], [-20, -36], [-30, 24], [26, -13], [4, -29], [4, -2], [10, -9], [-4, 22], [-24, 16], [0, 0], [-22, -20], [26, 21], [-32, 22], [44, -9], [-48, -20], [-42, -32], [-38, 12], [34, 2], [-36, 1], [-60, -12], [30, 16], [-2, 39], [-68, 6], [44, 8], [6, 28], [-22, 4], [-14, 47], [-46, -10], [2, 36], [22, 21], [-20, 48], [-18, -26], [18, 38], [4, -2], [6, 20], [-14, -57], [6, 28], [-6, -40], [24, -15], [18, 21], [-10, -49], [74, 14], [22, 10], [-2, -10], [-6, -30], [30, -20], [-34, 36], [50, -4], [76, 2], [10, -36], [30, 16], [-24, -5], [-64, -16], [-50, 4], [12, 57], [4, -3], [12, -8], [18, -12], [-26, -11], [20, -40], [-10, 10], [22, 24], [-6, -41], [46, -2], [-12, 22], [-68, 15], [0, -60], [-32, -2], [16, 6], [-66, -7], [-68, 24], [-34, 21], [-42, 15], [-68, 18], [10, -20], [-30, -22], [50, 12], [42, -16], [-28, 44], [-48, 16], [-2, -30], [40, 24], [38, -16], [-24, -2], [-14, -19], [-44, 41], [24, 35], [-64, 14], [24, -12], [-52, 4], [42, 16], [-6, -28], [-48, 10], [24, 50], [-82, 2], [-4, -8]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4410_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4410_2_a_bx();". // 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_4410_2_a_bx(:prec:=2) chi := MakeCharacter_4410_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_4410_2_a_bx();". // 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_4410_2_a_bx( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4410_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![2, 4, 1]>,<13,R![8, 8, 1]>,<17,R![2, 4, 1]>,<19,R![-4, 4, 1]>,<29,R![14, 8, 1]>,<31,R![34, 12, 1]>],Snew); return Vf; end function;