// Make newform 6400.2.a.bd in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_6400_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_6400_2_a_bd();". // 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_6400_2_a_bd();". // 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 := [-6, 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_6400_a();" function MakeCharacter_6400_a() N := 6400; order := 1; char_gens := [4351, 4101, 5377]; 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_6400_a_Hecke(Kf) return MakeCharacter_6400_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], [-1, 1], [0, 0], [0, 0], [3, -1], [0, 0], [-3, -2], [1, 3], [0, 0], [0, 0], [0, 0], [0, 0], [-3, -4], [-10, 0], [0, 0], [0, 0], [-6, 0], [0, 0], [7, 3], [0, 0], [-1, 6], [0, 0], [-9, -1], [9, 2], [-10, 0], [0, 0], [0, 0], [-3, -7], [0, 0], [9, -4], [0, 0], [-18, 0], [3, -8], [11, 3], [0, 0], [0, 0], [0, 0], [-1, -9], [0, 0], [0, 0], [9, 7], [0, 0], [0, 0], [-11, 6], [0, 0], [0, 0], [-7, 9], [0, 0], [-30, 0], [0, 0], [-30, 0], [0, 0], [-13, 6], [3, -11], [30, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-18, 0], [-11, -9], [0, 0], [-17, -3], [0, 0], [10, 0], [0, 0], [-13, -9], [-7, 12], [-3, 13], [0, 0], [30, 0], [0, 0], [0, 0], [0, 0], [19, -3], [0, 0], [0, 0], [0, 0], [3, 14], [-11, 12], [9, -13], [0, 0], [0, 0], [-19, -6], [0, 0], [21, -1], [-21, 2], [-13, -12], [0, 0], [0, 0], [-30, 0], [0, 0], [0, 0], [42, 0], [14, 0], [0, 0], [0, 0], [-3, 16], [-19, 9], [0, 0], [23, -3], [0, 0], [-30, 0], [21, 8], [-22, 0], [17, -12], [-3, -17], [9, 16], [0, 0], [-23, 6], [0, 0], [0, 0], [-30, 0], [26, 0], [0, 0], [42, 0], [50, 0], [0, 0], [0, 0], [9, 17], [0, 0], [-10, 0], [0, 0], [21, -11], [-23, -9], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-34, 0], [0, 0], [0, 0], [0, 0], [-27, 4], [11, 18], [0, 0], [50, 0], [0, 0], [6, 0], [-38, 0], [0, 0], [0, 0], [-27, 7], [0, 0], [0, 0], [0, 0], [27, 8], [-29, 3], [0, 0], [0, 0], [18, 0], [-1, 21], [0, 0], [-10, 0], [0, 0], [0, 0], [-54, 0], [-17, -18], [0, 0], [-30, 0], [-21, 16], [0, 0], [27, 11], [-3, -22], [0, 0], [0, 0], [0, 0], [19, -18], [0, 0], [-21, 17], [0, 0], [0, 0], [31, -6], [0, 0], [9, 22], [-13, 21], [0, 0], [0, 0], [0, 0], [0, 0], [33, -1], [0, 0], [-33, -2], [0, 0], [0, 0], [0, 0], [31, 9], [29, 12], [0, 0], [1, 24], [21, 19], [17, 21], [0, 0], [-33, -7], [-30, 0], [7, -24], [0, 0], [33, -8], [0, 0], [0, 0], [0, 0], [0, 0], [31, -12], [-6, 0], [0, 0], [0, 0], [-30, 0], [-66, 0], [58, 0], [70, 0], [0, 0], [0, 0], [-27, 17], [0, 0], [13, 24], [0, 0], [3, -26], [0, 0], [0, 0], [0, 0], [0, 0], [-21, 22], [0, 0], [-33, 13], [0, 0], [-9, -26], [0, 0], [0, 0], [27, -19], [0, 0], [1, -27], [0, 0], [33, 14], [-10, 0], [0, 0], [-29, 18], [0, 0], [-21, -23], [0, 0], [39, 1], [-37, 9], [0, 0], [0, 0], [-39, 4], [0, 0], [0, 0], [78, 0], [11, -27], [0, 0], [0, 0], [-33, 16], [0, 0], [-31, -18], [0, 0], [-39, -7], [0, 0], [13, 27], [0, 0], [-37, 12], [0, 0], [-33, -17], [0, 0], [0, 0], [-27, 22], [41, 3], [0, 0], [78, 0], [29, 21], [0, 0], [0, 0], [-17, 27], [-41, 6], [0, 0], [-50, 0], [0, 0], [-27, -23], [0, 0], [-2, 0], [33, 19], [0, 0], [0, 0], [0, 0], [0, 0], [-43, 3], [0, 0], [70, 0], [0, 0], [0, 0], [66, 0], [0, 0], [-30, 0], [39, -14], [3, -31], [0, 0], [0, 0], [0, 0], [0, 0], [-54, 0], [23, 27], [-29, 24], [0, 0], [0, 0], [-9, -31], [43, 9], [37, 18], [-90, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [27, 26], [50, 0], [0, 0], [-34, 0], [-39, -17], [0, 0], [-31, -24], [-9, -32], [-82, 0], [43, 12], [0, 0], [0, 0], [90, 0], [-38, 0], [1, 33], [-10, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [39, -19], [-37, -21], [-90, 0], [0, 0], [30, 0], [-47, -6], [0, 0], [0, 0], [27, 28], [0, 0], [0, 0], [0, 0], [-66, 0], [0, 0], [13, -33], [0, 0], [0, 0], [-47, 9], [-70, 0], [0, 0], [0, 0], [0, 0], [-9, 34], [0, 0], [27, -29], [-90, 0], [0, 0], [0, 0], [33, -26], [0, 0], [-6, 0], [-17, -33], [-49, -6], [0, 0], [0, 0], [37, -24], [-18, 0], [19, -33], [0, 0], [0, 0], [0, 0], [0, 0], [-39, 23], [0, 0], [1, -36], [51, -2], [10, 0], [0, 0], [51, 4], [0, 0], [33, -28], [94, 0], [0, 0], [0, 0], [0, 0], [70, 0], [0, 0], [-49, -12], [0, 0], [51, -7], [23, -33], [0, 0], [11, -36], [0, 0], [-51, -8], [43, -21], [0, 0], [0, 0], [21, 34], [0, 0], [27, -32], [0, 0], [0, 0], [0, 0], [-102, 0], [50, 0], [9, 37], [41, 24], [0, 0], [-51, 11], [-98, 0], [47, 18], [0, 0], [0, 0], [0, 0], [-3, 38], [0, 0], [0, 0], [0, 0], [0, 0], [51, 13], [19, 36], [0, 0], [-30, 0], [9, -38], [-53, -9], [0, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_6400_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_6400_2_a_bd();". // 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_6400_2_a_bd(:prec:=2) chi := MakeCharacter_6400_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_6400_2_a_bd();". // 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_6400_2_a_bd( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_6400_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-5, 2, 1]>,<7,R![0, 1]>,<11,R![3, -6, 1]>,<13,R![0, 1]>,<17,R![-15, 6, 1]>,<29,R![0, 1]>,<31,R![0, 1]>],Snew); return Vf; end function;