// Make newform 605.2.a.k in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_605_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_605_2_a_k();". // 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_605_2_a_k();". // 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, 1, -3, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-1, -1, 1, 0], [1, -2, -1, 1]]; Rf_basisdens := [1, 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_605_a();" function MakeCharacter_605_a() N := 605; order := 1; char_gens := [122, 486]; v := [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_605_a_Hecke(Kf) return MakeCharacter_605_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], [-1, 0, 1, 1], [-1, 0, 0, 0], [0, 1, 2, -1], [0, 0, 0, 0], [0, -3, 0, 2], [-1, -1, -1, 3], [5, 2, -1, 0], [2, -1, -1, 0], [5, -2, -1, -2], [0, -5, 0, 0], [3, 3, 3, -7], [1, -1, 3, 1], [6, -1, -2, 0], [3, -1, -3, 0], [-6, 1, 5, 1], [2, -1, -4, 5], [2, -2, 2, 1], [-5, -3, -4, 6], [3, -3, 0, -2], [1, 3, 2, 0], [9, 4, 3, -6], [-1, -1, 3, -6], [1, 0, -6, 0], [5, 0, -2, 8], [2, -3, -7, 1], [-1, 0, 4, 6], [-2, 8, 3, -7], [5, -1, -4, 6], [1, -1, 1, -2], [5, 2, -3, -3], [6, -1, -6, 2], [-8, 9, 4, -3], [8, -5, -5, 7], [1, 6, 5, -7], [12, -2, -9, -5], [6, 3, 6, -10], [4, 2, -9, -4], [-7, -5, -6, 7], [-11, -1, -2, 3], [-5, 12, 2, 0], [0, 3, -8, 4], [-1, 5, 6, 0], [7, 5, -9, -2], [2, 5, -1, -7], [-4, -2, 3, -2], [17, 1, 2, -12], [-6, -2, 2, 10], [10, 2, -7, -7], [1, 1, 13, 2], [-12, 13, 6, 3], [-4, -7, 7, 16], [2, -10, 0, 9], [9, 2, -1, -9], [7, -1, -3, -14], [-1, -7, 0, -4], [-4, 5, 6, 4], [7, -7, 0, 5], [6, 2, 0, -1], [-6, -8, 3, 16], [17, -9, -10, 3], [-10, 1, -2, -7], [0, 5, 7, 7], [-9, -13, -7, 16], [-6, 8, -5, -2], [6, -13, -7, 4], [-2, 20, 4, -12], [11, 6, -6, 5], [5, -3, 0, 1], [10, 7, 0, -16], [-4, -17, 2, 7], [-5, 6, 8, 0], [-3, -2, -11, 8], [1, -6, -5, -11], [2, -2, 4, 9], [-16, -5, -5, 7], [11, 22, 6, -18], [-11, -2, 0, -3], [-15, -11, -3, 13], [-9, -8, -2, 3], [-11, -9, 9, 5], [5, 15, 0, -17], [2, 11, 13, -15], [8, 18, -7, -17], [0, 0, 5, -8], [-3, -17, 2, -3], [16, 0, -13, -7], [9, 8, -6, -5], [-8, 5, 3, -2], [19, -2, -9, -17], [18, 4, -6, 9], [-8, 13, 1, -2], [-12, 12, -3, -1], [15, 5, 4, 0], [10, -12, -11, -5], [-38, 11, 5, 0], [3, 1, 6, -9], [-5, -13, 0, -3], [-12, 2, 5, 12], [-4, -2, -4, 1], [2, -2, 1, 14], [-5, -20, 5, 22], [-9, -2, 9, -2], [3, 5, -13, -20], [6, 1, 0, 16], [26, -13, -5, -8], [-11, -10, 15, 16], [-36, 2, 1, 6], [-5, 3, 3, -12], [-24, 9, 0, 16], [-2, -11, 13, -6], [-11, 15, 11, -4], [0, 5, 7, -15], [21, -10, -17, -5], [-15, -16, -12, 20], [-21, -12, 6, 12], [9, -8, -17, -4], [1, 4, 2, -29], [9, -25, 3, 6], [10, -7, -2, 9], [-18, -14, 9, 8], [4, 20, -13, -7], [19, -4, -22, 6], [-15, 11, -1, -8], [10, -2, -5, 15], [21, -23, -9, 1], [-15, 10, 7, 2], [-19, 9, 15, -6], [19, 0, -18, -6], [-18, 4, 14, -11], [4, 0, 17, 8], [2, 4, -6, -30], [-5, -4, 12, 13], [-24, 0, 6, 6], [7, -7, -9, 16], [-5, 18, -10, -20], [9, 19, -12, -7], [24, -6, -8, -11], [-17, -6, 5, 7], [3, -2, -10, 8], [4, -17, 3, -5], [30, 7, 0, -31], [-4, 24, -4, 1], [26, 1, -3, -35], [-7, -5, 5, 5], [12, -20, -3, 8], [-11, -10, -17, 12], [29, -20, -7, -7], [7, -3, -11, -2], [5, 2, 8, -13], [13, 14, 4, -14], [14, 10, 17, -15], [0, 13, -8, 19], [11, -16, -7, -17], [2, 23, -2, -8], [8, 8, -2, -20], [-24, 8, 6, 22], [-8, 14, -11, 13], [12, -7, -13, 12], [38, 8, -5, -4], [9, -11, -6, 19], [-17, -10, 9, 13], [12, 11, -4, -7], [-17, -7, 2, 3], [-6, 15, 17, 1], [23, 8, -14, 17], [15, 7, 1, 0], [-8, -19, -2, 17]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_605_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_605_2_a_k();". // 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_605_2_a_k(:prec:=4) chi := MakeCharacter_605_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(997) - 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_605_2_a_k();". // 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_605_2_a_k( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_605_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![1, 1, -3, -1, 1]>,<3,R![-1, 5, -6, 0, 1]>],Snew); return Vf; end function;