// Make newform 990.2.n.i in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_990_n();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_990_n_Hecke();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_990_2_n_i();". // 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_990_2_n_i();". // 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 := [841, -87, -198, 75, 21, 5, -2, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [20083080, -10517880, -2170527, 621001, -119321, 210024, -168690, 3622], [23823587, -4202349, -2218842, 66153, 136080, 209410, -90523, 4915], [949895, -93476, -39405, 96671, 3050, 11167, -3211, 908], [-7703821, 11658276, 298519, -1154253, 415469, -28223, 137254, -31462], [-912398, 360035, -187200, -91661, 17019, -19751, 3143, -3648], [105038, -703386, 337956, 84213, -18791, 4739, -7492, 5692]]; Rf_basisdens := [1, 1, 11583209, 11583209, 399421, 11583209, 399421, 399421]; 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_990_n();" function MakeCharacter_990_n() N := 990; order := 5; char_gens := [551, 397, 541]; v := [5, 5, 4]; // 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; // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_990_n_Hecke();" function MakeCharacter_990_n_Hecke(Kf) N := 990; order := 5; char_gens := [551, 397, 541]; char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -1, 1, 0, -1, 0, 0]]; assert UnitGenerators(DirichletGroup(N)) eq char_gens; values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field F := Universe(values);// the Hecke field chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values); return chi; 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, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0], [0, -1, 0, 0, 1, 1, 2, 1], [-2, 1, 0, 2, 0, 0, 0, -1], [-1, 1, -1, 0, -2, -1, 0, -2], [1, 0, 0, 0, 1, 3, -1, 0], [-3, 1, -3, 2, -1, -1, -1, -2], [0, 2, 0, -1, -2, -1, -1, 0], [-5, -1, 0, 0, 1, -4, 2, 1], [0, 1, 4, 0, -2, -1, -1, -1], [-2, -2, -3, 1, -2, -2, 0, 0], [3, 1, 2, 2, -2, -2, -3, -4], [1, 2, 3, 0, -1, 2, -1, -1], [-3, 2, -2, 4, -1, -1, 0, -2], [0, -1, -4, 3, 2, -2, 1, 1], [6, 0, -1, 1, 0, 6, 0, 0], [-4, -2, -2, 4, 0, 0, 0, -2], [4, -4, 2, -1, 1, 4, 2, 3], [0, 0, 0, -2, -2, -2, 2, 0], [2, 0, -6, 6, 0, 2, 0, 0], [3, 0, 4, -3, 0, 3, -3, 3], [0, 0, 0, 0, 0, 8, 0, 0], [5, -4, -3, -1, 1, -1, 2, 3], [0, -2, -2, 4, 4, -2, 2, 2], [-1, 0, 6, -11, 0, 11, 1, -1], [-7, 3, -6, 9, 3, -7, 0, 0], [0, -2, 0, 10, 2, 2, 2, 4], [4, 0, 8, 0, 0, 8, 0, 0], [2, -1, 1, 0, 0, 0, -1, 0], [6, 2, 2, -3, 3, -1, -3, 2], [-12, 0, -6, 1, 1, -6, 0, -1], [1, 1, 1, 1, 2, 3, -2, 1], [4, 1, -8, 8, -1, 3, -2, -1], [-5, 0, 0, 4, -1, 5, 1, 0], [0, 4, 2, -8, -2, -2, 0, -4], [-7, 2, -5, -4, 0, 0, 2, 0], [-1, -2, -2, 9, 4, -7, 3, 1], [1, -1, 1, -6, 2, 7, 0, 2], [-3, -3, -4, -2, 2, 2, 1, 4], [-9, 0, -9, -4, 0, 0, 0, 0], [-4, 0, 0, 4, 0, 12, 0, 0], [10, 4, 4, -4, -4, 6, -8, -4], [8, 0, 0, -10, -2, 10, 2, 0], [5, -8, -3, -3, 1, 1, 4, 7], [3, -6, -3, 2, 5, 0, 3, 1], [-4, -2, 0, 4, 4, -2, 6, -2], [1, -1, 0, -4, 0, 0, -1, 0], [4, 2, 8, -8, -2, 2, -4, -2], [0, 2, 2, 0, -4, -2, -2, -2], [0, 1, -4, -4, -2, 3, -1, -1], [-6, 8, -2, 2, -4, -4, 0, -8], [13, -4, 3, -1, 1, 5, 2, 3], [1, -3, -4, 8, 6, -5, 2, 4], [-6, -4, -2, 2, 4, -2, 8, 4], [-2, 8, -7, 0, -4, -11, -4, -4], [-5, -3, -3, 3, -2, -3, 2, -3], [1, 3, 5, -4, -1, -1, -4, -2], [-1, -3, 8, 4, 6, -1, 4, 2], [4, -4, -4, -4, 0, -2, 0, -4], [4, -3, 5, -4, 4, 4, 5, 8], [10, -1, 10, -12, -3, 9, -2, -1], [-18, 4, -4, -3, -5, -6, -2, 1], [-8, 6, -4, -4, -2, -2, 2, -4], [-8, -4, -4, 6, -2, 0, 2, -4], [1, 0, 3, 16, 0, -16, -1, 1], [11, 2, -2, 4, 3, -3, -1, -5], [6, 6, 10, -8, -2, 2, -8, -4], [8, 4, 4, -6, 2, 0, -2, 4], [4, 4, 6, 8, -2, -2, 0, -4], [-1, 2, 3, 0, -1, 2, -1, -1], [16, 0, 6, -6, 0, 16, 0, 0], [2, -2, -12, 12, 2, 4, 4, 2], [1, -2, -8, 0, 1, -7, 1, 1], [-18, 1, 1, 16, -2, -6, 2, 1], [0, 2, -8, 5, -4, -7, -2, -2], [9, 1, -13, 12, -3, 7, -4, -2], [-6, 6, 14, 1, -2, 11, -3, -4], [-18, 0, 0, 19, 1, -7, -1, 0], [-7, 1, 1, -4, -2, 3, 6, -8], [-8, -4, -5, 4, 6, -3, 2, -2], [4, 0, 6, -8, 2, 2, 4, 4], [0, -2, -22, 10, 4, -8, 2, 2], [0, -6, 0, 0, 6, 6, 12, 6], [4, -4, -2, -1, 1, 0, 2, 3], [-4, -6, -6, 6, 4, 4, 2, 8], [1, -2, -1, 8, 4, -6, 1, 3], [18, -2, -2, -18, 0, 22, 0, -2], [-13, -2, -3, 2, 3, -2, 1, -1], [1, -2, 12, 0, 1, 13, 1, 1], [10, 4, 4, -12, -2, 10, 2, 4], [-14, 2, 2, 16, 2, 2, -2, 2], [-4, -4, -8, 16, 0, 0, -4, 0], [30, 2, 24, -25, -4, 27, -6, -3], [-18, -7, -20, 16, -1, -15, 6, 3], [3, -4, 0, -12, 1, 1, -2, 2], [-18, -3, -17, 8, 4, 4, 5, 8], [6, -7, 4, -8, -1, 9, 6, 3], [-4, 2, 2, 0, -4, 12, 4, 2], [6, 0, -16, 10, 0, -10, -6, 6], [-4, 7, 1, 8, -2, -2, 3, -4], [5, -7, -10, 5, -3, 7, 4, 2], [2, 2, 0, 14, -4, -16, -4, 0], [-7, -7, -14, -8, 0, 0, -7, 0], [7, 4, -1, 3, 1, -3, -2, -5], [-12, 2, 2, 12, 0, -26, 0, 2], [2, 2, 10, -8, 2, 2, 0, 0], [-2, -16, -10, 1, 9, -2, 8, 7], [-14, 2, 2, 16, 2, -8, -2, 2], [-11, 3, -16, 15, -5, -15, -8, -4], [2, -6, -16, -2, 12, 8, 4, 8], [-20, -4, -24, 14, 0, 0, -4, 0], [18, 4, -2, 2, 0, -4, -2, -4], [-3, 11, -1, -2, -9, -9, -7, -18], [23, 5, 8, -10, -9, 16, -14, -7], [-11, 9, -7, -8, -5, -5, -1, -10], [11, -1, -1, -1, 10, 5, -10, -1], [-3, 0, 22, -1, 0, 1, 3, -3], [-11, -3, -12, 9, -3, -11, 0, 0], [5, -8, -9, -1, 3, -5, 4, 5], [-2, -8, -2, -6, -2, 2, 4, 10], [-6, -2, 6, -12, 4, 14, 8, -4], [18, -2, -2, -22, -4, 8, 4, -2], [-8, 8, -6, 4, 0, -10, -4, -8], [2, 5, -8, 15, -10, -20, -7, -3], [-6, -2, -4, -2, 4, 4, 6, 8], [8, -2, -2, -4, 4, 18, -4, -2], [28, -2, 16, -14, 6, 32, 8, 4], [-19, -4, -11, -1, 1, -9, 2, 3], [-3, 5, -3, 8, 5, -3, 0, 0], [0, 1, -12, -13, -2, 12, -1, -1], [25, -4, -4, -25, 0, 13, 0, -4], [20, 3, 22, -29, -1, -1, 1, -2], [5, -5, -27, 8, 10, -3, 0, 10], [4, 0, -10, -4, 0, 4, -4, 4], [12, 0, 13, -4, -4, 13, 0, 4], [-7, 15, -4, 4, -12, -12, -9, -24], [1, -2, -2, -4, -3, -27, 3, -2], [-18, 4, 4, 19, 1, -7, -1, 4], [-2, 0, 0, 5, 3, -9, -3, 0], [-5, -5, -8, 18, 2, 2, -1, 4], [-8, 6, 2, 4, 6, -8, 0, 0], [3, -2, 11, 8, 4, -6, -1, 5], [2, -2, 32, -16, 4, 18, 0, 4], [8, 2, 22, -22, -2, 6, -4, -2], [-14, 0, -18, 8, -4, -4, -8, -8], [2, 1, 1, -4, -2, 12, 2, 1], [-11, 10, 11, -2, -7, 6, -5, -3], [5, 20, 5, -1, -11, -5, -10, -9], [5, 4, 4, -1, 4, 26, -4, 4], [15, -2, 15, -25, 2, 2, 2, 4], [21, -4, 9, -3, -1, 11, 2, 5], [12, 1, 10, -7, -3, -3, -5, -6], [5, 2, 11, -4, 12, 10, 10, 5], [3, 2, 2, -2, 1, -43, -1, 2], [2, -4, 24, -4, 8, 8, 2, 6], [7, 0, 0, -2, 5, -15, -5, 0], [-2, 1, 1, 4, 2, -14, -2, 1], [-2, 0, 20, 6, 0, -6, 2, -2], [-7, 1, 1, 13, 6, -9, -6, 1], [8, 4, 2, 6, 4, 0, -2, -8], [7, 7, -9, 10, -5, 1, -12, -6], [-27, -2, 2, 2, 3, 3, 1, -1], [1, -15, -6, 2, 8, 8, 1, 16], [4, -4, 26, -4, 8, 8, 0, 8], [-16, -11, -10, 2, -5, -13, 6, 3], [9, 6, 3, -2, -5, 0, -3, -1], [29, -3, 5, -12, -11, 25, -8, -4]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_990_n_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_990_2_n_i();". // 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_990_2_n_i(:prec:=8) chi := MakeCharacter_990_n(); 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_990_2_n_i();". // 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_990_2_n_i( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_990_n(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![13456, 7192, 808, -896, 505, -76, 38, -3, 1]>,<13,R![14641, 7381, -898, -2097, 1505, -123, 62, -1, 1]>,<17,R![256, -192, 608, -1104, 1105, 276, 38, 3, 1]>],Snew); return Vf; end function;