// Make newform 5184.2.a.cf in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_5184_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_5184_2_a_cf();". // 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_5184_2_a_cf();". // 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, -6, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [-1, -7, -1, 1], [-4, -6, 0, 1], [1, -9, -3, 2]]; 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_5184_a();" function MakeCharacter_5184_a() N := 5184; order := 1; char_gens := [2431, 325, 1217]; 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_5184_a_Hecke(Kf) return MakeCharacter_5184_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, 0, 0], [0, 0, 0, 0], [1, 1, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0], [-1, 1, 0, 0], [0, -1, 0, 0], [0, 0, 1, -1], [0, 0, 0, 1], [3, 1, 0, 0], [0, 0, 2, -1], [-4, 0, 0, 0], [1, 0, 0, 0], [0, 0, -1, 0], [0, 0, 0, 1], [4, 0, 0, 0], [0, 0, -1, -2], [-5, 3, 0, 0], [0, 0, 1, 2], [0, 0, 0, 2], [8, 1, 0, 0], [0, 0, -2, 1], [0, 0, 2, 1], [8, 2, 0, 0], [9, 0, 0, 0], [5, -3, 0, 0], [0, 0, 0, -1], [0, 0, 1, -1], [-4, 4, 0, 0], [15, 1, 0, 0], [0, 0, 2, -2], [0, 0, 2, -1], [1, 4, 0, 0], [0, 0, 1, -4], [15, 1, 0, 0], [0, 0, 0, -1], [-7, -3, 0, 0], [0, 0, 0, -4], [0, 0, -2, 1], [-3, -1, 0, 0], [0, 0, 0, -4], [-12, -6, 0, 0], [0, 0, 4, -3], [1, 0, 0, 0], [6, 2, 0, 0], [0, 0, 2, 4], [0, 0, 2, -3], [0, 0, 0, 1], [0, 0, -3, 4], [1, -1, 0, 0], [-14, -5, 0, 0], [0, 0, 4, -3], [7, 2, 0, 0], [0, 0, 3, -3], [7, 4, 0, 0], [0, 0, -4, 3], [4, 4, 0, 0], [0, 0, 4, -4], [-3, 3, 0, 0], [9, -5, 0, 0], [0, 0, -2, 3], [17, -1, 0, 0], [0, 0, 3, 5], [0, 0, -4, 5], [-1, -6, 0, 0], [-5, 1, 0, 0], [0, 0, 6, -1], [-9, 0, 0, 0], [0, 0, 1, 2], [13, -3, 0, 0], [1, 6, 0, 0], [0, 0, 6, -6], [0, 0, -2, -7], [7, -3, 0, 0], [0, 0, 1, 1], [0, 0, 2, 5], [-11, -9, 0, 0], [4, 0, 0, 0], [1, -8, 0, 0], [-25, -2, 0, 0], [0, 0, 2, -3], [19, -1, 0, 0], [0, 0, 2, -2], [-8, 3, 0, 0], [0, 0, 2, 5], [0, 0, 3, -4], [-8, 3, 0, 0], [1, 8, 0, 0], [19, -3, 0, 0], [0, 0, -6, -1], [0, 0, -5, -3], [0, 0, 0, -1], [0, 0, 2, -2], [0, 0, -3, 8], [0, 0, -5, 2], [0, 0, 4, -6], [-17, -9, 0, 0], [14, 9, 0, 0], [0, 0, 4, -2], [-2, 2, 0, 0], [0, 0, -1, 2], [12, -2, 0, 0], [0, 0, -3, -2], [-33, 2, 0, 0], [0, 0, 3, -8], [8, -3, 0, 0], [0, 0, -1, 8], [16, 10, 0, 0], [0, 0, -2, -7], [-23, -6, 0, 0], [0, 0, 6, 1], [4, 12, 0, 0], [-31, 2, 0, 0], [0, 0, -1, 2], [0, 0, 0, 0], [7, -2, 0, 0], [0, 0, 3, 0], [0, 0, -6, -2], [-31, 3, 0, 0], [0, 0, -2, -9], [-7, -3, 0, 0], [-7, -9, 0, 0], [29, 9, 0, 0], [0, 0, 5, 3], [0, 0, 6, 5], [-28, 2, 0, 0], [-11, -7, 0, 0], [0, 0, -4, -4], [0, 0, -4, -5], [-33, -3, 0, 0], [0, 0, -3, 1], [0, 0, 8, -9], [0, 0, 2, 5], [-6, -12, 0, 0], [17, -1, 0, 0], [-25, -3, 0, 0], [12, -2, 0, 0], [0, 0, 6, 3], [-9, 11, 0, 0], [-16, -1, 0, 0], [0, 0, -9, 9], [-5, 1, 0, 0], [0, 0, -2, -5], [0, 0, 0, -4], [-12, -6, 0, 0], [0, 0, 2, -1], [11, -3, 0, 0], [-7, 11, 0, 0], [0, 0, 3, 0], [0, 0, 0, 8], [-17, -9, 0, 0], [-16, -6, 0, 0], [0, 0, -1, -7], [0, 0, -2, -5], [0, 0, -7, 6], [0, 0, 10, -3], [0, 0, -2, -6], [33, -1, 0, 0], [-8, -6, 0, 0], [9, -11, 0, 0], [0, 0, -5, -2], [-8, 7, 0, 0], [0, 0, -8, 9], [0, 0, -12, 10], [31, 6, 0, 0], [0, 0, 2, 13], [0, 0, -8, 0], [-11, 9, 0, 0], [-8, -10, 0, 0], [-1, -9, 0, 0], [0, 0, 5, 2], [-1, -9, 0, 0], [0, 0, -6, 7], [17, 10, 0, 0], [0, 0, -8, 7], [-17, 9, 0, 0], [0, 0, -1, 8], [-28, 2, 0, 0], [0, 0, -4, 4], [5, -3, 0, 0], [0, 0, 4, 5], [0, 0, 3, 4], [-31, -5, 0, 0], [-32, 9, 0, 0], [0, 0, 8, 7], [19, -5, 0, 0], [34, 2, 0, 0], [0, 0, -10, 5], [15, -6, 0, 0], [0, 0, 4, 12], [8, 1, 0, 0], [0, 0, 5, -4], [0, 0, -11, 11], [-51, 1, 0, 0], [0, 0, 3, -3], [-1, 0, 0, 0], [17, 18, 0, 0], [-27, 9, 0, 0], [-33, -8, 0, 0], [0, 0, 6, -6], [25, -5, 0, 0], [0, 0, 4, 5], [9, 9, 0, 0], [-41, -4, 0, 0], [0, 0, -5, -3], [-52, -6, 0, 0], [0, 0, -8, -8], [0, 0, -11, 8], [-25, 0, 0, 0], [0, 0, 5, 8], [32, 15, 0, 0], [15, 5, 0, 0], [0, 0, 12, -11], [0, 0, -2, 7], [0, 0, 10, -11], [31, 6, 0, 0], [0, 0, -6, 7], [33, 16, 0, 0], [0, 0, 10, -2], [-41, 7, 0, 0], [-41, 5, 0, 0], [0, 0, 10, -3], [9, -16, 0, 0], [0, 0, -2, 2], [0, 0, 3, -6], [29, -9, 0, 0], [-49, -3, 0, 0], [0, 0, 4, -12], [0, 0, -8, -1], [0, 0, -6, -13], [17, 19, 0, 0], [0, 0, 0, 12], [0, 0, 0, -15], [7, 4, 0, 0], [0, 0, 2, 3], [0, 0, -8, 1], [-7, 7, 0, 0], [4, 18, 0, 0], [0, 0, 3, 2], [0, 0, 6, 2], [0, 0, -9, 0], [0, 0, 0, 4], [0, 0, -10, 9], [44, -6, 0, 0], [-31, -6, 0, 0], [0, 0, -2, 3], [0, 0, -4, 12], [0, 0, -7, 10], [0, 0, -6, 5], [0, 0, -6, -10], [-17, 17, 0, 0], [0, -19, 0, 0], [0, 0, 10, -9], [-25, -11, 0, 0], [21, 11, 0, 0], [0, 0, 7, -5], [-4, 4, 0, 0], [0, 0, -2, -9], [38, -10, 0, 0], [40, 3, 0, 0], [0, 0, 2, 5], [0, 0, -11, -6], [1, -19, 0, 0], [-28, -6, 0, 0], [25, 3, 0, 0], [0, 0, 1, -8], [-12, -16, 0, 0], [31, -8, 0, 0], [0, 0, 9, 2], [-49, -7, 0, 0], [7, 3, 0, 0], [0, 0, -8, -12], [63, 0, 0, 0], [0, 0, -12, -5], [17, 10, 0, 0], [0, 0, 12, -4], [0, 0, 6, 3], [43, 9, 0, 0], [8, -3, 0, 0], [0, 0, 5, 4], [0, 0, 0, -17], [0, 0, -8, 7], [0, 0, 4, -13], [-11, -9, 0, 0], [0, 0, 3, -6], [0, 0, 2, 0], [32, 22, 0, 0], [25, -15, 0, 0], [0, 0, 2, -1], [42, -5, 0, 0], [35, 3, 0, 0], [0, 0, 5, -11], [-9, 16, 0, 0], [0, 0, -5, -8], [5, -3, 0, 0], [-15, -5, 0, 0], [0, 0, -12, 3], [3, -7, 0, 0], [0, 0, -8, 12], [0, 0, 10, -7], [31, 6, 0, 0], [28, -6, 0, 0], [0, 0, 8, 8], [0, 0, -10, 3], [0, 0, 2, -9], [-24, 6, 0, 0], [0, 0, -10, 15], [9, 9, 0, 0], [0, 0, -4, -5], [-20, -12, 0, 0], [0, 0, -2, 11], [20, 12, 0, 0], [15, -11, 0, 0], [0, 0, 6, 17], [0, 0, -12, -4], [24, -15, 0, 0], [0, 0, -15, 0], [0, 0, -6, -1], [-49, 4, 0, 0], [-7, -21, 0, 0], [0, 0, -13, 14], [-17, -6, 0, 0], [20, 8, 0, 0], [0, 0, -6, 6], [41, 15, 0, 0], [10, -19, 0, 0], [0, 0, -3, 13], [0, 0, 11, -10], [0, 0, -6, -1], [-52, -6, 0, 0], [-3, 21, 0, 0], [33, 5, 0, 0], [0, 0, -2, 3], [0, 0, 9, -10], [0, 0, -3, -5], [0, 0, 0, 12], [14, 0, 0, 0], [-39, -22, 0, 0], [9, -18, 0, 0], [0, 0, 8, -2], [-13, -21, 0, 0], [31, -15, 0, 0], [-15, -19, 0, 0], [0, 0, 10, -1], [-13, 17, 0, 0], [0, 0, 5, 5], [-20, -16, 0, 0], [0, 0, 1, -10], [0, 0, 2, -11], [26, 20, 0, 0], [0, 0, 2, 13], [-80, -1, 0, 0], [55, 1, 0, 0], [0, 0, -2, 9], [23, 3, 0, 0], [6, -16, 0, 0], [0, 0, 10, -1], [0, 0, -8, -4], [47, -3, 0, 0], [0, 0, -8, -9], [-13, -3, 0, 0], [-1, -4, 0, 0], [0, 0, -12, 2], [0, 0, 9, 8], [0, 0, 15, -15], [-7, -18, 0, 0], [67, 3, 0, 0], [0, 0, -2, -4], [8, 3, 0, 0], [0, 0, 14, -17], [0, 0, 9, -9], [0, 0, -18, 9], [-13, -1, 0, 0], [0, 0, 0, -9], [4, 14, 0, 0], [0, 0, -5, -16], [0, 0, -10, -8], [64, -6, 0, 0], [-16, -33, 0, 0], [1, -1, 0, 0], [-11, 7, 0, 0], [-9, -2, 0, 0], [0, 0, -8, 16], [33, -10, 0, 0], [0, 0, -7, 12], [0, 0, -4, -20], [0, 0, -4, 13], [-55, 1, 0, 0], [0, 0, -12, 0], [0, 0, 6, 17], [17, 10, 0, 0], [-29, 7, 0, 0], [0, 0, -13, 11], [0, 0, -21, 12], [0, 0, 0, -7], [17, 19, 0, 0], [0, 0, 4, -10], [81, 4, 0, 0], [0, 0, -11, -16], [15, 23, 0, 0], [49, 15, 0, 0], [-80, 3, 0, 0], [0, 0, 2, 7], [9, 7, 0, 0], [42, 4, 0, 0], [0, 0, -16, 18], [13, 23, 0, 0], [63, 14, 0, 0], [0, 0, 7, -12], [0, 0, -6, 9], [-7, 18, 0, 0], [21, -17, 0, 0], [0, 0, 17, -11], [0, 0, 1, -16], [-23, -13, 0, 0], [36, 10, 0, 0], [0, 0, 10, -2], [0, 0, -8, -9], [-16, -1, 0, 0], [0, 0, -6, -3], [61, 9, 0, 0], [-4, 24, 0, 0], [0, 0, 2, 15], [0, 0, -1, 6], [40, -18, 0, 0], [39, -5, 0, 0], [0, 0, 5, 18], [0, -10, 0, 0], [0, 0, -8, -12], [0, 0, 2, 5]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_5184_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_5184_2_a_cf();". // 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_5184_2_a_cf(:prec:=4) chi := MakeCharacter_5184_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_5184_2_a_cf();". // 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_5184_2_a_cf( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_5184_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-8, -1, 1]>,<7,R![108, 0, -27, 0, 1]>,<11,R![27, 0, -36, 0, 1]>],Snew); return Vf; end function;