// Make newform 9408.2.a.eh in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_9408_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_9408_2_a_eh();". // 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_9408_2_a_eh();". // 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 := [-3, -6, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0], [0, 2, 0], [-4, -1, 1]]; Rf_basisdens := [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_9408_a();" function MakeCharacter_9408_a() N := 9408; order := 1; char_gens := [1471, 6469, 3137, 4609]; v := [1, 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_9408_a_Hecke(Kf) return MakeCharacter_9408_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], [-1, 0, 0], [0, 0, -1], [0, 0, 0], [0, 1, 1], [1, 1, -1], [2, 0, 2], [-1, -1, 1], [-2, 0, -2], [-4, 0, 1], [1, -1, 0], [-1, -1, -3], [-2, 2, 2], [-5, -1, -1], [4, 0, 2], [-2, -2, -1], [-4, 1, 1], [6, -2, 0], [-3, -1, 1], [0, 0, -2], [11, -1, -1], [-9, 1, 2], [-6, -1, 3], [-4, 0, -2], [0, -1, -1], [-6, -2, 0], [5, -1, -1], [-8, -1, 3], [-3, 1, 5], [8, 0, 0], [5, -1, -2], [0, 1, 1], [-4, 0, 2], [-1, 1, 3], [-4, 0, 2], [-4, 0, 5], [-6, 2, 2], [0, 2, -4], [-4, -2, -6], [6, 2, 0], [-8, 0, 4], [9, -1, -1], [0, -4, 0], [3, 2, 6], [-14, -2, 0], [-12, 0, -4], [0, 2, 2], [-20, 0, -1], [4, 3, -5], [5, 3, 7], [12, 2, 0], [-2, 0, -2], [8, 3, -5], [-12, 1, -3], [-12, 0, -2], [-8, 0, -2], [-6, 2, 3], [-12, 0, -3], [1, -1, -1], [18, -2, 0], [19, -1, -1], [-8, 0, 1], [-3, -3, -9], [6, -2, -6], [3, 2, 2], [0, 0, -3], [-7, -1, 3], [1, -4, -4], [-10, 2, -2], [14, -4, 0], [-4, 4, 4], [-12, 6, 2], [-9, -3, 0], [-5, 5, -3], [-15, -1, -1], [-2, -4, -4], [-4, -4, 2], [3, 1, 1], [2, 6, -2], [1, -2, 6], [-14, -2, -10], [-7, -5, 1], [-8, 0, 4], [9, 1, 5], [4, 4, 5], [-24, 1, -3], [8, 2, -8], [-11, -6, 2], [-20, 0, -2], [11, 5, 1], [-8, 4, -4], [18, -2, 0], [3, 5, -8], [-10, 3, 3], [15, -3, 3], [14, 0, 2], [4, -4, 1], [-4, 2, 2], [9, 5, 15], [-11, -1, 5], [-8, 2, 4], [-18, -2, -7], [6, -9, -9], [2, -4, 6], [-13, -5, 7], [-17, 0, 4], [-14, 1, 1], [-26, 4, -2], [-24, -2, -2], [3, 0, -4], [7, 1, 0], [22, 0, -2], [-6, 4, 0], [9, 1, 9], [0, 0, -13], [-2, 8, 12], [3, 5, 3], [10, 2, -8], [22, -2, -7], [10, 6, -2], [3, 1, 11], [-25, 6, 6], [-32, 0, -1], [-8, -5, -5], [-3, -1, -13], [18, 6, 3], [-2, -4, -4], [10, 0, -4], [3, 5, -6], [5, 3, 1], [-9, 1, 11], [2, 4, 8], [-3, -1, -10], [-2, 6, -8], [10, 4, -2], [11, 6, -2], [-6, -6, 0], [20, 2, 2], [-10, -6, -7], [-24, 4, 8], [20, -8, -2], [8, 4, 13], [-16, 0, 12], [-22, -1, -9], [17, 1, 7], [8, -2, 12], [-13, -1, -1], [-12, 6, 2], [0, -2, -2], [10, 4, -4], [10, -2, -10], [-22, 2, -4], [21, -5, -5], [-2, 8, -2], [-11, 5, -9], [22, 2, 4], [-29, 5, 1], [8, -6, 2], [3, -8, -8], [18, 2, 1], [6, 2, -6], [-22, -4, 8], [-17, 1, -2], [-12, -1, 7], [-2, -8, -4], [2, -10, 2], [5, -1, -8], [23, 3, 11], [-27, -2, 2], [-8, 0, 11], [2, 1, 17], [10, 6, 4], [8, 2, -8], [-19, -4, -8], [13, -9, -10], [4, -8, 0], [-19, -3, -15], [-16, 8, -1], [0, -4, 4], [15, -3, -5], [5, 3, 4], [14, -3, 9], [25, 1, -7], [-2, -10, -4], [-10, 0, -4], [-4, 0, 3], [22, -2, -8], [17, -5, 9], [-19, 2, -6], [18, 4, 6], [-11, 4, -12], [-18, 1, 13], [24, 2, 16], [-6, 6, 4], [-24, -3, -7], [16, -4, -2], [17, 1, -11], [-10, 2, 0], [-22, 0, -12], [26, 2, 12], [-42, -2, -3], [-5, -3, -15], [-22, -8, -8], [34, 4, 16], [-18, 10, 10], [38, -2, -1], [0, 4, 4], [16, 7, 15], [0, -6, -4], [-4, -8, -6], [1, 5, 17], [-20, -4, -5], [-44, 0, 1], [4, 3, -17], [-46, 4, 6], [49, -3, 1], [28, 4, 3], [-32, 2, 12], [-30, -4, 8], [8, -4, 2], [13, 3, 3], [29, -1, 7], [-8, -6, -14], [20, -8, -3], [28, 3, 7], [10, -6, 6], [-12, 4, -6], [42, 2, -6], [23, -3, -6], [-8, -4, -4], [25, -1, -7], [-4, 4, -12], [15, 5, -2], [6, 10, -12], [23, 5, -9], [-12, 10, -4], [13, 2, -10], [-2, -2, -11], [-16, 4, -4], [24, 2, -4], [12, -8, 0], [19, 1, -11], [-43, -5, 2], [-2, 4, 16], [-20, 2, 12], [-34, -6, -2], [23, 1, -6], [24, 0, -12], [4, -12, -12], [-8, -8, 8], [-39, 5, -3], [10, 0, 8], [14, 0, 8], [15, 7, 3], [24, -8, 5], [58, -1, -5], [-31, -1, -7], [-31, -1, -5], [-20, -8, -11], [-24, -7, 5], [19, 5, 17], [18, -9, -9], [37, -3, -3], [23, 1, 21], [2, 4, -16], [-29, 1, -9], [-26, 10, 5], [-32, -6, 8], [-16, 8, 12], [26, -2, -7], [-21, 1, 5], [-8, 0, 6], [21, -12, -8], [27, 5, 0], [12, -7, -7], [-16, 8, -4], [-18, -7, 1], [7, 3, -3], [25, -7, -3], [-6, -2, -10], [0, -8, -18], [27, -7, -10], [30, -4, -14], [43, 5, 1], [-4, -4, -2], [14, 4, 14], [-15, -6, 2], [0, 8, -13], [21, -1, -2], [-28, 0, -8], [-48, 4, 6], [-18, 9, -3], [28, 4, -8], [76, 0, 0], [27, 3, -3], [10, 6, 1], [-31, -1, 3], [-38, -6, 4], [-8, -8, 4], [-11, 9, 17], [-34, 0, -8], [30, 2, 17], [-20, -4, -9], [-26, 3, -13], [11, -5, -5], [11, -5, 7], [10, 7, -5], [-31, -7, -7], [16, -14, -2], [-5, 7, -9], [-30, 4, -12], [-12, -8, -2], [-2, -4, 2], [20, 12, 4], [-12, -6, 0], [19, -1, 3], [22, 11, -17], [-6, 8, 2], [-11, 4, -12], [50, -2, -10], [3, 5, -3], [13, -3, -19], [50, -2, 7], [-5, -15, -7], [-30, 2, -18], [-7, -2, 10], [13, 1, 13], [3, -7, -9], [-4, 2, -2], [16, 12, 11], [-27, -1, -11], [-20, -4, 3], [25, 11, 3], [-6, -2, 14], [53, -1, 1], [-28, -8, 0], [-34, -8, 2], [-40, -10, -2], [-43, 0, 0], [-15, -5, 18], [-1, -11, -17], [-4, 2, 4], [14, 6, 27], [41, 7, 18], [8, -8, 13], [36, -1, -9], [-3, 9, -3], [-12, 0, -2], [-34, -12, 12], [30, 2, 11], [-12, 6, 4], [48, -1, -5], [-46, -6, -24], [40, 8, -17], [-5, -3, -5], [24, 2, 18], [-8, -12, -2], [36, -3, 9], [22, -2, 14], [2, 0, -20], [-30, 14, 16], [-8, 0, -32], [18, 0, 8], [26, 6, -6], [-41, -9, -3], [-44, 3, 3], [20, 0, -10], [49, 3, 11], [28, -3, -3], [4, 3, -5], [-41, 13, 9], [-12, 18, 18], [26, -14, -1], [43, 9, -7], [-35, -7, -9], [34, -2, -2], [58, -4, 6], [4, 3, 23], [32, -12, -10], [11, -9, -13], [-18, 10, 0], [-20, -10, -22], [-68, -4, 8], [34, -12, -2], [-36, 6, 10], [-32, -4, -18], [-41, 13, 13], [-26, -6, -18], [-47, -7, -1], [-32, -14, 24], [69, 0, 4], [-10, 2, -24], [-50, -1, -5], [13, 13, 5], [22, 4, 0], [-24, -13, -9], [4, 12, 3], [-10, -6, 8], [33, 9, 11], [-12, 4, -1], [22, -8, 2], [-30, 4, 18], [16, -4, -24], [-2, -10, 16], [-30, 2, -5], [5, 11, -6], [-42, 4, -4], [68, -2, 6], [-9, 11, -9], [34, 3, 3], [-12, -9, 23], [-40, -4, -5], [18, -3, -15], [-20, -2, -14], [19, -1, 15], [-34, 2, -9], [-26, -14, 6], [-19, -5, -9], [-56, -2, -12], [-42, 6, -8], [-10, 6, 1], [-21, -9, -1], [-24, 4, 22], [58, 7, 19], [14, -4, 12], [-28, -4, -26], [-14, -7, -11], [0, -2, 28], [-12, -4, 10], [-12, 8, 0], [19, 15, 19], [4, -7, -27], [-39, 7, 15], [58, -10, -8], [-21, -9, -17], [-70, -4, 10], [9, 0, 0], [11, 3, -3], [-1, 9, -13], [36, -8, -7], [-2, 3, -5], [-60, -6, 6], [35, -7, -7], [-18, -14, 20], [4, 1, -15], [18, -2, -12], [-8, 14, 12], [6, -16, 10], [-6, 4, -8], [13, 13, 13], [-13, 3, -13], [-28, 14, 6], [-30, 7, -5], [36, 12, -16], [25, 12, 4], [34, 2, 13], [9, 5, -1], [64, 7, -1], [38, -2, 2], [22, -16, -10], [-8, 10, -10], [47, 5, -11], [-50, -5, -5], [-69, 11, 1], [-24, 6, 22], [-4, -3, -35], [-65, 1, -7], [-10, -5, 3], [12, 12, -12], [-37, -1, -21], [-31, 3, -24], [-4, 12, 8], [-40, 16, 6], [-15, -8, 20], [36, -4, -4], [-58, 6, -4], [6, 14, -3], [31, 5, 29], [-44, 4, 4], [14, 2, -19], [-43, -2, -14], [0, 8, -6], [-7, -10, 6], [-34, -14, 11], [-20, 4, -15], [-10, 1, 1], [-39, 3, 11], [-14, 7, 23], [-24, -6, -18], [-16, 0, 13], [42, -10, 12], [-2, -6, -14], [-9, 12, 12], [58, 10, -20], [-30, -5, -9], [7, 1, 23], [25, -13, -13], [-30, -2, -21], [43, 5, -10], [67, -3, 11], [-14, -2, -17]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_9408_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_9408_2_a_eh();". // 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_9408_2_a_eh(:prec:=3) chi := MakeCharacter_9408_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(3581) - 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_9408_2_a_eh();". // 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_9408_2_a_eh( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_9408_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-4, -9, 0, 1]>,<11,R![-38, -27, 0, 1]>,<13,R![112, -36, -3, 1]>,<17,R![96, -24, -6, 1]>,<19,R![-112, -36, 3, 1]>,<31,R![47, -21, -3, 1]>],Snew); return Vf; end function;