// Make newform 7098.2.a.bl in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_7098_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_7098_2_a_bl();". // 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_7098_2_a_bl();". // 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 := [-4, -1, 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_7098_a();" function MakeCharacter_7098_a() N := 7098; order := 1; char_gens := [4733, 5071, 6931]; 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_7098_a_Hecke(Kf) return MakeCharacter_7098_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 := [[-1, 0], [1, 0], [-1, -1], [1, 0], [-1, 1], [0, 0], [1, -3], [-3, 3], [-3, -1], [-1, 3], [4, -4], [5, 1], [-4, 2], [-9, 1], [0, 0], [2, 4], [8, -4], [-1, 3], [2, 2], [-8, 0], [1, 1], [-6, -2], [-14, 2], [-10, 0], [6, -8], [0, -2], [7, -3], [-14, 2], [-1, 7], [6, -4], [-10, 2], [-7, -1], [1, 1], [12, 0], [-8, 10], [1, -5], [-13, -1], [22, -2], [-5, -7], [10, 4], [-4, 8], [10, -4], [-3, 7], [-4, -6], [-16, 2], [-11, 7], [-3, -5], [-2, 10], [2, 2], [-18, -4], [12, 6], [-16, 0], [14, 0], [5, -5], [6, -4], [-20, -4], [10, 4], [-10, 2], [-2, -8], [-6, -4], [8, 4], [6, -12], [-16, -4], [-18, 2], [12, -10], [-8, 2], [2, -6], [7, -1], [-20, 0], [-18, 4], [-10, -8], [-4, 12], [-16, 0], [8, -10], [-24, -4], [11, -15], [2, -12], [18, 0], [2, -4], [-13, 7], [-29, -3], [14, -4], [18, 6], [-16, -6], [11, -7], [10, -6], [-3, -3], [-12, -6], [5, 9], [1, 3], [-11, 3], [1, -13], [8, 0], [-14, 2], [-36, 0], [2, -2], [-11, 9], [-5, 3], [32, -4], [15, 7], [-28, 0], [-12, 6], [17, -9], [-10, 20], [8, 4], [30, 0], [2, -6], [10, 4], [19, 1], [-16, 2], [-21, 1], [-1, -17], [-5, 15], [-27, 3], [5, 7], [-10, -4], [-29, 5], [-4, -4], [1, -7], [6, 6], [-10, 4], [-21, -5], [-18, 8], [-43, 3], [4, -8], [-2, 0], [6, 4], [38, 2], [11, -7], [16, 10], [-14, 18], [18, -18], [4, -4], [14, 8], [0, -2], [19, -1], [-35, -7], [13, -13], [16, 6], [18, -8], [-39, -1], [-40, 10], [-16, 8], [-9, -7], [31, 3], [-20, 12], [12, 6], [-14, -8], [16, -4], [2, 6], [30, 4], [9, -11], [-1, 1], [-32, -8], [-28, 8], [27, -15], [14, 2], [0, -10], [-6, -16], [34, 0], [-13, 21], [-4, 6], [19, 1], [-8, 4], [15, 3], [-15, 11], [18, -10], [10, 12], [-10, -8], [-44, 10], [27, 5], [10, -8], [24, 0], [-6, 12], [-27, -1], [-18, 12], [6, -10], [-32, 18], [-16, 24], [17, -23], [35, -7], [0, 20], [-14, 4], [18, 4], [8, 8], [-16, -2], [-12, 2], [-11, -5], [18, -12], [-17, -3], [-32, -6], [-33, 1], [52, 0], [51, 3], [-6, 18], [5, 1], [32, -10], [-38, 20], [14, 0], [0, -24], [-24, 2], [23, -11], [6, 20], [26, 8], [-8, -4], [46, 0], [0, -16], [-8, 12], [11, -1], [36, 8], [-19, -15], [36, -6], [-2, 18], [-14, -6], [15, 21], [-21, -1], [-26, 2], [-39, 5], [24, -16], [8, -14], [12, -14], [-12, -12], [-39, 17], [-4, 20], [-21, 5], [42, -4], [37, -15], [-8, 24], [15, 21], [17, -9], [-42, -8], [18, -6], [19, 9], [12, 6], [2, 2], [63, -3], [-25, 11], [48, -14], [6, -18], [-16, -8], [-20, 32], [24, -12], [-13, 33], [14, -12], [18, -4], [32, -8], [34, -10], [-24, 20], [46, -2], [-61, 9], [21, -7], [40, -18], [8, 16], [-46, -8], [8, -34], [-15, -17], [28, 18], [36, -16], [42, -4], [-11, 13], [-16, -16], [12, -24], [-2, 12], [11, 7], [32, -10], [6, -26], [46, -12], [-6, 12], [24, 12], [-2, -24], [-2, -16], [54, -18], [-19, -11], [24, -24], [20, 6], [-20, -4], [41, -17], [21, -7], [20, -38], [36, -16], [0, 0], [7, -35], [68, 4], [16, -30], [-15, 7], [28, -12], [-13, 3], [-14, -8], [40, -16], [-77, 3], [17, -23], [19, -3], [-23, 9], [0, 20], [58, 4], [56, -2], [16, 16], [46, -8], [-1, 9], [-29, 5], [-32, 2], [68, -10], [-10, 18], [-32, 20], [-23, 7], [-56, -2], [-8, 20], [-51, -3], [-26, -6], [2, -12], [-28, 28], [36, -18], [48, 10], [-17, 33], [-62, -2], [48, -14], [20, 8], [36, 12], [71, -5], [42, -24], [14, -18], [-24, 6], [52, -14], [-23, 11], [37, -19], [-22, 8], [52, -8], [-7, 15], [-51, -9], [-23, -23], [5, 25], [45, -19], [-72, 8], [-22, -6], [4, -32], [18, 26], [-3, 17], [28, 10], [-2, -8], [7, -27], [-46, -8], [-28, 30], [27, 19], [-22, 38], [10, -8], [14, 14], [-19, -11], [-30, 26], [-8, 8], [58, -4], [-16, 12], [0, 30], [40, 2], [44, -12], [-10, 32], [5, -15], [-67, 7], [46, -2], [34, 16], [-56, 0], [11, -13], [13, 25], [-35, -9], [68, 0], [10, 18], [30, -36], [-23, -11], [-33, 5], [57, -3], [38, 22], [-64, -4], [-12, 12], [35, 7], [-31, -5], [-20, 10], [-4, 32], [-21, -15], [-37, -9], [33, -3], [34, 24], [-6, 24], [31, -5], [-37, 9], [38, -8], [12, 0], [-41, -19], [-53, 1], [-41, 3], [-56, -4], [18, -10], [-10, 40], [-26, 20], [-56, -4], [24, -20], [-20, -20], [2, -8], [-30, -10], [21, -15], [3, -27], [46, -20], [48, -22], [-82, 4], [33, -29], [-52, 2], [-2, -4], [-29, 17], [-78, 0], [-23, 17], [25, -9], [-8, -28], [-14, 32], [39, 19], [-28, -24], [-23, 47], [69, -15], [-32, 38], [18, 22], [-59, -9], [46, -16], [-4, 12], [17, -7], [26, -32], [8, -24], [-52, -16], [-3, 21], [22, 28], [38, -26], [34, -12], [5, -13], [-48, 8]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_7098_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_7098_2_a_bl();". // 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_7098_2_a_bl(:prec:=2) chi := MakeCharacter_7098_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_7098_2_a_bl();". // 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_7098_2_a_bl( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_7098_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-2, 3, 1]>,<11,R![-4, 1, 1]>,<17,R![-38, 1, 1]>],Snew); return Vf; end function;