// Make newform 8030.2.a.l in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_8030_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_8030_2_a_l();". // 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_8030_2_a_l();". // 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, -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_8030_a();" function MakeCharacter_8030_a() N := 8030; order := 1; char_gens := [1607, 2191, 881]; 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_8030_a_Hecke(Kf) return MakeCharacter_8030_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], [0, 1], [-1, 0], [-2, 1], [1, 0], [-1, 3], [3, -3], [-4, 0], [-1, 1], [-4, 1], [-2, -2], [5, -3], [-6, 0], [0, -4], [-2, -4], [0, 0], [-6, 6], [4, -2], [0, -1], [8, -5], [1, 0], [6, -4], [8, 1], [-3, 9], [-8, -5], [4, -1], [-14, -2], [11, -5], [4, -8], [6, 0], [10, -2], [7, 5], [0, 3], [-7, 3], [-4, -2], [-1, 3], [-12, -4], [8, 0], [12, 0], [2, 4], [2, 4], [-4, 3], [-4, 10], [5, 3], [14, 1], [-4, -6], [-8, 4], [17, 3], [-12, -6], [-18, 2], [-10, -8], [19, -1], [14, 0], [6, -15], [-6, 12], [-2, -1], [-20, 5], [-20, 1], [-10, -3], [-21, 3], [-6, 8], [12, -12], [0, -1], [-24, 0], [-8, -8], [-9, -9], [2, -12], [8, 9], [6, -12], [10, 4], [12, -3], [-4, 10], [-11, 19], [-20, -2], [4, -8], [-4, 4], [15, -3], [-7, -3], [-26, -1], [-2, -11], [22, -1], [8, 0], [-7, 1], [14, -6], [36, -4], [-16, 10], [-12, 0], [-22, 12], [6, 0], [7, 13], [12, 0], [28, -4], [22, -11], [6, -9], [19, -11], [-30, 0], [11, 1], [18, 6], [-10, 12], [24, 5], [-4, 12], [-18, 12], [-4, 7], [-14, 5], [1, -5], [-6, 8], [-4, -5], [18, 6], [-18, 6], [-13, 3], [-22, 0], [-7, 9], [-26, 2], [-29, 1], [-34, -6], [-10, 1], [6, 2], [-14, 2], [2, 1], [-14, 14], [-21, -7], [-30, 14], [-5, -13], [16, -4], [-20, -2], [30, 0], [-8, 16], [-6, 12], [6, -1], [26, 6], [-24, 2], [18, -3], [-6, -4], [14, -3], [1, 17], [-23, 13], [6, -18], [38, -12], [-3, 21], [2, -14], [4, 4], [32, 4], [-14, 10], [24, -12], [-20, 10], [36, -12], [4, 16], [-14, 2], [-18, -10], [-10, 4], [34, -8], [-10, 4], [-30, 14], [-2, 11], [-28, 0], [31, -13], [-8, 16], [6, -12], [-14, 16], [20, -8], [-28, 4], [-10, -20], [52, 4], [28, 14], [-8, 2], [0, -24], [0, 20], [-10, -18], [-26, 19], [-22, -8], [-12, -18], [28, -11], [0, 0], [4, -23], [-34, -12], [-13, -17], [-50, 7], [-30, 6], [10, -26], [4, 7], [22, 7], [26, -14], [-21, 23], [12, 12], [43, 5], [-38, -10], [-22, 12], [-48, -4], [6, 17], [11, -17], [50, -12], [53, 1], [-5, -11], [12, -6], [-18, -12], [-26, -4], [-25, 3], [26, 18], [36, -12], [-16, -2], [-36, 12], [44, 0], [9, 17], [-12, 32], [31, -7], [-41, -1], [-24, -1], [-11, -7], [-12, -18], [-35, 7], [-9, 5], [-40, 10], [-4, 6], [-18, 24], [33, -3], [20, 6], [34, 13], [-38, -4], [36, 0], [30, -18], [26, 6], [-44, 16], [30, 12], [11, 9], [-12, -3], [27, -7], [21, -9], [24, -24], [-5, 13], [-30, 12], [16, 16], [22, -14], [31, -5], [18, 0], [-5, -17], [48, -6], [52, -14], [-25, 7], [-6, 0], [36, -12], [60, 0], [10, -26], [6, -10], [20, -9], [-60, -3], [28, -1], [-34, 0], [7, -7], [16, -32], [-12, 6], [-12, 2], [-30, 3], [52, 8], [-1, -33], [-2, 8], [-24, -24], [-64, -6], [37, 13], [22, -28], [-34, 0], [34, -26], [6, 18], [-6, -4], [-42, -1], [48, -3], [8, -36], [-22, 10], [-18, -12], [13, -11], [24, -12], [-26, 10], [26, 12], [-12, 41], [-36, 17], [42, 14], [-6, -10], [4, 23], [-48, 26], [-30, 20], [-48, 24], [-6, 6], [-19, -3], [46, -13], [54, 2], [41, -3], [-24, 15], [64, -17], [31, -37], [32, 12], [48, -6], [-12, -27], [-40, 1], [-4, 25], [49, -13], [44, -30], [-3, 3], [24, -34], [-33, 27], [-36, 6], [-2, -20], [-36, 17], [-9, -9], [11, -21], [37, 17], [56, -6], [26, 0], [8, -20], [-4, 6], [-54, 6], [20, -18], [-42, -6], [-8, 29], [-28, 28], [-20, 4], [57, -21], [-62, 7], [-16, -20], [-42, -18], [-34, 24], [14, -20], [-26, -5], [48, -4], [46, 14], [-58, 12], [52, 2], [-50, 4], [-30, 2], [-20, -20], [42, -15], [-52, 16], [18, -16], [2, 10], [-56, -8], [-20, 35], [44, -6], [-13, 43], [-26, -8], [70, -16], [-54, -7], [30, -16], [-42, 14], [-22, 4], [42, 12], [80, -3], [6, -27], [-2, 32], [64, -2], [-70, -6], [-16, 37], [-46, -20], [-40, -6], [11, -27], [-30, -12], [-22, 9], [64, -5], [-68, 8], [-24, 12], [30, 24], [-40, 28], [-26, 26], [6, 11], [-36, -12], [26, -5], [34, 26], [-4, 24], [7, 1], [10, -40], [-2, 37], [17, -27], [63, 9], [13, -29], [-32, 26], [-8, 8], [-24, 50], [-46, 21], [-57, -15], [0, 6], [-20, -14], [-2, -34], [-56, 19], [-14, 20], [40, 2], [-20, 28], [15, -3], [38, 21], [10, -7], [38, 6], [26, -12], [38, 0], [20, 16], [-91, 9], [27, 3], [1, 11], [17, 9], [48, -12], [50, 12], [-48, 26], [-8, 14], [30, 8], [-83, -1], [-58, 0], [64, -22], [52, 4], [-11, -13], [49, -25], [-33, -25], [26, -3], [-42, 24], [-52, 0], [4, 38], [-39, -1], [-57, 27], [27, -9], [-16, -36], [-13, -27], [18, 18], [54, -24], [20, 24], [11, 37], [-48, 0], [42, 24], [-12, -16], [12, 30], [42, 0], [18, 32], [42, -36], [-28, -8], [-90, -6], [-42, 14], [-24, 30]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_8030_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_8030_2_a_l();". // 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_8030_2_a_l(:prec:=2) chi := MakeCharacter_8030_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_8030_2_a_l();". // 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_8030_2_a_l( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_8030_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-3, -1, 1]>,<7,R![-1, 3, 1]>],Snew); return Vf; end function;