// Make newform 8015.2.a.g in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_8015_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_8015_2_a_g();". // 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_8015_2_a_g();". // 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, -3, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0], [0, 1, 0], [-2, -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_8015_a();" function MakeCharacter_8015_a() N := 8015; order := 1; char_gens := [3207, 4581, 4586]; 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_8015_a_Hecke(Kf) return MakeCharacter_8015_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, 1, 0], [0, -1, 1], [-1, 0, 0], [-1, 0, 0], [-1, -1, 1], [-2, 3, 2], [-2, 2, 0], [0, 0, 0], [-1, -1, 3], [3, 1, -1], [1, 3, -3], [-3, 3, 3], [-9, 1, -1], [6, -2, -2], [-3, 1, 1], [1, -1, -1], [-4, -3, -4], [2, -5, 1], [-5, 3, 3], [5, 3, -1], [-4, -3, -4], [3, -4, -3], [-2, -1, -1], [0, -6, -8], [-8, 0, -4], [-1, -5, -3], [0, 1, 3], [-4, 0, -4], [1, -3, -7], [4, 5, -6], [8, -4, -2], [1, 3, 3], [3, 7, 9], [-5, 2, 5], [2, 1, -1], [-9, -7, 1], [3, -6, -5], [-5, -3, -5], [0, 8, 2], [10, -10, -2], [12, 1, -2], [-3, -6, -4], [-6, 13, 0], [-15, -5, -1], [-7, -4, -7], [0, -3, 10], [6, 3, -8], [-8, 4, 2], [-2, 2, 2], [-1, 0, 0], [8, -4, -2], [9, -8, 5], [-1, -6, 8], [6, 1, -6], [-10, 5, 4], [13, 7, -7], [2, 10, 2], [-11, 1, -5], [12, 2, 10], [2, -10, 0], [7, -5, 1], [-2, 4, -4], [-1, -4, -4], [-15, 5, 7], [1, 12, 1], [-9, -1, 7], [1, -2, -7], [-13, 3, -3], [12, 3, 1], [9, -3, -9], [-8, -3, 10], [10, 12, 2], [-6, 2, 8], [-13, -3, 3], [-9, 9, 1], [-11, -8, -8], [-7, 3, 7], [-8, -4, -10], [-9, -6, 0], [2, -12, 10], [-9, 6, -13], [-28, -3, -7], [8, -10, 2], [1, 7, 1], [-11, 15, 5], [16, 8, 4], [26, -9, -3], [9, -17, -9], [-2, -13, 7], [0, -4, -10], [-9, 8, -2], [-19, -4, -9], [11, -11, 1], [7, 9, 19], [-17, 16, 1], [-12, -3, 3], [-2, -13, 5], [14, 4, 14], [-11, -3, -9], [-33, 0, -4], [31, -3, -3], [-1, -5, -5], [2, 2, -4], [-22, 17, 3], [-3, -1, -5], [-25, 15, 9], [-19, -8, 0], [-22, 16, 12], [21, -5, -5], [1, 15, 5], [17, -16, -8], [9, 1, -7], [-9, -3, 5], [2, 8, -8], [26, -8, -6], [22, -11, 1], [-5, 0, -12], [12, -2, -2], [-10, 13, -2], [3, 4, -5], [6, 9, 5], [-7, 1, 9], [-7, 9, -5], [22, 6, -8], [-15, -13, 7], [2, -11, 11], [9, 5, 1], [3, 9, -11], [-15, 7, -7], [-24, -8, 6], [-13, -6, -5], [11, 10, 0], [-2, -16, -2], [-9, -9, -13], [3, 3, -3], [-13, 14, 20], [17, -6, -5], [-47, -2, 0], [-25, 28, 9], [31, 1, 11], [23, -12, -11], [27, 0, 8], [-1, -11, -3], [-9, -15, -11], [4, -24, -16], [32, 1, -8], [23, -6, 1], [12, -5, -2], [11, 9, 1], [-4, -11, -13], [-7, 20, 15], [3, -7, -15], [-7, 20, 10], [-17, -15, -11], [-10, -3, 3], [43, -5, -1], [2, 0, -8], [15, 9, 15], [3, 19, -9], [3, -16, -4], [-10, 4, -16], [30, 4, -8], [-8, 8, 24], [14, 8, -10], [4, -26, 8], [14, -12, 16], [18, 8, -6], [3, 9, -1], [41, -3, -3], [-26, 0, -6], [-18, -6, -24], [-6, 10, 2], [14, 9, 10], [21, 5, -5], [27, -23, -3], [-14, 8, -6], [-32, 10, 8], [3, 17, 9], [22, -22, 4], [10, 3, 27], [14, -13, -23], [13, -5, -17], [49, 1, 7], [3, -27, -5], [0, 0, 24], [33, 2, -6], [-5, 11, -11], [-15, -1, -13], [24, 9, -5], [-7, -17, -9], [-13, 3, -7], [-21, 3, 5], [17, -23, -9], [32, -18, -16], [38, -9, 7], [-35, 9, -1], [-21, -2, -14], [-5, -5, 1], [3, -21, 13], [-1, -12, 6], [-4, -10, -10], [6, -11, 14], [-21, -6, -1], [-16, -3, 23], [37, 7, 23], [-5, -9, 13], [9, -15, 13], [-14, -27, 3], [-2, 16, 20], [33, -3, -9], [10, -3, -6], [25, -5, 15], [-6, 7, -17], [1, -7, 13], [-34, -4, -16], [-18, 12, -18], [-8, 30, 18], [28, 18, -8], [-41, -1, -5], [-26, 4, -8], [4, -2, -12], [3, 7, -7], [22, -16, -14], [-24, 3, -15], [-27, 6, -14], [-61, 10, 4], [22, -5, -12], [3, 16, -19], [-22, -2, -8], [-3, 7, 31], [29, -19, 11], [7, 11, 29], [3, -5, 25], [-11, 9, -23], [25, -3, -1], [-3, 11, -5], [-26, -16, 10], [-9, 2, 21], [-21, 27, -3], [-30, 17, -6], [-8, 28, 28], [-23, -9, -5], [-32, -9, 15], [21, -31, -19], [1, -32, -5], [-25, 35, 21], [-42, 9, 1], [-32, -17, -10], [19, 4, 29], [43, 12, -10], [20, 17, 16], [6, -2, 6], [16, -24, -22], [14, 22, 26], [-10, 3, -14], [-33, 21, 9], [8, 2, -30], [14, 6, -20], [10, 23, -10], [31, 10, 13], [-6, 25, 13], [-26, 10, -6], [44, 10, -2], [-12, 17, -4], [20, -8, 24], [29, 19, 5], [21, -11, -19], [-10, 8, 2], [5, 11, 5], [31, 17, 3], [-41, -2, 10], [-32, -5, -3], [29, -7, -13], [30, -25, -20], [-3, -11, -9], [30, 9, -5], [52, -4, -2], [22, 25, -7], [-8, 0, 2], [-41, 9, 17], [29, -46, -18], [7, 13, 19], [2, -21, 7], [-7, 29, 23], [23, 9, -13], [35, -24, -13], [-9, 27, 7], [56, -4, 0], [23, -2, 7], [14, -25, 13], [-21, 11, 27], [-73, 2, 0], [-19, 23, 17], [-13, 25, 5], [37, -19, -9], [-2, -14, 36], [-31, 19, 23], [13, -24, -13], [58, 12, 0], [-51, 27, 19], [-26, 22, 26], [12, -12, 28], [-21, 9, 27], [-1, -14, -12], [15, 4, -11], [1, -9, 37], [-8, -4, 8], [18, -22, -14], [12, -23, 10], [-4, 27, 6], [12, 14, -18], [19, 9, 17], [18, -8, -10], [16, -22, 14], [-17, 14, -16], [-40, -10, 6], [29, -10, 17], [26, -31, 0], [-1, -20, -17], [21, -4, 18], [-44, -2, 20], [8, 2, 18], [-4, -34, -2], [43, -34, -26], [27, 12, 35], [-8, 1, 9], [-1, 17, 15], [-10, -4, 6], [-2, 25, -19], [-6, -17, 22], [-5, -1, 17], [13, -39, -3], [4, 4, -18], [-34, 4, -8], [29, 7, 17], [-8, -6, -24], [16, 24, -22], [12, 22, 28], [34, -31, -13], [-44, -20, 0], [-2, -16, -32], [39, 18, 5], [25, 4, 12], [-35, 8, 25], [-40, 40, 24], [-15, 35, 21], [-1, -8, 0], [58, -17, -17], [47, -3, 9], [2, 15, 44], [20, -20, -26], [20, -7, 12], [-18, -4, 34], [-50, 8, 6], [-18, -11, 27], [-45, 5, -9], [5, 3, -11], [16, -30, 16], [-2, 3, -12], [-72, -6, -16], [-23, -15, -21], [50, 16, 8], [11, -41, 5], [10, -18, -6], [-14, 22, 20], [-67, -9, -5], [8, -14, -14], [-30, 22, 12], [-27, 13, -13], [-20, -12, -30], [-15, -25, 31], [11, 11, -7], [68, 8, 4], [-25, 24, 13], [-29, 31, -3], [43, -1, -5], [-1, 21, 5], [-31, 41, 15], [8, -14, -8], [39, 0, 27], [-11, -29, -5], [-31, -26, 19], [-8, 24, 30], [-32, -14, 8], [43, 13, 33], [-12, -4, -24], [-44, -7, 12], [-19, -1, 29], [6, -4, -22], [-28, -31, 17], [38, -14, 26], [3, -1, -23], [-3, 27, 29], [26, 20, -12], [25, -23, -9], [-38, -15, -22], [5, 3, 3], [42, -57, -29], [-1, 23, -25], [37, -7, -9], [6, -3, -17], [-36, 4, -10], [21, 38, -24], [-13, -12, -3], [-6, -16, 28], [5, 3, 29], [-12, 10, 10], [13, 28, 4], [-28, 0, 16], [38, 12, 28], [16, 28, 4], [1, -3, -25], [-24, -20, -14], [24, -28, 2], [-7, -18, -15], [-19, 5, -21], [13, -30, -21], [10, 12, 40], [3, -23, 13], [-32, 32, -8], [6, -28, 26], [-30, 14, 4], [40, 0, -22], [-3, -34, 9], [51, -13, 3], [-43, 43, 19], [6, -7, -13], [-5, -29, -19], [27, -27, -37], [67, -23, -21]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_8015_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_8015_2_a_g();". // 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_8015_2_a_g(:prec:=3) chi := MakeCharacter_8015_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_8015_2_a_g();". // 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_8015_2_a_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_8015_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![1, -3, -1, 1]>,<3,R![-13, -9, 1, 1]>],Snew); return Vf; end function;