// Make newform 8281.2.a.ba in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_8281_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_8281_2_a_ba();". // 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_8281_2_a_ba();". // 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, 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_8281_a();" function MakeCharacter_8281_a() N := 8281; order := 1; char_gens := [1522, 3382]; v := [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_8281_a_Hecke(Kf) return MakeCharacter_8281_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, 1], [-1, 2], [-1, 2], [0, 0], [3, 0], [0, 0], [5, -4], [3, 0], [-5, -2], [-2, 4], [5, 0], [5, -6], [2, -4], [-8, 0], [-1, -4], [-1, -4], [5, -4], [-3, 0], [3, 0], [-4, 8], [7, -6], [7, -6], [0, 0], [-1, 2], [-10, 12], [9, 0], [-7, 6], [-13, 2], [-7, 6], [-10, 8], [8, -12], [5, 2], [5, 2], [4, 12], [9, -6], [-1, -12], [-7, 0], [-3, 12], [16, 4], [-3, 12], [9, -18], [6, 12], [-5, 10], [9, -6], [22, -8], [11, 6], [4, 0], [-4, 0], [-7, 8], [-5, 18], [-1, 8], [0, -12], [-5, 6], [-8, 4], [13, -8], [-3, 18], [-7, -4], [-1, -12], [19, -12], [-14, -8], [3, -18], [-2, -8], [0, 12], [31, -2], [25, -12], [5, 2], [21, -12], [18, 0], [-19, 26], [-10, -12], [27, -6], [1, -8], [3, 6], [-21, 12], [-4, 12], [15, 0], [-5, 4], [3, -18], [-11, -2], [-1, 6], [-20, -8], [6, -12], [-1, 20], [-10, -12], [5, -18], [-1, 2], [2, 20], [-23, 18], [2, 20], [24, 0], [13, -14], [-15, 24], [-17, 24], [-32, 4], [-7, -12], [-8, -8], [-17, -2], [5, 8], [-23, -6], [-11, 6], [-24, 12], [9, 6], [23, -22], [19, 4], [-7, -18], [-29, 6], [-32, 16], [31, -2], [25, -26], [22, 0], [5, -18], [-11, 18], [-2, 4], [17, 0], [16, -24], [-1, 8], [-4, 24], [-17, -2], [11, -4], [4, -8], [3, -6], [10, -12], [-23, 16], [7, 4], [-13, -24], [10, -20], [21, 18], [-15, -6], [0, 24], [19, 6], [-21, 24], [-20, -8], [19, -18], [-14, -24], [15, -30], [46, 0], [-5, -26], [1, 12], [-22, 8], [3, 12], [24, -24], [-35, -2], [5, 18], [12, -24], [25, 0], [20, -16], [6, -36], [-31, 20], [-11, -6], [-11, -8], [-3, -18], [2, -28], [20, 0], [17, 2], [9, 6], [24, -36], [-9, 18], [-41, 10], [-6, -12], [47, 2], [29, -4], [10, -32], [-36, 12], [21, -6], [25, -14], [1, 4], [21, 6], [-7, -12], [30, -12], [-35, 28], [35, -4], [-38, 36], [27, -30], [17, -30], [-1, -30], [10, -8], [16, 0], [17, 26], [20, 24], [5, -24], [29, -12], [12, 0], [-14, 36], [-41, -2], [43, -32], [5, -16], [-25, 0], [-35, 36], [-23, 42], [53, -4], [-11, 0], [8, 20], [31, 18], [11, 14], [-35, -2], [-15, -12], [-23, -6], [-45, 12], [30, -12], [-7, 2], [17, 2], [4, -24], [19, 6], [-23, 36], [-16, 8], [33, -12], [-1, 12], [-25, -22], [10, -20], [31, -6], [43, 12], [-38, 40], [-4, 0], [15, -24], [53, -4], [-17, -6], [-7, -18], [13, 16], [37, -44], [-54, 24], [7, -24], [-24, 12], [1, 10], [-27, 48], [-20, 4], [-42, 24], [13, 28], [25, 10], [9, 18], [17, -16], [3, 12], [-9, 6], [4, -24], [-1, 32], [36, -24], [37, -20], [-21, 18], [-7, -10], [-44, 28], [52, -20], [39, -48], [37, 18], [-13, 54], [13, 18], [10, -8], [41, -46], [4, -12], [25, 4], [31, -12], [-8, -32], [34, -12], [-37, 14], [-25, 20], [-74, 0], [13, 28], [21, -24], [-1, 48], [-3, -12], [18, -48], [7, -18], [73, -6], [28, -32], [31, -30], [-22, 12], [31, -50], [-55, 18], [46, -32], [30, -36], [16, 12], [-5, -8], [25, -24], [23, -48], [-45, -6], [-13, 18], [62, -12], [43, 12], [-39, 72], [65, -6], [-39, 6], [-3, -6], [27, -30], [27, -48], [24, 12], [-14, -24], [23, -48], [19, -26], [35, 12], [30, -12], [-55, 0], [-26, 4], [31, -8], [-17, -26], [-7, 50], [52, -20], [-50, 24], [-11, 40], [61, -6], [-70, 8], [41, -22], [-44, 12], [-51, 24], [5, 26], [-3, -42], [-48, 0], [15, 6], [2, -36], [-71, 10], [18, -36], [61, -32], [19, -48], [-3, -42], [57, -30], [-25, -40], [-27, -6], [-20, 4], [53, 0], [16, -44], [-53, 28], [38, -60], [10, -32], [-37, 18], [13, -30], [26, -16], [40, -36], [-55, 38], [5, -48], [-23, 36], [-1, 0], [35, -34], [74, -16], [41, -18], [-65, 28], [16, -36], [1, 4], [-25, 0], [24, -24], [22, 12], [-53, -14], [-46, 0], [-11, -18], [-11, -6], [34, -32], [-6, 36], [32, 0], [-3, -6], [36, 24], [5, -48], [21, 12], [-4, 56], [57, -48], [7, 24], [9, -18], [-58, 44], [-1, 18], [-73, 12], [-30, 36], [83, -4], [25, 16], [43, 16], [0, -24], [34, 36], [-55, 44], [-3, -6], [-79, 8], [11, -18], [-1, 24], [50, -4], [-37, 12], [70, -36], [49, -38], [81, 6], [-19, -40], [-54, 60], [-13, 18], [3, 48], [-19, 20], [8, 8], [-5, -42], [-27, 48], [14, -24], [-65, -14], [94, 4], [-36, 0], [-47, -2], [-44, 48], [69, -12], [-5, -24], [29, -12], [-43, 0], [-28, 56], [-46, 60], [-41, 22], [43, 16], [-11, 6], [33, 36], [-17, -12], [69, -24], [6, 0], [12, -12], [57, -42], [-57, -6], [-1, 32], [-43, -36], [31, -38], [71, -10], [21, 18], [9, 18], [-46, -16], [-43, 24], [79, -20], [-67, 48], [-5, -44], [-28, -40], [75, 18], [-58, 24], [-47, 4], [21, -36], [7, 30], [-10, -4], [21, -66], [27, 24], [-49, 30], [-20, -8], [16, 40], [-10, -48], [-41, 22], [-29, -2], [42, -24], [53, -60], [47, 2]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_8281_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_8281_2_a_ba();". // 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_8281_2_a_ba(:prec:=2) chi := MakeCharacter_8281_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_8281_2_a_ba();". // 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_8281_2_a_ba( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_8281_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![1, -3, 1]>,<3,R![-5, 0, 1]>,<5,R![-5, 0, 1]>,<11,R![-3, 1]>,<17,R![-11, -6, 1]>],Snew); return Vf; end function;