// Make newform 8001.2.a.j in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_8001_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_8001_2_a_j();". // 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_8001_2_a_j();". // 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 := [-2, 0, 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_8001_a();" function MakeCharacter_8001_a() N := 8001; order := 1; char_gens := [3557, 1144, 7750]; 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_8001_a_Hecke(Kf) return MakeCharacter_8001_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], [0, -1], [1, 0], [0, 2], [-2, 0], [3, -2], [-4, 3], [0, 2], [3, 1], [4, 0], [3, -6], [3, -4], [-10, 0], [6, 2], [9, 1], [0, -5], [-8, -3], [0, -3], [-6, 1], [2, 9], [-11, 0], [-6, 4], [-12, -2], [7, -3], [0, -2], [-8, 6], [0, -1], [-4, -3], [-6, -4], [-1, 0], [15, -4], [-6, -8], [1, 3], [6, 0], [0, 3], [-6, -9], [3, -6], [-18, 1], [-6, 5], [6, 12], [-9, 3], [6, -11], [-2, -15], [-6, -7], [-8, -6], [-10, 6], [-7, 3], [-6, 10], [5, -15], [-3, 9], [-21, 5], [21, -3], [-9, -2], [-6, -1], [-6, 1], [3, 12], [6, 3], [-12, -6], [9, 11], [-7, 15], [0, -11], [-11, 3], [0, -3], [3, 9], [-6, -10], [2, -9], [-22, -3], [15, 5], [9, 15], [-6, 0], [21, -3], [-6, -6], [-8, 0], [12, 12], [-3, 4], [-12, 11], [-8, -12], [-9, 1], [-18, -6], [6, -18], [18, -3], [-6, 17], [-14, 3], [21, 9], [0, 5], [-36, -4], [-25, 0], [0, 11], [10, 12], [12, 18], [-3, -10], [26, -9], [0, 4], [-14, -9], [27, -4], [15, -14], [-12, -12], [-10, 21], [-8, 6], [-20, 0], [12, 8], [-24, -2], [-6, -10], [26, -3], [-26, 3], [-3, -16], [-12, 6], [-21, -11], [-7, 15], [4, 21], [-26, 12], [9, 17], [7, 3], [2, 0], [15, -3], [12, 21], [-18, 14], [-12, -3], [0, 8], [30, 9], [17, 18], [9, 20], [0, -18], [-17, 15], [-18, -14], [-9, 0], [27, -12], [-11, 9], [4, 27], [33, 6], [27, -17], [-4, -9], [3, -6], [-36, -5], [7, 15], [21, -16], [8, -9], [36, -8], [-6, -7], [-20, -12], [33, 3], [5, -12], [-9, -13], [20, 18], [-30, 15], [-11, -9], [-6, 20], [-17, -27], [-27, -1], [-31, 6], [-18, 21], [-2, 24], [-24, -4], [-1, 36], [18, -8], [7, 30], [36, -16], [-16, 6], [21, 6], [57, 3], [6, 39], [-2, -27], [12, 0], [-24, 8], [21, 4], [-14, 3], [28, -18], [11, 24], [48, -2], [-6, 0], [-9, 3], [42, 1], [4, -24], [-21, 3], [-54, 0], [-1, 0], [12, 10], [2, -36], [-9, 15], [-41, 6], [-18, -1], [4, -9], [0, -2], [-42, -7], [-18, -2], [-24, 3], [40, -18], [-13, 18], [33, -10], [-5, 15], [9, -15], [44, -9], [-3, 38], [-42, -11], [3, 4], [8, -3], [-1, 18], [-15, -4], [24, 5], [-6, -14], [-50, 12], [28, 12], [-23, -21], [-6, -7], [12, -21], [30, 3], [24, -26], [36, 15], [2, 24], [-20, -3], [27, 20], [24, 12], [-3, 36], [24, -4], [-8, 42], [18, -33], [6, 5], [21, 1], [-36, -19], [4, -9], [22, 27], [9, -13], [17, -18], [9, 22], [-39, -18], [0, 16], [-30, -11], [-22, 12], [-27, 5], [-6, 12], [-2, -36], [-38, 18], [-24, 11], [-7, 3], [12, 13], [-23, 15], [21, 13], [-9, 9], [-6, -16], [-63, -1], [40, -18], [56, 3], [-23, 30], [0, 14], [-9, 38], [1, -15], [-39, 20], [14, -33], [51, 7], [-45, 12], [0, 20], [-15, -31], [1, -27], [36, -16], [-18, 28], [-36, 15], [6, 24], [27, 26], [-14, 24], [-18, 9], [-54, -15], [-38, -30], [-34, 6], [36, -6], [8, 0], [-27, -31], [-39, 10], [-7, -30], [36, -2], [-33, -15], [24, 18], [14, 12], [46, -6], [9, -27], [-1, -39], [12, -27], [11, -48], [22, -18], [0, -23], [18, 10], [-50, 21], [-39, -20], [14, 0], [62, -12], [-60, -6], [8, -45], [42, -19], [33, -3], [-36, 3], [-3, -25], [54, 10], [6, -4], [-36, -20], [34, -27], [-12, -40], [-23, -15], [-9, -34], [9, -18], [-46, -15], [6, 33], [-6, -16], [5, -6], [18, -8], [48, 24], [-16, -9], [-6, -4], [-40, 3], [6, -8], [-60, 0], [12, -28], [-18, -10], [78, -6], [-44, -30], [-18, 44], [18, -30], [30, -38], [0, 46], [68, 6], [-42, 2], [-6, -30], [24, 18], [18, -38], [12, 27], [-18, -2], [22, 33], [-10, 54], [-12, 36], [21, 49], [36, -18], [-16, 0], [27, -19], [-69, 3], [-18, -16], [60, 6], [48, 6], [-16, 3], [60, -11], [-18, -42], [4, -9], [36, 15], [18, 31], [-60, 10], [21, -6], [-30, 20], [33, 13], [38, 12], [23, -30], [-24, -19], [-21, 28], [48, 0], [-16, 15], [27, -9], [21, -21], [47, -6], [24, 18], [21, 26], [30, 4], [18, -43], [45, -27], [-34, -3], [18, 55], [-60, -10], [-21, 37], [31, -33], [7, -60], [66, 12], [20, 6], [48, -15], [-39, 5], [-49, 33], [-3, -31], [-36, -14], [-44, 3], [20, 36], [12, -4], [27, -45], [-83, 9], [3, -18], [-6, -36], [-33, 24], [9, 29], [-33, -36], [36, -10], [59, -9], [-18, -1], [-40, -24], [-25, -21], [-50, 12], [-54, -15], [-1, 0], [60, 2], [-30, -8], [-14, 36], [-60, 30], [-6, 54], [30, -48], [3, 26], [50, 0], [42, -25], [42, -30], [45, 27], [-2, -6], [-6, 9], [-12, 26], [10, -36], [12, 30], [-33, 13], [-26, -15], [-30, 22], [33, 15], [27, 25], [-48, -10], [-20, -12], [-96, -3], [-42, -2], [-63, 15], [37, -3], [-66, 8], [-36, -33], [-42, 15], [-2, -24], [-18, 40], [-21, 32], [-36, -36], [15, 28], [24, 54], [-21, 53], [-50, -12], [84, -1]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_8001_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_8001_2_a_j();". // 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_8001_2_a_j(:prec:=2) chi := MakeCharacter_8001_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_8001_2_a_j();". // 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_8001_2_a_j( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_8001_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![-2, 0, 1]>,<5,R![-2, 0, 1]>],Snew); return Vf; end function;