// Make newform 5824.2.a.bk in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_5824_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_5824_2_a_bk();". // 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_5824_2_a_bk();". // 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_5824_a();" function MakeCharacter_5824_a() N := 5824; order := 1; char_gens := [2367, 1093, 4161, 4929]; v := [1, 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_5824_a_Hecke(Kf) return MakeCharacter_5824_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, 0], [0, 1], [-3, 1], [-1, 0], [0, 3], [1, 0], [0, 1], [-3, -3], [3, 2], [-3, 2], [1, -3], [2, -3], [6, 2], [-5, 0], [-3, 1], [3, -2], [6, -4], [-6, 0], [-6, -6], [6, 5], [-5, -3], [-7, -6], [9, 3], [3, -1], [-1, 9], [-6, -3], [8, 0], [-6, -10], [-4, -9], [-9, 8], [-2, 0], [0, -2], [-6, 1], [2, 3], [-12, -3], [14, -3], [8, -3], [0, 6], [3, -13], [-12, -9], [-9, 0], [6, 9], [-18, 2], [-6, -6], [-18, 4], [8, -3], [1, -12], [-5, 3], [24, 2], [-4, -18], [-9, -4], [-12, 6], [1, -15], [18, -1], [-18, -1], [15, -12], [-12, 2], [-20, 0], [-1, 6], [0, 11], [0, -6], [3, -13], [9, 3], [6, 1], [2, 15], [0, 8], [-18, 0], [-33, 0], [0, -4], [13, 9], [0, -6], [18, 7], [-6, -3], [0, -6], [28, -3], [12, 6], [-24, -4], [-3, 15], [12, -4], [-1, 3], [6, -19], [6, -9], [18, -4], [-8, -12], [26, 6], [-15, 8], [24, -2], [14, -15], [-6, -2], [0, 3], [6, -7], [-15, 15], [-14, 18], [-6, 16], [-26, 9], [6, 16], [15, -7], [-12, 4], [14, 12], [-14, 15], [27, 6], [-12, 0], [-12, 4], [-9, 10], [17, 18], [10, -12], [9, -11], [15, -11], [9, 14], [-22, -12], [14, 9], [16, 0], [18, -4], [-16, 12], [-2, 0], [21, -4], [-4, -6], [18, -23], [-6, -8], [-21, 4], [15, -3], [7, -24], [6, 25], [-24, 2], [-1, 21], [-9, 14], [12, -9], [6, -3], [-42, 0], [-13, 21], [12, -21], [24, 4], [-7, -6], [-13, -6], [39, -3], [31, 15], [30, 2], [-29, 3], [30, -4], [9, 6], [12, -24], [6, 10], [-2, -6], [-24, -18], [-38, -9], [36, -2], [-27, -9], [-18, -26], [14, 12], [0, -20], [6, 3], [-18, 22], [2, 0], [12, -22], [-15, 12], [27, 12], [-24, 0], [27, 7], [24, 15], [-33, -5], [18, -2], [-3, 10], [12, -21], [18, 18], [0, 17], [-9, 17], [-18, 12], [20, -6], [12, 15], [12, 13], [12, 7], [-10, 6], [-15, 24], [22, -27], [-16, -18], [-24, 20], [-16, -6], [-42, -10], [-22, 9], [-10, 15], [8, 6], [18, -15], [-4, 18], [-3, 25], [0, -17], [6, -13], [43, -12], [-17, -21], [2, 30], [15, 25], [-16, 9], [-30, 22], [-47, -12], [9, -23], [15, 22], [-12, -3], [-16, -21], [-33, -24], [-15, 9], [-42, 2], [-18, 5], [-32, 15], [17, 3], [8, -36], [-21, -7], [-12, 30], [49, 9], [27, 8], [30, -7], [-22, 24], [19, 24], [36, 19], [40, 0], [12, 30], [-21, -29], [-5, -27], [-11, -30], [-12, -10], [12, -2], [18, 9], [-19, 18], [-6, 12], [0, -22], [30, -24], [-30, -10], [3, -24], [0, -16], [27, 20], [66, 3], [-24, 19], [24, 12], [-24, 21], [40, 0], [12, -28], [-18, -6], [-15, 29], [-6, -18], [-12, 32], [-33, 28], [-42, -14], [48, 12], [2, -6], [-16, 24], [-26, 18], [45, 1], [0, -19], [-43, 9], [-9, 19], [-22, -9], [9, -22], [16, -36], [-30, 2], [-6, 2], [2, -18], [6, -20], [12, -18], [-11, 12], [60, 0], [-24, 6], [-11, 27], [11, -18], [-3, -32], [-19, 9], [14, -21], [18, 28], [10, -36], [30, 34], [-9, -3], [8, -33], [24, 26], [-50, -3], [10, 39], [6, 24], [13, -18], [-68, 3], [20, -24], [-30, 3], [2, 21], [-24, 12], [36, 24], [-6, 18], [-30, 36], [-12, 24], [17, 15], [50, 24], [33, -16], [-25, -24], [12, 0], [-8, -6], [-48, 5], [-9, -34], [36, -22], [30, 16], [-27, 19], [-7, -18], [-48, 4], [-2, 6], [-12, -4], [-24, 25], [-17, -15], [6, 21], [-6, -43], [3, 12], [33, -6], [3, 0], [28, -3], [6, -2], [24, 15], [42, -20], [23, -24], [-36, -3], [-48, 9], [36, 32], [-39, 18], [54, 7], [-20, -36], [39, 13], [-24, 31], [-49, 21], [-3, 10], [4, 42], [32, -30], [-54, 17], [-32, 18], [0, -32], [-56, -9], [-4, 0], [17, 27], [-6, -8], [-48, -20], [-4, 51], [75, 8], [34, -33], [27, -11], [16, 27], [-15, 51], [-38, 9], [-54, 22], [8, 54], [-14, 30], [-26, -21], [12, 22], [12, 18], [-26, -36], [6, 3], [30, 0], [2, -33], [-42, 27], [27, 37], [-18, 3], [-11, -15], [-48, 15], [-54, -1], [-22, 36], [4, -42], [-6, -51], [27, -17], [-15, 31], [-24, -2], [12, -21], [-62, -6], [-42, -2], [-39, -30], [-6, 25], [34, 45], [11, 12], [-63, 7], [-4, -9], [-47, 0], [42, 10], [30, -27], [-54, -31], [-15, 12], [32, -9], [-3, 0], [-12, -10], [-42, 10], [16, 18], [-48, -18], [-14, 0], [9, 23], [18, -46], [-78, -9], [-30, -7], [7, -63], [-21, -15], [-14, -3], [8, 60], [-26, 9], [12, 11], [-38, -18], [15, -7], [-24, 2], [-28, -30], [-60, -12], [55, 12], [-12, -12], [6, -12], [9, 0], [0, 18], [-21, -33], [-39, 14], [-52, -21], [-39, 13], [-45, -23], [-12, 33], [48, 6], [18, -53], [23, -3], [27, 7], [40, -12], [33, -16], [-9, 8], [-6, 18], [-7, -66], [-12, -23], [-24, 42], [-20, 18], [-54, -4], [0, 9], [27, 48], [-8, 18], [-54, 13], [0, 35], [-34, 12], [15, 11], [-33, 16], [-24, -18], [59, 9], [42, 20]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_5824_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_5824_2_a_bk();". // 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_5824_2_a_bk(:prec:=2) chi := MakeCharacter_5824_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_5824_2_a_bk();". // 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_5824_2_a_bk( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_5824_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-2, 0, 1]>,<5,R![7, 6, 1]>,<11,R![-18, 0, 1]>,<17,R![-2, 0, 1]>,<19,R![-9, 6, 1]>],Snew); return Vf; end function;