// Make newform 5472.2.a.bf in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_5472_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_5472_2_a_bf();". // 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_5472_2_a_bf();". // 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 := [-4, -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_5472_a();" function MakeCharacter_5472_a() N := 5472; order := 1; char_gens := [4447, 2053, 1217, 3745]; 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_5472_a_Hecke(Kf) return MakeCharacter_5472_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, 0], [0, 1], [3, 0], [2, -1], [1, -1], [5, -2], [1, 0], [-3, -1], [3, -3], [4, 2], [2, 2], [-4, 0], [8, -1], [-4, 3], [7, -1], [7, -1], [2, 5], [-3, 1], [-6, 4], [1, -4], [10, 0], [2, 6], [0, 6], [2, 2], [-10, 4], [-6, 8], [1, -7], [-1, -1], [-10, 2], [10, 2], [16, -1], [1, -6], [8, -1], [-8, -1], [-2, -8], [-18, 4], [-4, 8], [0, 0], [10, 4], [8, -4], [10, -4], [1, 0], [-2, 8], [4, -6], [7, -12], [13, 1], [14, -6], [-9, -3], [-10, -1], [2, 7], [-15, 0], [-4, -4], [2, -3], [-16, -4], [12, -9], [-2, 8], [-13, 9], [-4, -3], [16, -6], [26, 1], [-23, 3], [20, -6], [1, 2], [17, -7], [-19, 1], [7, -1], [0, -8], [2, 1], [6, 3], [11, 7], [-33, 2], [-20, 0], [3, -7], [-7, 9], [-14, -4], [-14, -5], [-8, 1], [-8, -6], [24, -2], [16, 4], [-3, 9], [26, 2], [14, 10], [12, -4], [6, -3], [10, 2], [-31, -2], [16, 7], [-8, 9], [-10, -3], [4, 4], [14, -8], [12, 8], [4, -3], [5, -1], [-6, 10], [-8, -6], [11, -17], [-38, 1], [4, 8], [0, 7], [20, 4], [-4, 12], [-8, 0], [-5, 4], [16, 7], [-2, -12], [-8, -14], [-20, 6], [-18, 2], [2, -17], [-14, -3], [-42, -2], [-12, -13], [22, -18], [-6, 13], [-47, 0], [4, -5], [-11, 17], [7, -11], [-16, -2], [5, -1], [28, -4], [-22, -9], [12, -10], [30, 8], [-3, -8], [7, 6], [-12, 6], [4, -1], [-28, 2], [-12, 0], [-2, 9], [5, 14], [-3, 6], [9, 9], [25, -1], [-5, 3], [-1, -2], [27, -15], [-12, 3], [5, 6], [-17, -1], [-25, 17], [-4, -10], [22, 4], [-28, 6], [-18, -7], [-38, 6], [-5, 3], [-42, -3], [18, -7], [-14, 20], [-13, -1], [4, 2], [9, -13], [27, 11], [-23, 4], [15, -13], [4, -16], [-2, 4], [-24, 0], [44, -8], [28, 10], [14, -10], [-8, -4], [12, -23], [-12, -2], [28, -9], [17, 3], [1, 5], [0, -20], [-45, -1], [4, -8], [6, -21], [0, 4], [-42, 9], [-8, 16], [-16, 11], [9, -4], [25, -9], [18, -12], [-20, 2], [1, -4], [6, 13], [-9, -5], [-8, -14], [22, 10], [29, 3], [-2, 6], [-20, 13], [-8, -10], [17, 3], [-52, 8], [4, 12], [1, 12], [8, -6], [25, -11], [-17, 24], [26, -14], [12, -16], [-23, 17], [20, 16], [-46, -6], [-24, -5], [-21, -6], [-41, 9], [27, 4], [-11, -15], [41, -7], [20, -14], [8, -13], [45, 7], [-32, 4], [26, 2], [7, -19], [-6, -6], [18, -8], [-2, 11], [-42, -4], [-54, 2], [62, -6], [15, 18], [-35, -15], [-2, -7], [-10, 20], [-30, 16], [26, -4], [10, -13], [-54, -7], [-51, 1], [-4, -14], [-6, 16], [18, 3], [43, 10], [-22, 11], [12, -7], [2, 10], [-28, -6], [12, 22], [-2, 23], [17, -13], [-47, 9], [-36, 16], [-16, 11], [-12, 13], [-28, 22], [44, -4], [13, -14], [10, -28], [-43, 8], [17, -25], [2, -12], [32, -11], [-30, -6], [-14, 11], [-12, 24], [2, 6], [-66, 4], [56, -8], [20, 6], [-16, 11], [-19, -7], [18, -5], [-7, 21], [-7, 13], [37, 0], [-37, 11], [14, 15], [-15, 5], [-49, -11], [1, 32], [17, 22], [-6, -8], [-11, 15], [-26, 7], [-63, -3], [64, -6], [4, 5], [-2, -16], [31, -11], [25, -26], [13, -15], [-8, 23], [-12, -23], [33, -22], [30, -14], [25, 10], [30, -6], [-38, 0], [-16, -15], [-34, 26], [-20, 2], [66, 4], [-20, -6], [-26, 34], [18, -17], [29, 5], [-30, 7], [58, -13], [5, 17], [3, 24], [27, -33], [40, -19], [-2, -20], [-25, 1], [19, 9], [-12, 13], [16, -22], [76, 1], [-30, -4], [22, 14], [4, 8], [24, 16], [-64, 10], [10, 9], [-22, -20], [-37, -4], [-26, 21], [-21, 5], [-33, 30], [-53, 3], [34, -20], [-35, 21], [26, -8], [-55, -13], [39, -19], [4, 0], [-16, -5], [0, 15], [45, -19], [-49, -3], [2, 7], [-10, 5], [20, 3], [27, -41], [-4, -16], [-1, 8], [69, -13], [19, 7], [-63, -2], [43, 17], [6, 24], [-25, -17], [20, 0], [-14, -8], [-5, -7], [14, 10], [-24, -17], [32, 12], [6, 6], [60, -7], [20, 18], [42, 0], [28, -8], [25, -13], [-40, 8], [-31, 10], [50, -6], [-4, 21], [10, 0], [38, 10], [-9, -17], [-38, 7], [-16, -8], [34, -40], [74, -16], [-70, -4], [38, 6], [3, -11], [37, -18], [-13, 5], [0, 3], [-12, 21], [-35, -13], [8, -24], [-70, -3], [19, 10], [36, 0], [38, -24], [58, -1], [-19, -8], [-10, -5], [29, -1], [74, -18], [65, -17], [16, 14], [38, 19], [48, 8], [38, -32], [-11, 17], [62, 14], [-22, 19], [-16, 2], [34, -12], [-62, -4], [32, 0], [76, 4], [-12, -6], [-18, -12], [34, 19], [18, -10], [-32, -18], [-38, -12], [7, -26], [26, -21], [36, -8], [-9, -15], [-44, 15], [-48, -18], [-58, 15], [-37, -25], [66, 11], [-47, -11], [-32, 26], [-74, 6], [-28, 42], [-15, 33], [-4, 34], [70, -12], [47, 9], [43, 0], [-59, -9], [-26, 14], [-43, 11], [51, -3], [5, 12], [48, -17], [-47, -10]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_5472_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_5472_2_a_bf();". // 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_5472_2_a_bf(:prec:=2) chi := MakeCharacter_5472_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_5472_2_a_bf();". // 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_5472_2_a_bf( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_5472_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-4, -1, 1]>,<7,R![-3, 1]>,<11,R![-2, -3, 1]>,<13,R![-4, -1, 1]>,<23,R![8, 7, 1]>],Snew); return Vf; end function;