// Make newform 4400.2.a.bu in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4400_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_4400_2_a_bu();". // 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_4400_2_a_bu();". // 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 := [-5, -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_4400_a();" function MakeCharacter_4400_a() N := 4400; order := 1; char_gens := [2751, 3301, 177, 1201]; 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_4400_a_Hecke(Kf) return MakeCharacter_4400_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], [0, 0], [3, -1], [1, 0], [-1, 0], [-2, 1], [-1, -2], [1, 1], [4, 1], [3, 2], [3, -2], [-7, 2], [10, 0], [7, -2], [3, 3], [5, 2], [7, -1], [4, 0], [-6, 6], [-5, -1], [-4, 7], [-4, 5], [2, -1], [-9, 1], [-9, -3], [8, -5], [-3, -6], [-2, -1], [7, -2], [-8, -3], [9, -3], [8, -7], [7, -6], [-8, 4], [-5, 0], [6, 4], [13, 3], [11, 2], [3, -6], [3, -9], [5, -3], [-20, 1], [4, -4], [7, 7], [10, -3], [-5, 0], [15, 2], [8, -7], [-16, 3], [13, 1], [5, -1], [2, -9], [-8, 7], [18, 0], [-13, -4], [10, -11], [1, -6], [-3, -2], [3, -6], [0, 11], [-14, 4], [30, -1], [17, -4], [-10, 6], [0, -3], [5, -8], [-22, 0], [-21, 3], [-9, -2], [7, 4], [6, 0], [-17, 3], [3, -8], [31, 0], [18, 0], [-12, 0], [10, -1], [-26, 4], [17, 0], [-27, 0], [17, 3], [17, -10], [9, -8], [2, -5], [-28, 2], [29, -1], [5, 3], [9, -12], [19, -6], [-5, -2], [11, -4], [13, 6], [13, -2], [-1, -5], [-22, 2], [-15, 9], [-4, 2], [14, 4], [-19, 3], [1, -3], [-1, -16], [15, 9], [5, 5], [9, -13], [-2, 5], [-2, -5], [-17, -2], [-14, -11], [9, 7], [34, 0], [-14, 17], [-12, -6], [-23, 6], [7, -9], [-35, 4], [10, 12], [1, -8], [-4, 11], [-20, 7], [25, 2], [-11, -10], [4, -2], [-7, -10], [7, 9], [32, -4], [-4, 12], [27, -12], [-2, 9], [-19, 18], [-17, -9], [-37, -1], [-16, 1], [15, -8], [8, 8], [-25, -6], [29, -1], [1, 6], [-2, 13], [-12, -12], [11, 16], [4, 16], [-8, 0], [16, -2], [-24, -5], [-7, 11], [-22, 9], [3, -12], [30, -10], [5, 8], [28, -16], [27, -9], [-15, -4], [17, 8], [12, -16], [-10, 8], [-40, 4], [-3, 6], [24, 4], [-11, -14], [-23, 10], [-42, 0], [-24, -7], [-5, -5], [-9, -6], [4, 16], [-14, -15], [32, 3], [-37, 0], [-24, 18], [-35, 1], [15, 7], [36, 3], [-7, 9], [-11, -12], [6, 12], [-4, 4], [43, 4], [-26, 6], [12, -8], [28, -6], [7, -11], [35, -6], [-18, 0], [-47, -2], [1, 4], [38, -6], [-41, -9], [-25, 18], [-52, -4], [-36, -5], [36, 0], [-25, 10], [10, -5], [-41, 7], [-3, -6], [-16, -6], [1, -22], [-13, 5], [8, -7], [-19, -13], [-9, 11], [12, 1], [4, -13], [38, 8], [37, 1], [20, -8], [-1, -16], [2, 8], [19, 3], [-13, 12], [-9, 6], [31, -3], [-20, 10], [-35, -2], [-22, 6], [-12, 20], [30, -15], [-33, -9], [-49, 5], [-48, -5], [-11, 0], [4, -8], [-2, -3], [-15, 6], [-13, 12], [-9, 21], [-12, 6], [-8, 12], [-10, 29], [11, 9], [-35, 12], [-3, 8], [3, 12], [-10, -5], [-3, 12], [30, -2], [-47, 1], [-28, -1], [-41, 1], [41, -16], [10, 6], [-10, -2], [23, 0], [-6, -18], [-3, -9], [25, 3], [-7, 5], [-32, 3], [-32, -2], [44, -6], [1, -2], [38, -10], [-37, 0], [-21, -3], [34, 10], [-19, -18], [61, 0], [49, 1], [-10, 12], [-8, 12], [18, -6], [25, 14], [4, -25], [13, -14], [-14, 6], [-15, -24], [-63, 3], [7, -18], [-45, -12], [32, 18], [-51, 8], [8, 18], [-5, 18], [-14, 17], [15, 14], [12, 0], [-4, 3], [-29, 17], [-35, 16], [-2, 19], [52, 0], [-19, -19], [23, 12], [13, -21], [20, -28], [-16, 15], [-33, -9], [-19, -11], [-10, 23], [-47, -8], [17, -22], [48, -12], [-6, 9], [35, 0], [41, -19], [15, 2], [-63, 6], [34, 4], [-31, 4], [39, -2], [-6, -12], [-33, -7], [1, 10], [-38, 15], [13, -10], [-8, 10], [-5, -19], [4, 7], [57, 4], [34, 7], [2, 11], [7, 7], [-17, 18], [-53, 10], [-14, -10], [22, 4], [-18, 6], [-48, 22], [27, -15], [47, -5], [17, 21], [57, 6], [47, -20], [14, -25], [58, -16], [-44, 15], [39, 5], [-28, 26], [-22, 8], [2, -6], [-55, 8], [4, 3], [-11, -2], [-50, 6], [54, 0], [-24, 34], [-46, -10], [-29, 5], [-41, 21], [74, 6], [29, 20], [-25, 26], [-3, 17], [61, -8], [82, -5], [-26, 11], [5, 28], [10, -29], [71, 8], [-11, 3], [15, -23], [21, -18], [-32, -21], [8, 0], [3, -12], [-14, 4], [-65, 13], [-57, 18], [-36, 0], [1, 17], [-47, 22], [13, -8], [-76, 11], [15, -7], [5, -6], [24, -30], [-29, -6], [-28, 24], [52, 7], [68, -8], [33, -15], [-42, -15], [46, -9], [3, 16], [15, -9], [-27, 21], [-10, -24], [-1, 11], [73, -1], [-12, 30], [-97, 2], [2, -14], [-6, 21], [-33, -4], [-48, -9], [-11, -9], [-40, -12], [-5, 12], [50, 8], [43, -7], [-8, -32], [31, -14], [16, -30], [-15, -24], [11, -18], [64, -9], [-6, 18], [-7, -2], [29, -7], [-4, 24], [23, 11], [25, -3], [-65, 4], [81, -6], [49, -12], [-26, 14], [25, 16], [28, 0], [-56, -14], [2, -3], [-13, 32], [5, 20], [55, -3], [-22, 0], [1, -2], [39, 0], [92, -5], [27, -18], [-72, 3], [45, 6], [13, 26], [3, 24], [-41, 16], [14, -18], [-36, -12], [-14, -20], [14, -25], [38, -32], [0, 27]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4400_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4400_2_a_bu();". // 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_4400_2_a_bu(:prec:=2) chi := MakeCharacter_4400_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_4400_2_a_bu();". // 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_4400_2_a_bu( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4400_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-5, -1, 1]>,<7,R![1, -5, 1]>,<13,R![1, 1]>],Snew); return Vf; end function;