// Make newform 4800.2.f.l in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4800_f();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4800_f_Hecke();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_4800_2_f_l();". // 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_4800_2_f_l();". // 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, 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_4800_f();" function MakeCharacter_4800_f() N := 4800; order := 2; char_gens := [4351, 901, 1601, 577]; v := [2, 2, 2, 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; // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4800_f_Hecke();" function MakeCharacter_4800_f_Hecke(Kf) N := 4800; order := 2; char_gens := [4351, 901, 1601, 577]; char_values := [[1, 0], [1, 0], [1, 0], [-1, 0]]; assert UnitGenerators(DirichletGroup(N)) eq char_gens; values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field F := Universe(values);// the Hecke field chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values); return chi; 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], [0, 3], [-2, 0], [0, -1], [0, -2], [-5, 0], [0, 6], [10, 0], [-3, 0], [0, 2], [-8, 0], [0, -1], [0, -2], [0, 4], [-10, 0], [-7, 0], [0, -3], [-8, 0], [0, -14], [0, 0], [0, -6], [0, 0], [0, -17], [-12, 0], [0, -4], [0, 12], [5, 0], [0, -4], [0, 8], [-12, 0], [0, 18], [20, 0], [10, 0], [7, 0], [0, -13], [0, -11], [0, -12], [0, -6], [-10, 0], [-17, 0], [22, 0], [0, 11], [0, -18], [5, 0], [13, 0], [0, -19], [0, -8], [-15, 0], [0, -24], [-20, 0], [-23, 0], [-12, 0], [0, -12], [0, 16], [-10, 0], [-8, 0], [0, -3], [-18, 0], [0, 9], [0, -6], [0, 7], [-18, 0], [0, 11], [0, -8], [-12, 0], [0, 23], [0, 2], [10, 0], [0, 6], [0, 0], [0, -27], [0, 29], [25, 0], [0, 36], [0, 0], [0, 7], [12, 0], [-5, 0], [-20, 0], [-22, 0], [-18, 0], [0, -29], [35, 0], [0, 24], [-20, 0], [0, -22], [-12, 0], [0, -24], [0, -38], [-30, 0], [0, 13], [8, 0], [-5, 0], [0, 16], [-10, 0], [22, 0], [0, -31], [3, 0], [0, -8], [0, 42], [0, -6], [-30, 0], [13, 0], [0, 13], [0, 12], [0, 16], [20, 0], [-13, 0], [0, 8], [0, 14], [0, -12], [-25, 0], [-23, 0], [12, 0], [0, -36], [0, 28], [0, 14], [-40, 0], [38, 0], [0, 6], [0, 42], [0, 24], [8, 0], [-22, 0], [25, 0], [-30, 0], [0, 43], [0, 34], [-20, 0], [0, -4], [-8, 0], [0, -23], [12, 0], [-35, 0], [0, 24], [0, 7], [0, 52], [20, 0], [-27, 0], [18, 0], [0, 41], [0, -28], [-30, 0], [-10, 0], [0, -51], [0, 28], [40, 0], [0, 16], [0, 27], [32, 0], [0, -41], [0, 18], [0, 12], [-58, 0], [-55, 0], [50, 0], [0, 33], [-22, 0], [0, -18], [0, 56], [0, -32], [-42, 0], [0, -2], [0, 36], [17, 0], [0, 42], [-5, 0], [0, 14], [-30, 0], [53, 0], [-18, 0], [0, 21], [5, 0], [0, 0], [-27, 0], [18, 0], [0, -19], [-5, 0], [0, -17], [8, 0], [0, -1], [0, 38], [0, -54], [-10, 0], [0, -63], [0, 29], [-5, 0], [2, 0], [0, 1], [0, 64], [3, 0], [58, 0], [0, 42], [0, 46], [22, 0], [0, -6], [0, -42], [0, 46], [0, 0], [12, 0], [0, 42], [-10, 0], [60, 0], [0, 42], [-35, 0], [0, -26], [-60, 0], [43, 0], [0, 13], [38, 0], [0, -39], [0, -28], [30, 0], [-58, 0], [0, -37], [42, 0], [0, 8], [0, -6], [-37, 0], [25, 0], [30, 0], [0, 16], [0, -8], [35, 0], [0, -54], [-10, 0], [0, 53], [-42, 0], [0, -21], [-20, 0], [47, 0], [-58, 0], [0, 4], [0, -72], [30, 0], [0, 44], [-30, 0], [12, 0], [0, -66], [3, 0], [0, -29], [15, 0], [0, 6], [10, 0], [0, 73], [-2, 0], [5, 0], [0, -34], [0, 22], [2, 0], [0, -12], [-45, 0], [0, 14], [-60, 0], [-37, 0], [0, -28], [0, 12], [0, -62], [0, 1], [0, 22], [-10, 0], [0, 69], [0, -42], [40, 0], [-70, 0], [-8, 0], [0, 4], [0, -26], [13, 0], [0, -8], [0, 26], [15, 0], [0, -57], [0, 41], [0, -18], [75, 0], [-33, 0], [48, 0], [0, 16], [-8, 0], [0, 38], [33, 0], [0, -33], [-78, 0], [0, 1], [0, -48], [40, 0], [0, 0], [-82, 0], [0, 52], [0, -84], [-72, 0], [0, -71], [0, 0], [77, 0], [0, 24], [-30, 0], [0, -28], [0, 6], [0, -28], [-25, 0], [0, 54], [-52, 0], [0, 43], [0, 12], [5, 0], [-60, 0], [0, -6], [0, -54], [60, 0], [12, 0], [0, 19], [0, -22], [-25, 0], [-30, 0], [12, 0], [0, -39], [-30, 0], [-32, 0], [0, 13], [28, 0], [0, 61], [0, 66], [-38, 0], [-85, 0], [0, 29], [0, 78], [0, -56], [3, 0], [0, 62], [-65, 0], [0, -46], [13, 0], [0, 52], [-25, 0], [0, -44], [-43, 0], [0, 48], [0, -31], [0, 8], [-30, 0], [-93, 0], [0, -86], [30, 0], [-77, 0], [0, -53], [12, 0], [0, -68], [-27, 0], [0, -42], [-22, 0], [0, -89], [-45, 0], [0, 6], [-40, 0], [78, 0], [0, -42], [0, -34], [0, -53], [-58, 0], [0, -42], [80, 0], [0, 92], [0, 91], [0, 72], [0, -84], [47, 0], [-12, 0], [-75, 0], [0, 26], [60, 0], [72, 0], [0, -23], [80, 0], [42, 0], [0, -49], [60, 0], [0, -37], [-22, 0], [0, 6], [0, 8], [0, 8], [60, 0], [0, -94], [-43, 0], [0, -58], [0, -36], [0, 78], [-45, 0], [0, 64], [20, 0], [0, 77], [2, 0], [0, -79], [-5, 0], [60, 0], [83, 0], [-12, 0], [10, 0], [0, 36], [0, -67], [0, 68], [-20, 0], [-8, 0], [0, 17], [-8, 0], [0, 69], [60, 0], [0, -9], [0, 12], [0, -26], [-27, 0], [0, -77], [-92, 0], [-10, 0], [0, -67], [0, -72], [0, -44], [40, 0], [0, 7], [0, 18], [-70, 0], [0, -34], [0, 32], [0, -16], [-60, 0], [-77, 0], [-50, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4800_f_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4800_2_f_l();". // 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_4800_2_f_l(:prec:=2) chi := MakeCharacter_4800_f(); 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_4800_2_f_l();". // 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_4800_2_f_l( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4800_f(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![9, 0, 1]>,<11,R![2, 1]>,<13,R![1, 0, 1]>,<19,R![5, 1]>,<23,R![36, 0, 1]>,<31,R![3, 1]>],Snew); return Vf; end function;