// Make newform 1050.2.i.s in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1050_i();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1050_i_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_1050_2_i_s();". // 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_1050_2_i_s();". // 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, -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_1050_i();" function MakeCharacter_1050_i() N := 1050; order := 3; char_gens := [701, 127, 451]; v := [3, 3, 2]; // 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_1050_i_Hecke();" function MakeCharacter_1050_i_Hecke(Kf) N := 1050; order := 3; char_gens := [701, 127, 451]; char_values := [[1, 0], [1, 0], [0, -1]]; 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, 1], [1, -1], [0, 0], [-3, 2], [1, -1], [-7, 0], [-4, 4], [0, -1], [0, 1], [-8, 0], [-6, 6], [0, -3], [9, 0], [4, 0], [0, -3], [-1, 1], [-12, 12], [0, 4], [12, -12], [-14, 0], [-14, 14], [0, -4], [-12, 0], [0, 2], [16, 0], [0, 0], [0, 16], [0, -18], [10, -10], [6, 0], [-5, 0], [0, 13], [-2, 2], [-4, 0], [0, 4], [2, -2], [15, -15], [0, 8], [5, 0], [0, -21], [-13, 13], [-12, 0], [0, 10], [26, -26], [3, 0], [12, -12], [-15, 0], [4, 0], [-20, 20], [0, 22], [0, 26], [-6, 0], [-7, 7], [-3, 0], [0, -8], [-16, 16], [0, 0], [0, -16], [-2, 2], [3, 0], [-2, 2], [-9, 0], [-8, 0], [-16, 16], [0, -24], [0, 10], [0, 9], [0, 0], [-34, 34], [-28, 0], [-8, 8], [0, -36], [19, -19], [0, 26], [1, 0], [0, 13], [14, -14], [0, 18], [0, 17], [-10, 10], [11, 0], [-14, 0], [12, -12], [-40, 0], [0, -16], [0, -36], [-25, 0], [0, 10], [-28, 0], [-33, 0], [0, 12], [-26, 26], [-8, 8], [-12, 0], [0, -24], [-28, 0], [0, -30], [21, -21], [0, -14], [0, -2], [-36, 0], [45, -45], [14, -14], [0, 37], [8, -8], [-14, 14], [-42, 0], [0, 12], [-6, 6], [22, 0], [0, 25], [-15, 15], [-8, 0], [7, -7], [14, 0], [23, -23], [-26, 0], [-15, 15], [0, 29], [-12, 0], [-8, 8], [12, 0], [0, 1], [12, -12], [0, -12], [6, 0], [0, -4], [0, -26], [-17, 0], [0, 37], [-41, 41], [9, 0], [0, 26], [2, 0], [0, 17], [-29, 0], [43, -43], [-22, 22], [6, 0], [-53, 53], [37, 0], [0, -34], [48, -48], [-10, 0], [-12, 12], [44, 0], [-1, 0], [-22, 22], [0, 20], [0, 29], [0, 23], [-25, 0], [58, 0], [0, 20], [46, -46], [-58, 0], [0, 44], [0, -31], [-16, 0], [30, -30], [0, 46], [-44, 0], [20, 0], [0, -43], [18, -18], [-33, 33], [-10, 10], [10, -10], [-29, 0], [0, 39], [20, -20], [22, 0], [0, 2], [-44, 44], [16, -16], [30, 0], [37, 0], [-30, 30], [15, 0], [0, 22], [0, 3], [-9, 0], [47, 0], [0, -2], [-39, 39], [-12, 12], [23, -23], [24, -24], [0, 31], [-30, 30], [0, 2], [-2, 0], [0, -15], [0, 54], [-32, 32], [36, -36], [-5, 5], [0, 26], [38, 0], [0, 12], [0, 0], [-2, 0], [0, -22], [35, -35], [0, 0], [17, -17], [0, 56], [0, -24], [-5, 0], [37, -37], [0, -52], [66, 0], [-40, 0], [0, 10], [16, -16], [0, 41], [13, -13], [33, -33], [0, 35], [51, 0], [0, 4], [-6, 0], [0, 51], [0, -59], [-8, 0], [-6, 0], [0, -6], [-36, 36], [0, -53], [0, -68], [19, -19], [23, -23], [22, 0], [-19, 19], [30, 0], [-64, 64], [0, -18], [0, -45], [-11, 0], [2, 0], [-60, 60], [0, -67], [-24, 24], [0, -38], [54, 0], [0, 6], [51, 0], [41, -41], [28, -28], [-29, 0], [50, 0], [0, -17], [-19, 19], [-57, 0], [-2, 2], [0, -1], [38, -38], [-26, 26], [-23, 0], [0, -52], [-59, 59], [66, 0], [-14, 14], [-14, 0], [-24, 24], [0, 5], [-20, 0], [-3, 0], [22, 0], [-3, 3], [0, -40], [22, -22], [14, -14], [0, -32], [-58, 0], [0, 43], [0, -20], [-44, 44], [0, 47], [0, 11], [55, -55], [-44, 44], [-9, 0], [72, 0], [0, 50], [0, -4], [56, -56], [-25, 0], [-20, 20], [-31, 0], [-40, 40], [10, -10], [0, -36], [-53, 0], [-9, 0], [-48, 48], [0, -18], [-70, 0], [0, 71], [46, 0], [0, 4], [0, 73], [18, -18]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1050_i_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1050_2_i_s();". // 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_1050_2_i_s(:prec:=2) chi := MakeCharacter_1050_i(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 2)); S := BaseChange(S, Kf); maxprec := NextPrime(1999) - 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_1050_2_i_s();". // 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_1050_2_i_s( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1050_i(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![1, -1, 1]>,<13,R![7, 1]>],Snew); return Vf; end function;