// Make newform 2106.2.e.ba in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2106_e();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2106_e_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_2106_2_e_ba();". // 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_2106_2_e_ba();". // 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_2106_e();" function MakeCharacter_2106_e() N := 2106; order := 3; char_gens := [1379, 1783]; v := [2, 3]; // 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_2106_e_Hecke();" function MakeCharacter_2106_e_Hecke(Kf) N := 2106; order := 3; char_gens := [1379, 1783]; char_values := [[0, -1], [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 := [[1, -1], [0, 0], [0, 3], [1, -1], [-6, 6], [0, -1], [-3, 0], [2, 0], [0, 0], [-6, 6], [0, 4], [-7, 0], [0, 0], [1, -1], [-3, 3], [0, 0], [0, 6], [-8, 8], [0, -14], [-3, 0], [2, 0], [-8, 8], [-12, 12], [-6, 0], [10, -10], [12, -12], [0, 4], [12, 0], [-7, 0], [0, 6], [20, 0], [0, 21], [0, 0], [0, 13], [0, 6], [-17, 17], [0, -14], [-16, 0], [0, 0], [0, 0], [3, 0], [20, 0], [18, -18], [0, 4], [3, 0], [2, 0], [0, 13], [19, -19], [0, 0], [0, 13], [-27, 0], [0, -15], [10, -10], [24, 0], [0, -9], [12, -12], [24, 0], [11, 0], [28, -28], [6, -6], [0, 4], [0, -21], [2, 0], [0, 30], [1, -1], [6, -6], [-8, 8], [0, -23], [0, -3], [19, -19], [-24, 24], [0, 0], [-26, 26], [0, 4], [20, 0], [0, -21], [6, -6], [-34, 0], [0, -36], [0, -32], [0, -9], [-17, 17], [-33, 0], [-25, 0], [-26, 26], [-21, 21], [6, 0], [10, -10], [-9, 9], [0, 40], [36, 0], [21, -21], [-16, 0], [0, 9], [0, 40], [-30, 0], [0, 18], [-9, 0], [20, 0], [11, 0], [-17, 17], [3, 0], [0, -39], [-15, 15], [0, -5], [38, 0], [-24, 24], [18, 0], [0, -6], [19, -19], [0, -14], [38, 0], [0, 24], [28, -28], [29, 0], [18, -18], [0, -14], [-6, 0], [0, 0], [36, -36], [0, 22], [19, -19], [-48, 48], [24, 0], [-8, 8], [0, 0], [-26, 26], [6, 0], [10, -10], [0, -23], [20, 0], [0, 9], [0, 40], [-16, 0], [0, 6], [0, -32], [-39, 0], [0, 40], [0, 42], [-33, 0], [20, 0], [3, -3], [0, -14], [18, 0], [38, 0], [0, 0], [37, -37], [42, -42], [0, 4], [-45, 0], [0, 13], [21, 0], [29, 0], [0, 0], [37, -37], [-30, 30], [-16, 0], [-36, 36], [-34, 0], [0, 21], [-6, 6], [15, 0], [0, 31], [-3, 0], [0, 54], [-39, 39], [2, 0], [46, -46], [38, 0], [0, -6], [-30, 30], [0, -23], [0, 24], [-8, 8], [0, -14], [0, 33], [-44, 44], [-54, 0], [20, 0], [-8, 8], [-8, 8], [-12, 12], [0, 22], [-18, 0], [0, 36], [-54, 54], [38, 0], [10, -10], [0, 4], [24, 0], [29, 0], [-36, 36], [65, 0], [30, -30], [-3, 0], [0, 54], [0, 40], [-44, 44], [-48, 48], [-54, 0], [0, -21], [-8, 8], [0, 13], [-17, 17], [-24, 0], [-6, 0], [-7, 0], [0, -12], [-42, 42], [0, 31], [38, 0], [0, -6], [64, -64], [18, -18], [0, -57], [28, -28], [0, 4], [27, -27], [12, 0], [0, 21], [0, 4], [0, -41], [0, -60], [-7, 0], [0, -9], [46, -46], [-15, 15], [-12, 0], [-62, 62], [-30, 30], [0, 58], [-52, 0], [0, -32], [0, -51], [-53, 53], [33, -33], [0, 22], [-33, 0], [0, -48], [30, 0], [66, -66], [-25, 0], [0, -32], [-61, 0], [0, 42], [-12, 12], [-43, 0], [0, 36], [0, 22], [66, 0], [0, -41], [-6, 0], [0, 3], [1, -1], [-48, 48], [-30, 0], [-16, 0], [46, -46], [-36, 0], [-34, 0], [-44, 44], [21, -21], [0, 49], [2, 0], [0, 24], [-35, 35], [-39, 0], [24, -24], [0, 4], [0, -12], [0, -68], [2, 0], [-26, 26], [0, -32], [0, -14], [-16, 0], [0, 6], [-62, 62], [20, 0], [63, -63], [0, -36], [0, 67], [42, -42], [-26, 26], [0, -32], [18, 0], [65, 0], [0, 57], [-71, 71], [42, 0], [12, -12], [15, 0], [0, 0], [0, 42], [-26, 26], [0, -60], [46, -46], [12, -12], [-15, 0], [82, -82], [0, -23], [9, 0], [38, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2106_e_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2106_2_e_ba();". // 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_2106_2_e_ba(:prec:=2) chi := MakeCharacter_2106_e(); 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_2106_2_e_ba();". // 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_2106_2_e_ba( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2106_e(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![9, -3, 1]>,<7,R![1, -1, 1]>,<11,R![36, 6, 1]>,<19,R![-2, 1]>],Snew); return Vf; end function;