// Make newform 196.4.a.f in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_196_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_196_4_a_f();". // 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_196_4_a_f();". // 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 := [-2, 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_196_a();" function MakeCharacter_196_a() N := 196; order := 1; char_gens := [99, 101]; v := [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_196_a_Hecke(Kf) return MakeCharacter_196_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 := 4; raw_aps := [[0, 0], [0, 1], [0, -2], [0, 0], [-26, 0], [0, -24], [0, 73], [0, -67], [-148, 0], [-118, 0], [0, 210], [-254, 0], [0, -65], [122, 0], [0, -218], [-170, 0], [0, -215], [0, 430], [420, 0], [420, 0], [0, -575], [1052, 0], [0, 1025], [0, 725], [0, -223], [0, -1014], [0, -26], [884, 0], [598, 0], [-1548, 0], [2080, 0], [0, -1521], [-2100, 0], [0, -247], [-2990, 0], [656, 0], [0, 1520], [-1742, 0], [0, 998], [0, -876], [-4620, 0], [0, -1560], [-676, 0], [800, 0], [-1270, 0], [0, 1890], [-1236, 0], [0, -3456], [0, -3657], [0, 2340], [840, 0], [-2028, 0], [0, -1401], [0, 5369], [0, -3705], [2172, 0], [0, -3952], [0, 4160], [4246, 0], [-1664, 0], [0, -4199], [0, 3806], [0, 2863], [0, -3016], [0, 351], [3874, 0], [-1790, 0], [9794, 0], [-10086, 0], [0, -1992], [0, 1729], [-7040, 0], [0, 7852], [-2990, 0], [-11570, 0], [0, 4082], [-9838, 0], [0, -10312], [-9906, 0], [0, 2013], [0, 1495], [1686, 0], [8844, 0], [0, -195], [0, 8172], [-5212, 0], [-3042, 0], [-360, 0], [0, -1300], [-4160, 0], [0, -7007], [0, -2642], [11964, 0], [-9636, 0], [-12172, 0], [0, -2716], [0, 12248], [0, 2041], [0, -4785], [11746, 0], [16990, 0], [12090, 0], [0, -14495], [-8390, 0], [-6170, 0], [0, -9633], [0, 12309], [0, 15701], [25232, 0], [0, 5019], [0, -10452], [-11622, 0], [16246, 0], [0, -4693], [21404, 0], [-3622, 0], [0, 8489], [0, 13650], [-19578, 0], [-1962, 0], [0, -1830], [-26460, 0], [0, -14508], [5460, 0], [0, -10545], [-15306, 0], [-29822, 0], [0, 7462], [0, 3830], [0, 17106], [-35958, 0], [23920, 0], [-30724, 0], [22010, 0], [0, -6211], [0, -11373], [0, 25454], [0, 9097], [0, -5208], [-15952, 0], [0, 8541], [-7674, 0], [-5720, 0], [41964, 0], [0, -19370], [0, -16022], [0, -1752], [0, 25355], [0, 18355], [39260, 0], [46874, 0], [0, -3765], [-10540, 0], [0, -4030], [-9524, 0], [15560, 0], [22048, 0], [0, 14545], [0, 31433], [0, -5930], [9990, 0], [7990, 0], [-26364, 0], [0, 34945], [13884, 0], [0, -43066], [16640, 0], [0, 598]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_196_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_196_4_a_f();". // 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_196_4_a_f(:prec:=2) chi := MakeCharacter_196_a(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 4)); S := BaseChange(S, Kf); maxprec := NextPrime(997) - 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_196_4_a_f();". // 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_196_4_a_f( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_196_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<3,R![-2, 0, 1]>,<5,R![-8, 0, 1]>],Snew); return Vf; end function;