// Make newform 1764.2.a.l in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1764_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_1764_2_a_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_1764_2_a_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 := [-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_1764_a();" function MakeCharacter_1764_a() N := 1764; order := 1; char_gens := [883, 785, 1081]; v := [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_1764_a_Hecke(Kf) return MakeCharacter_1764_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, 0], [0, 1], [0, 0], [-4, 0], [0, -3], [0, 1], [0, -2], [4, 0], [-8, 0], [0, 0], [-8, 0], [0, -5], [-4, 0], [0, 4], [-10, 0], [0, 10], [0, 5], [0, 0], [0, 0], [0, 5], [8, 0], [0, -10], [0, 5], [0, 1], [0, -9], [0, 8], [-8, 0], [-8, 0], [-6, 0], [-20, 0], [0, -6], [0, 0], [0, -2], [-10, 0], [-4, 0], [0, -5], [4, 0], [0, -4], [0, -3], [0, 0], [0, 15], [16, 0], [-10, 0], [10, 0], [0, 0], [24, 0], [0, -12], [0, 6], [0, 15], [0, 0], [12, 0], [0, -9], [0, 14], [0, 15], [24, 0], [0, -13], [0, -20], [22, 0], [-16, 0], [0, 2], [0, 23], [0, 14], [0, -16], [0, -3], [2, 0], [-20, 0], [8, 0], [12, 0], [0, 3], [0, 7], [20, 0], [0, -4], [10, 0], [-20, 0], [0, -4], [-8, 0], [0, -11], [-24, 0], [0, -27], [0, 10], [6, 0], [36, 0], [0, 15], [0, 12], [16, 0], [18, 0], [30, 0], [0, 5], [40, 0], [0, -14], [0, -8], [-12, 0], [-24, 0], [32, 0], [0, -28], [0, -13], [0, -29], [0, -30], [-14, 0], [-20, 0], [-30, 0], [0, 10], [-40, 0], [-20, 0], [0, -9], [0, 18], [0, -7], [-8, 0], [0, 21], [0, 24], [24, 0], [8, 0], [0, 22], [-16, 0], [-8, 0], [0, -2], [0, 0], [24, 0], [-12, 0], [0, 15], [0, 0], [0, 9], [0, 0], [0, 30], [-24, 0], [-8, 0], [0, 28], [0, -20], [0, 27], [-12, 0], [20, 0], [-4, 0], [-40, 0], [0, 29], [0, -33], [0, 17], [0, -34], [0, -21], [-2, 0], [0, -6], [-6, 0], [40, 0], [-48, 0], [0, 5], [0, -8], [0, -9], [0, 5], [0, -10], [40, 0], [8, 0], [0, -15], [-40, 0], [0, 20], [-8, 0], [-20, 0], [-8, 0], [0, 25], [0, -11], [0, -5], [-60, 0], [-10, 0], [12, 0], [0, 10], [-48, 0], [0, 32], [-40, 0], [0, -1], [-38, 0], [0, -25], [-24, 0], [0, -19], [-24, 0], [16, 0], [0, -28], [0, -5], [-24, 0], [-24, 0], [0, 24], [0, -5], [-20, 0], [0, -10], [24, 0], [0, -37], [20, 0], [0, -7], [-30, 0], [0, -18], [32, 0], [0, -40], [0, -9], [-24, 0], [-16, 0], [0, -5], [0, 0], [0, 13], [-24, 0], [40, 0], [0, -39], [0, 28], [30, 0], [0, -24], [0, 25], [0, 7], [0, -2], [0, 29], [0, 32], [-60, 0], [34, 0], [0, -30], [40, 0], [0, 39], [-20, 0], [0, -26], [0, -28], [0, -5], [32, 0], [0, -25], [-48, 0], [24, 0], [-40, 0], [0, 12], [-18, 0], [-4, 0], [0, 34], [40, 0], [0, -43], [-28, 0], [0, 4], [-60, 0], [-40, 0], [0, 18], [40, 0], [-6, 0], [0, -22], [0, -36], [0, 15], [70, 0], [-32, 0], [0, 16], [60, 0], [0, -30], [0, -12], [-30, 0], [0, -17], [0, -8], [0, -40], [0, -34], [68, 0], [64, 0], [2, 0], [0, -9], [40, 0], [0, 35], [0, 35], [8, 0], [40, 0], [0, -50], [0, 31], [0, 11], [-44, 0], [-52, 0], [0, -15], [0, -53], [0, -15], [0, 18], [-18, 0], [0, 21], [80, 0], [56, 0], [0, -45], [-20, 0], [0, 27], [-44, 0], [0, -25], [0, 8], [60, 0], [-34, 0], [-34, 0], [0, -14], [0, 28], [16, 0], [0, 20], [0, 9], [0, -14], [-36, 0], [34, 0], [58, 0], [0, 20], [0, 25], [-34, 0], [0, -14], [40, 0], [0, 50], [-34, 0], [0, 35], [0, 4], [0, -23], [0, 10], [0, 46], [0, -3], [-72, 0], [72, 0], [-60, 0], [-4, 0], [48, 0], [-12, 0], [0, -35], [-60, 0], [70, 0], [0, 20], [72, 0], [-32, 0], [4, 0], [-72, 0], [0, -35], [0, -38], [56, 0], [0, -7], [-48, 0], [0, 34], [-80, 0], [0, -19], [0, 0], [-66, 0], [0, 65], [68, 0], [0, 30], [-20, 0], [40, 0], [30, 0], [-40, 0], [0, -52], [0, 22], [-40, 0], [0, -26], [72, 0], [0, -55], [0, -21], [0, 24], [86, 0], [40, 0], [0, -7], [-24, 0], [-56, 0], [-20, 0], [0, -39], [-20, 0], [0, 4], [0, 9], [0, -30], [-10, 0], [14, 0], [0, 32], [38, 0], [0, -5], [0, 8], [0, 14], [18, 0], [-4, 0], [10, 0], [0, -5], [-60, 0], [-48, 0], [0, 30], [-56, 0], [0, -19], [-44, 0], [14, 0], [-96, 0], [0, 30], [80, 0], [-42, 0], [0, -24], [18, 0], [0, -30], [-84, 0], [0, 57], [0, 57], [0, 21], [0, 29], [-34, 0], [0, 0], [-62, 0], [0, 10], [0, 12], [96, 0], [0, -9], [40, 0], [0, -24], [8, 0], [0, -33], [-32, 0], [0, -30], [44, 0], [0, 0], [0, -40], [0, 35], [-24, 0], [-40, 0], [0, -45], [-30, 0], [32, 0], [0, 59], [0, -15], [0, 56], [-2, 0], [26, 0], [0, 30], [0, -30], [0, -33], [72, 0], [36, 0], [-80, 0], [-32, 0], [0, 75], [48, 0], [0, 68], [0, -55], [0, 28], [-72, 0], [0, 47], [-48, 0], [0, 10], [0, 27], [0, -61], [60, 0], [-80, 0], [0, -10], [0, 60]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1764_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1764_2_a_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_1764_2_a_l(:prec:=2) chi := MakeCharacter_1764_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_1764_2_a_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_1764_2_a_l( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1764_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-2, 0, 1]>,<11,R![4, 1]>,<13,R![-18, 0, 1]>],Snew); return Vf; end function;