// Make newform 700.2.g.f in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_700_g();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_700_g_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_700_2_g_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_700_2_g_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 := [1, 0, -1, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 1, 1, 0], [0, 0, 0, 1], [0, 1, -1, 0]]; Rf_basisdens := [1, 1, 1, 1]; Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf); Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]]; 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_700_g();" function MakeCharacter_700_g() N := 700; order := 2; char_gens := [351, 477, 101]; v := [1, 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_700_g_Hecke();" function MakeCharacter_700_g_Hecke(Kf) N := 700; order := 2; char_gens := [351, 477, 101]; char_values := [[-1, 0, 0, 0], [1, 0, 0, 0], [-1, 0, 0, 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, 1, -1, 0], [0, -1, 1, -1], [0, 0, 0, 0], [0, -1, -1, -1], [-1, 1, 2, -1], [-2, 2, 3, -2], [-2, 2, -3, -2], [6, 0, 0, 0], [-2, 2, 2, -2], [1, -4, 4, -4], [-6, 0, 0, 0], [6, -2, 2, -2], [2, -2, 0, 2], [0, 0, -2, 0], [0, 1, -1, 1], [-2, 0, 0, 0], [0, 2, -2, 2], [-2, 2, -6, -2], [2, -2, 0, 2], [2, -2, 4, 2], [4, -4, 6, 4], [5, -5, 6, 5], [-12, 2, -2, 2], [2, -2, -6, 2], [-6, 6, -3, -6], [2, -2, 12, 2], [12, 3, -3, 3], [6, -6, -8, 6], [9, 4, -4, 4], [-2, 2, -2, 2], [2, -2, 12, 2], [6, 2, -2, 2], [-10, 6, -6, 6], [0, -4, 4, -4], [10, 4, -4, 4], [3, -3, 10, 3], [8, -8, -6, 8], [12, -12, 0, 12], [0, -3, 3, -3], [-10, 10, 3, -10], [-6, 6, 4, -6], [4, -4, -6, 4], [3, -3, 2, 3], [6, -2, 2, -2], [-4, -10, 10, -10], [0, -2, 2, -2], [3, -3, -2, 3], [12, -1, 1, -1], [-12, 5, -5, 5], [-2, 2, 12, -2], [16, 4, -4, 4], [-15, 15, 2, -15], [-6, 6, -6, -6], [-12, 8, -8, 8], [0, 0, 6, 0], [2, -2, 8, 2], [0, 0, 12, 0], [6, -2, 2, -2], [4, 12, -12, 12], [-1, 4, -4, 4], [0, 7, -7, 7], [10, -10, 3, 10], [0, 1, -1, 1], [-6, -8, 8, -8], [2, -2, -21, 2], [10, 4, -4, 4], [-6, 6, -16, -6], [0, 0, 0, 0], [-14, 14, -10, -14], [10, -10, -12, 10], [-6, 6, 3, -6], [10, -10, 8, 10], [-24, -7, 7, -7], [14, 6, -6, 6], [-6, 6, 16, -6], [-24, 2, -2, 2], [-7, 8, -8, 8], [2, -2, 9, 2], [-17, -4, 4, -4], [8, -8, -18, 8], [0, 14, -14, 14], [19, 0, 0, 0], [-9, 9, 2, -9], [-8, 8, -18, -8], [-12, 16, -16, 16], [0, 0, 26, 0], [5, -4, 4, -4], [-24, -2, 2, -2], [-16, 16, 0, -16], [6, -6, 6, 6], [0, 13, -13, 13], [-6, -18, 18, -18], [12, -12, -8, 12], [7, -7, -22, 7], [9, -9, 10, 9], [0, -9, 9, -9], [-8, 8, -12, -8], [6, -6, -24, 6], [0, -14, 14, -14], [-13, -12, 12, -12], [-2, 2, -18, -2], [8, -8, 8, -8], [12, 18, -18, 18], [14, 4, -4, 4], [10, -10, -24, 10], [6, -6, 9, 6], [24, 2, -2, 2], [2, -2, 27, 2], [-7, 7, 2, -7], [12, -12, -6, 12], [24, -3, 3, -3], [10, 0, 0, 0], [16, -4, 4, -4], [18, 10, -10, 10], [19, -19, 6, 19], [2, -4, 4, -4], [12, 11, -11, 11], [0, -6, 6, -6], [-28, -6, 6, -6], [-3, 3, -26, -3], [-16, 16, -12, -16], [-18, -18, 18, -18], [-6, 6, -15, -6], [-10, 10, 10, -10], [12, 20, -20, 20], [-17, 12, -12, 12], [21, 0, 0, 0], [42, 2, -2, 2], [24, 6, -6, 6], [-2, 2, 15, -2], [1, -1, -6, 1], [-6, 6, -20, -6], [-9, 9, -10, -9], [-24, 8, -8, 8], [14, -14, 18, 14], [0, 0, -18, 0], [2, -2, -9, 2], [-24, 5, -5, 5], [-18, 18, 9, -18], [1, -4, 4, -4], [36, 4, -4, 4], [7, -12, 12, -12], [4, -4, -18, 4], [8, -8, -28, 8], [-14, 14, 24, -14], [-6, -6, 6, -6], [-12, 12, 18, -12], [16, -16, 30, 16], [12, -16, 16, -16], [8, -8, 26, 8], [8, -6, 6, -6], [-4, 4, 36, -4], [-6, 6, 34, -6], [12, 2, -2, 2], [6, -6, 22, 6], [-14, 14, 4, -14], [9, -9, -22, 9], [28, -28, 6, 28], [-6, 6, -15, -6], [-26, 26, 6, -26], [-12, 12, -2, -12], [-4, 10, -10, 10], [-18, 18, 0, -18], [-6, -12, 12, -12], [-20, -14, 14, -14], [12, -9, 9, -9], [18, -18, 16, 18], [-2, 2, 9, -2]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_700_g_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_700_2_g_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_700_2_g_f(:prec:=4) chi := MakeCharacter_700_g(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 2)); 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_700_2_g_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_700_2_g_f( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_700_g(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-3, 0, 1]>,<11,R![1, 0, 14, 0, 1]>,<19,R![-6, 1]>,<37,R![24, -12, 1]>],Snew); return Vf; end function;