// Make newform 684.3.h.e in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_684_h();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_684_h_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_684_3_h_e();". // 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_684_3_h_e();". // 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 := [39, -93, 112, -39, 22, -3, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0], [-15, 16, 43, -32, 5, -2], [-7, -8, 9, -2, 1, 0], [-17, 12, -13, 2, -1, 0], [-168, 392, -172, 128, -20, 8], [-510, 1114, -286, 204, -20, 8]]; Rf_basisdens := [1, 19, 2, 2, 19, 19]; 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_684_h();" function MakeCharacter_684_h() N := 684; order := 2; char_gens := [343, 533, 325]; v := [2, 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_684_h_Hecke();" function MakeCharacter_684_h_Hecke(Kf) N := 684; order := 2; char_gens := [343, 533, 325]; char_values := [[1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [-1, 0, 0, 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 := 3; raw_aps := [[0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, -1, 0, 0, 0], [4, 0, 1, 0, 0, 0], [0, -1, 0, 0, 1, 0], [8, 0, -1, 2, 0, 0], [-1, 0, 0, -2, 1, -1], [-6, 0, 3, 1, 0, 0], [0, 2, 0, 0, 2, 0], [0, -1, 0, 0, -1, -2], [0, 3, 0, 0, 1, 0], [0, 0, 0, 0, 3, -4], [-36, 0, 1, 2, 0, 0], [-4, 0, -4, 5, 0, 0], [0, -2, 0, 0, 1, -2], [0, 4, 0, 0, -5, -2], [32, 0, 7, 4, 0, 0], [0, -2, 0, 0, 0, -2], [0, -4, 0, 0, -1, -2], [12, 0, 1, 14, 0, 0], [0, 3, 0, 0, -8, 4], [66, 0, 0, 4, 0, 0], [0, 0, 0, 0, 3, -2], [0, 0, 0, 0, 7, -2], [30, 0, -15, 7, 0, 0], [0, 1, 0, 0, -6, 4], [0, 10, 0, 0, 4, 4], [0, 3, 0, 0, 12, -2], [0, 2, 0, 0, -4, -6], [0, -5, 0, 0, 10, -4], [-28, 0, 17, -16, 0, 0], [8, 0, 17, 8, 0, 0], [-32, 0, -15, -10, 0, 0], [-8, 0, 16, 9, 0, 0], [0, -3, 0, 0, -11, 2], [-110, 0, -18, 26, 0, 0], [-14, 0, -12, -28, 0, 0], [0, 2, 0, 0, -34, 8], [0, 0, 0, 0, -24, 8], [0, 6, 0, 0, 6, 6], [0, -1, 0, 0, -6, -2], [4, 0, -14, -33, 0, 0], [0, 0, 0, 0, -26, -8], [102, 0, 27, -11, 0, 0], [-120, 0, 13, 26, 0, 0], [0, 8, 0, 0, 27, 8], [0, -9, 0, 0, 9, 14], [0, 0, 0, 0, -3, 16], [168, 0, -25, -12, 0, 0], [-56, 0, -23, 24, 0, 0], [4, 0, -32, -39, 0, 0], [0, 22, 0, 0, -3, 10], [12, 0, 15, -14, 0, 0], [0, 2, 0, 0, 32, 0], [-84, 0, 0, -23, 0, 0], [0, 6, 0, 0, 27, 6], [122, 0, 24, 48, 0, 0], [-48, 0, -1, -36, 0, 0], [0, -2, 0, 0, -38, -10], [-104, 0, 17, -26, 0, 0], [0, 12, 0, 0, 6, 16], [0, -18, 0, 0, -42, -10], [156, 0, -42, -5, 0, 0], [-74, 0, -24, 20, 0, 0], [0, 10, 0, 0, -23, -14], [0, 2, 0, 0, -68, 2], [0, -4, 0, 0, 43, -22], [-132, 0, 57, -8, 0, 0], [120, 0, -11, -34, 0, 0], [-354, 0, 54, 10, 0, 0], [-444, 0, -12, 13, 0, 0], [50, 0, 12, -60, 0, 0], [0, 13, 0, 0, -58, 2], [0, 4, 0, 0, 14, -22], [0, 0, 0, 0, 75, -2], [-264, 0, 18, 31, 0, 0], [56, 0, -63, -6, 0, 0], [0, 2, 0, 0, -4, -18], [0, -36, 0, 0, -5, -22], [402, 0, 42, 58, 0, 0], [0, -3, 0, 0, -25, 24], [0, 26, 0, 0, -13, -10], [0, 2, 0, 0, 80, -12], [0, -11, 0, 0, 40, 16], [564, 0, -3, 28, 0, 0], [0, 2, 0, 0, 98, -8], [8, 0, 21, -42, 0, 0], [-544, 0, -22, 23, 0, 0], [352, 0, -19, 34, 0, 0], [-100, 0, -1, 26, 0, 0], [234, 0, 21, -17, 0, 0], [0, -1, 0, 0, 27, -38], [-102, 0, -78, 58, 0, 0], [68, 0, 13, -26, 0, 0], [-222, 0, 51, -23, 0, 0], [0, -32, 0, 0, -35, -6], [0, -34, 0, 0, 32, -32], [0, 0, 0, 0, 32, -26], [-232, 0, 129, -6, 0, 0], [0, -34, 0, 0, 37, 28], [280, 0, 10, 15, 0, 0], [0, 32, 0, 0, -49, 14], [0, -8, 0, 0, -41, -30], [-154, 0, 48, 84, 0, 0], [264, 0, 17, -6, 0, 0], [-468, 0, 3, 94, 0, 0], [366, 0, -138, 10, 0, 0], [0, 46, 0, 0, -32, 4], [0, -20, 0, 0, -59, 6], [0, -23, 0, 0, 5, 18], [-472, 0, 31, -44, 0, 0], [-704, 0, 7, 18, 0, 0], [-26, 0, -12, -64, 0, 0], [584, 0, 7, 40, 0, 0], [0, 16, 0, 0, -5, -18], [216, 0, -119, -34, 0, 0], [364, 0, 46, 51, 0, 0], [400, 0, 22, -165, 0, 0], [0, -50, 0, 0, 4, 34], [0, 33, 0, 0, -22, 2], [0, 16, 0, 0, 9, 38], [0, 32, 0, 0, 26, 20], [0, 44, 0, 0, 5, 0], [332, 0, 57, -30, 0, 0], [-594, 0, -57, -47, 0, 0], [98, 0, -12, -12, 0, 0], [468, 0, -60, -11, 0, 0], [-64, 0, -51, -150, 0, 0], [170, 0, 36, 108, 0, 0], [-416, 0, -87, 182, 0, 0], [0, 4, 0, 0, -68, 32], [0, -41, 0, 0, 86, 0], [-56, 0, 27, -112, 0, 0], [24, 0, 9, 88, 0, 0], [-132, 0, -67, -102, 0, 0], [0, -8, 0, 0, -5, 66], [0, -44, 0, 0, -30, -6], [0, 34, 0, 0, -74, 76], [384, 0, 99, 94, 0, 0], [0, 42, 0, 0, 69, -20], [520, 0, -2, -141, 0, 0], [-224, 0, 63, -136, 0, 0], [0, -22, 0, 0, -34, 10], [0, -23, 0, 0, -100, 30], [0, 24, 0, 0, -6, 24], [-574, 0, -36, -84, 0, 0], [0, -52, 0, 0, 65, -26], [380, 0, 97, -134, 0, 0], [0, 14, 0, 0, 29, 42], [0, 45, 0, 0, 86, -2], [1400, 0, 5, -4, 0, 0], [300, 0, 89, -54, 0, 0], [0, 36, 0, 0, -33, -54], [0, 18, 0, 0, -55, -24], [0, 18, 0, 0, 108, -28], [-110, 0, -6, 134, 0, 0], [-138, 0, -54, 94, 0, 0], [404, 0, -63, 30, 0, 0], [0, -70, 0, 0, -4, -32], [-318, 0, -30, -14, 0, 0], [0, 30, 0, 0, -162, 38], [-134, 0, 90, 158, 0, 0], [0, 2, 0, 0, 152, -34], [0, 56, 0, 0, -103, -52], [0, -28, 0, 0, 38, 40], [0, -9, 0, 0, 55, -42], [-704, 0, -21, -112, 0, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_684_h_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_684_3_h_e();". // 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_684_3_h_e(:prec:=6) chi := MakeCharacter_684_h(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 3)); 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_684_3_h_e();". // 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_684_3_h_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_684_h(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign))); Vf := Kernel([<5,R![-24, -44, -1, 1]>],Snew); return Vf; end function;