// Make newform 2304.3.e.l in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2304_e();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2304_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_2304_3_e_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_2304_3_e_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 := [1, 0, 0, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 0, 4, 0], [0, 3, 0, 3], [0, 6, 0, -6]]; 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_2304_e();" function MakeCharacter_2304_e() N := 2304; order := 2; char_gens := [1279, 2053, 1793]; 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_2304_e_Hecke();" function MakeCharacter_2304_e_Hecke(Kf) N := 2304; order := 2; char_gens := [1279, 2053, 1793]; 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 := 3; raw_aps := [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, -1], [0, 1, 0, 0], [18, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 2], [0, -9, 0, 0], [0, 0, -3, 0], [0, 0, 0, -1], [36, 0, 0, 0], [0, 0, 7, 0], [0, 0, 0, -8], [0, -9, 0, 0], [0, 0, -19, 0], [0, 20, 0, 0], [-36, 0, 0, 0], [0, 0, 0, 14], [0, 27, 0, 0], [56, 0, 0, 0], [0, 0, 0, -3], [0, -19, 0, 0], [0, 0, 21, 0], [104, 0, 0, 0], [0, 0, 35, 0], [0, 0, 0, 21], [0, 32, 0, 0], [126, 0, 0, 0], [0, 0, 21, 0], [0, 0, 0, 19], [0, -38, 0, 0], [0, 0, 49, 0], [0, 0, 0, -14], [0, 0, 23, 0], [0, 0, 0, -29], [-252, 0, 0, 0], [0, 0, 0, 12], [0, 0, 0, 0], [0, 0, -7, 0], [0, -14, 0, 0], [126, 0, 0, 0], [0, 0, 0, 0], [298, 0, 0, 0], [0, 0, -77, 0], [0, 0, 0, -7], [0, 0, 0, -6], [0, 0, 0, -33], [0, 71, 0, 0], [-126, 0, 0, 0], [0, 0, 67, 0], [0, -90, 0, 0], [-32, 0, 0, 0], [0, -5, 0, 0], [0, 0, 91, 0], [0, 18, 0, 0], [0, 0, -93, 0], [0, 0, 0, 11], [126, 0, 0, 0], [0, 0, 21, 0], [0, 0, 0, 24], [0, 0, 45, 0], [0, 0, 0, -34], [0, 126, 0, 0], [-58, 0, 0, 0], [0, 0, 43, 0], [0, 0, 0, 42], [-8, 0, 0, 0], [0, 7, 0, 0], [252, 0, 0, 0], [0, 0, 133, 0], [0, -63, 0, 0], [0, 0, 0, -1], [252, 0, 0, 0], [0, 0, 0, 70], [0, 126, 0, 0], [0, 0, -49, 0], [-756, 0, 0, 0], [0, 0, -119, 0], [136, 0, 0, 0], [0, -7, 0, 0], [270, 0, 0, 0], [0, 63, 0, 0], [-574, 0, 0, 0], [0, 0, 0, 11], [0, -179, 0, 0], [0, 0, 47, 0], [40, 0, 0, 0], [0, 0, 175, 0], [0, 0, 0, 77], [0, -13, 0, 0], [0, -171, 0, 0], [0, 0, 0, -27], [0, 110, 0, 0], [0, 0, 0, -56], [0, 225, 0, 0], [0, 0, -43, 0], [0, 0, 1, 0], [0, 0, 0, -30], [414, 0, 0, 0], [0, 0, 0, 8], [0, 0, -65, 0], [0, -223, 0, 0], [0, 0, -199, 0], [0, 0, 0, -52], [-326, 0, 0, 0], [0, -100, 0, 0], [0, 0, -213, 0], [0, -189, 0, 0], [-778, 0, 0, 0], [0, 0, 0, -45], [-180, 0, 0, 0], [0, 0, -107, 0], [0, 0, 0, -4], [0, 0, 0, 89], [0, 0, -55, 0], [0, 0, 0, 140], [0, 81, 0, 0], [0, 0, 209, 0], [0, 280, 0, 0], [180, 0, 0, 0], [302, 0, 0, 0], [0, 0, -183, 0], [0, 265, 0, 0], [0, 0, 0, 84], [0, 0, -67, 0], [882, 0, 0, 0], [0, 63, 0, 0], [0, 0, 0, 5], [162, 0, 0, 0], [0, 0, 0, 28], [0, 252, 0, 0], [0, 0, 0, 59], [1026, 0, 0, 0], [0, 0, -297, 0], [694, 0, 0, 0], [0, 0, -339, 0], [0, 0, 0, 102], [0, 0, -151, 0], [0, 0, -161, 0], [0, 0, 0, -154], [0, 0, 27, 0], [0, 0, 0, -77], [0, 194, 0, 0], [702, 0, 0, 0], [0, 153, 0, 0], [-252, 0, 0, 0], [0, 0, -153, 0], [0, 0, 0, 40], [0, 252, 0, 0], [1260, 0, 0, 0], [0, 0, -357, 0], [0, 0, 0, 112], [0, 54, 0, 0], [0, 0, 0, -88], [0, -36, 0, 0], [0, 0, 0, -77], [0, 0, 297, 0], [-1154, 0, 0, 0], [0, 0, -257, 0], [0, 280, 0, 0], [0, 0, -79, 0], [0, 0, 0, 37], [0, 193, 0, 0], [0, 0, -281, 0], [0, 126, 0, 0], [0, 0, 0, 43], [-756, 0, 0, 0], [-1240, 0, 0, 0], [0, 0, -111, 0], [0, -370, 0, 0], [-1404, 0, 0, 0], [0, 198, 0, 0], [976, 0, 0, 0], [0, 0, 0, -223], [0, 0, 21, 0], [0, 0, 0, 20], [0, 0, -233, 0], [0, 0, 0, 7], [-972, 0, 0, 0], [0, 0, 0, -31], [0, 215, 0, 0], [882, 0, 0, 0], [0, 0, 245, 0], [0, 126, 0, 0], [0, 0, -271, 0], [-558, 0, 0, 0], [0, 0, 0, 84], [656, 0, 0, 0], [0, -18, 0, 0], [128, 0, 0, 0], [0, -334, 0, 0], [0, 0, 0, -68], [0, 0, -217, 0], [0, -50, 0, 0], [0, 0, 91, 0], [442, 0, 0, 0], [-1854, 0, 0, 0], [0, 0, -19, 0], [0, -180, 0, 0], [0, 0, -47, 0], [0, 0, 0, 235], [-1890, 0, 0, 0], [616, 0, 0, 0], [0, -77, 0, 0], [0, 0, 51, 0], [0, 0, 0, 237], [0, 11, 0, 0], [0, 0, 467, 0], [0, 0, 0, 56], [-938, 0, 0, 0], [0, 0, -423, 0], [0, 0, 0, 187], [0, -550, 0, 0], [0, -207, 0, 0], [-14, 0, 0, 0], [0, 0, 0, 225], [0, 0, -41, 0], [0, 630, 0, 0], [0, 0, -111, 0], [126, 0, 0, 0], [0, 0, 0, 231], [0, 0, -31, 0], [0, 0, 0, -87], [0, -61, 0, 0], [2772, 0, 0, 0], [0, 0, 651, 0], [0, 315, 0, 0], [0, 0, 0, -81], [0, -280, 0, 0], [558, 0, 0, 0], [0, 0, 0, -252], [0, 0, 0, -327], [0, 0, 203, 0], [0, 0, 0, -252], [0, -81, 0, 0], [2168, 0, 0, 0], [0, 0, 279, 0], [0, -497, 0, 0], [0, 405, 0, 0], [0, -91, 0, 0], [0, 0, 0, 42], [0, 0, 0, 209], [-1260, 0, 0, 0], [0, 0, 611, 0], [0, 567, 0, 0], [0, 0, 0, 257], [0, -434, 0, 0], [0, 0, 0, 126], [0, 378, 0, 0], [2142, 0, 0, 0], [0, 0, -277, 0], [0, -567, 0, 0], [1552, 0, 0, 0], [0, 0, -373, 0], [0, 346, 0, 0], [558, 0, 0, 0], [0, 0, 0, 0], [0, 0, 101, 0], [-254, 0, 0, 0], [0, 0, 0, -103], [0, -170, 0, 0], [126, 0, 0, 0], [-1260, 0, 0, 0], [0, 0, 351, 0], [0, 0, 0, -154], [0, 0, -375, 0], [0, 0, -61, 0], [0, 0, 0, -154], [0, 0, -345, 0], [-756, 0, 0, 0], [0, 0, 0, 252], [256, 0, 0, 0], [0, 0, 0, 235], [1520, 0, 0, 0], [0, 0, 0, -217], [0, 277, 0, 0], [2700, 0, 0, 0], [-1078, 0, 0, 0], [0, -224, 0, 0], [0, 171, 0, 0], [0, 0, 0, -97], [0, -783, 0, 0], [-2772, 0, 0, 0], [0, 0, 0, -14], [0, 324, 0, 0], [592, 0, 0, 0], [0, 0, 319, 0], [0, 0, 0, -141], [0, 0, 245, 0], [0, 0, 609, 0], [0, -428, 0, 0], [0, 0, -91, 0], [0, 578, 0, 0], [900, 0, 0, 0], [0, 0, -33, 0], [0, 0, 0, -113], [0, 0, 419, 0], [0, -707, 0, 0], [0, 0, 0, 414], [2618, 0, 0, 0], [0, 0, -169, 0], [0, 0, 0, 169]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2304_e_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2304_3_e_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_2304_3_e_l(:prec:=4) chi := MakeCharacter_2304_e(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 3)); 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_2304_3_e_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_2304_3_e_l( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2304_e(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign))); Vf := Kernel([<5,R![18, 0, 1]>,<7,R![-72, 0, 1]>,<13,R![-18, 1]>,<19,R![-288, 0, 1]>,<31,R![-72, 0, 1]>],Snew); return Vf; end function;