// Make newform 882.3.s.i in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_882_s();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_882_s_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_882_3_s_i();". // 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_882_3_s_i();". // 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 := [81, 0, -72, 0, 55, 0, -8, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [576, 0, -440, 0, 55, 0, -8, 0], [0, 203, 0, 0, 0, 0, 0, 1], [-592, 0, 0, 0, 0, 0, -4, 0], [0, 1066, 0, 0, 0, 0, 0, 2], [6624, 0, -5060, 0, 880, 0, -92, 0], [0, 81, 0, -341, 0, 55, 0, -8], [0, 11376, 0, -8690, 0, 1210, 0, -158]]; Rf_basisdens := [1, 495, 165, 55, 165, 495, 297, 1485]; 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_882_s();" function MakeCharacter_882_s() N := 882; order := 6; char_gens := [785, 199]; v := [3, 4]; // 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_882_s_Hecke();" function MakeCharacter_882_s_Hecke(Kf) N := 882; order := 6; char_gens := [785, 199]; char_values := [[-1, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 0, 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, -1, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, -1, 1], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, 2, 0, 0, 0], [8, 0, 0, -1, 0, 0, 0, 0], [0, 0, -13, 0, 1, 0, 0, 0], [20, -20, 0, 0, 0, 0, 0, 0], [0, 0, 2, 0, 0, 0, 2, -2], [0, 0, 0, 0, 2, 0, -19, -2], [0, 4, 0, 0, 0, 2, 0, 0], [-38, 38, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 3, 0, -27, -3], [20, 0, 0, -6, 0, 0, 0, 0], [0, 0, 12, 0, 0, 0, 12, 0], [0, 0, -3, 0, -12, 0, 0, 0], [0, 0, -20, 0, -4, 0, 0, 0], [58, -58, 0, 4, 0, 4, 0, 0], [0, 48, 0, 0, 0, 8, 0, 0], [0, 0, 0, 0, -2, 0, -2, 2], [0, -24, 0, 0, 0, -5, 0, 0], [-76, 76, 0, -4, 0, -4, 0, 0], [0, 0, 0, 0, -4, 0, -64, 4], [0, 0, 51, 0, 0, 0, 51, -9], [72, 0, 0, -11, 0, 0, 0, 0], [0, 0, 49, 0, 5, 0, 0, 0], [68, -68, 0, -6, 0, -6, 0, 0], [0, 0, 90, 0, 0, 0, 90, -6], [0, 72, 0, 0, 0, -10, 0, 0], [0, 0, 0, 0, -4, 0, -1, 4], [172, 0, 0, 4, 0, 0, 0, 0], [0, 0, 20, 0, 0, 0, 20, 16], [0, 0, 65, 0, -2, 0, 0, 0], [-48, 0, 0, 16, 0, 0, 0, 0], [0, 0, 51, 0, 0, 0, 51, 12], [0, 72, 0, 0, 0, 2, 0, 0], [0, -58, 0, 0, 0, -12, 0, 0], [40, -40, 0, 12, 0, 12, 0, 0], [0, 0, 0, 0, 4, 0, 64, -4], [0, 0, -25, 0, 0, 0, -25, -17], [0, 0, -126, 0, -6, 0, 0, 0], [-136, 0, 0, 5, 0, 0, 0, 0], [0, 0, -66, 0, 0, 0, -66, -18], [0, -134, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -8, 0, 91, 8], [0, -88, 0, 0, 0, 18, 0, 0], [-148, 0, 0, -6, 0, 0, 0, 0], [-32, 0, 0, -18, 0, 0, 0, 0], [0, 0, 72, 0, 0, 0, 0, 0], [-216, 216, 0, 5, 0, 5, 0, 0], [0, 0, 119, 0, 0, 0, 119, -26], [0, 0, 0, 0, 18, 0, 30, -18], [0, 248, 0, 0, 0, 9, 0, 0], [0, 0, 0, 0, 0, 0, 252, 0], [0, 0, 69, 0, 0, 0, 69, 21], [0, 0, -90, 0, -18, 0, 0, 0], [0, 0, -119, 0, 35, 0, 0, 0], [-20, 20, 0, -2, 0, -2, 0, 0], [0, -32, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 10, 0, -65, -10], [0, 228, 0, 0, 0, -20, 0, 0], [0, 0, 0, 0, -51, 0, -9, 51], [108, 0, 0, -8, 0, 0, 0, 0], [0, 0, -140, 0, 44, 0, 0, 0], [-126, 126, 0, -16, 0, -16, 0, 0], [0, 0, -51, 0, 0, 0, -51, -24], [-44, 44, 0, 30, 0, 30, 0, 0], [-320, 0, 0, -2, 0, 0, 0, 0], [0, 0, -114, 0, -6, 0, 0, 0], [-138, 0, 0, 28, 0, 0, 0, 0], [0, 0, -147, 0, 3, 0, 0, 0], [0, 0, -122, 0, 0, 0, -122, -46], [0, 200, 0, 0, 0, -30, 0, 0], [318, -318, 0, -8, 0, -8, 0, 0], [420, 0, 0, 2, 0, 0, 0, 0], [0, 0, -188, 0, 0, 0, -188, -64], [0, 0, 389, 0, -2, 0, 0, 0], [202, -202, 0, 28, 0, 28, 0, 0], [0, 0, -105, 0, 0, 0, -105, 6], [0, -232, 0, 0, 0, 21, 0, 0], [0, 0, 0, 0, 0, 0, 240, 0], [-176, 0, 0, 40, 0, 0, 0, 0], [0, 0, -90, 0, -78, 0, 0, 0], [130, 0, 0, 48, 0, 0, 0, 0], [-128, 128, 0, -62, 0, -62, 0, 0], [0, 0, -154, 0, 0, 0, -154, 34], [0, 0, 0, 0, -76, 0, -121, 76], [-48, 48, 0, 28, 0, 28, 0, 0], [0, 0, 0, 0, 1, 0, -257, -1], [36, 0, 0, -4, 0, 0, 0, 0], [0, 0, 452, 0, 0, 0, 452, -32], [0, 0, -328, 0, -32, 0, 0, 0], [0, 400, 0, 0, 0, -30, 0, 0], [0, 0, 0, 0, 6, 0, 162, -6], [-12, 12, 0, 70, 0, 70, 0, 0], [0, 0, 0, 0, 84, 0, -24, -84], [0, 0, -687, 0, 0, 0, -687, 3], [0, 0, 267, 0, 45, 0, 0, 0], [-232, 232, 0, 0, 0, 0, 0, 0], [-208, 208, 0, -4, 0, -4, 0, 0], [592, 0, 0, 28, 0, 0, 0, 0], [0, 0, -139, 0, 16, 0, 0, 0], [0, 0, 380, 0, 64, 0, 0, 0], [0, 0, -129, 0, 0, 0, -129, -54], [0, -232, 0, 0, 0, -68, 0, 0], [0, -318, 0, 0, 0, 16, 0, 0], [0, 0, 0, 0, -52, 0, -40, 52], [0, 0, 21, 0, 0, 0, 21, -33], [0, 0, 90, 0, 6, 0, 0, 0], [418, 0, 0, 0, 0, 0, 0, 0], [352, -352, 0, -26, 0, -26, 0, 0], [0, 110, 0, 0, 0, 16, 0, 0], [0, 0, 0, 0, 22, 0, -137, -22], [0, -264, 0, 0, 0, -68, 0, 0], [-172, 0, 0, 12, 0, 0, 0, 0], [0, 0, 337, 0, -22, 0, 0, 0], [604, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 24, 0, 0, 0], [0, 0, 309, 0, 0, 0, 309, 6], [0, 0, 0, 0, -26, 0, -758, 26], [0, 198, 0, 0, 0, -68, 0, 0], [-406, 0, 0, 24, 0, 0, 0, 0], [0, 0, -417, 0, 0, 0, -417, -57], [0, 0, 198, 0, -6, 0, 0, 0], [584, -584, 0, 60, 0, 60, 0, 0], [0, 0, 0, 0, 38, 0, 101, -38], [-8, 8, 0, -6, 0, -6, 0, 0], [0, 0, 396, 0, 0, 0, 396, 60], [852, 0, 0, 46, 0, 0, 0, 0], [-152, 152, 0, 19, 0, 19, 0, 0], [0, -880, 0, 0, 0, 52, 0, 0], [0, 0, 0, 0, -66, 0, -270, 66], [-88, 88, 0, -22, 0, -22, 0, 0], [184, 0, 0, 10, 0, 0, 0, 0], [0, 0, -269, 0, 0, 0, -269, -145], [-306, 0, 0, 40, 0, 0, 0, 0], [0, 0, 79, 0, -73, 0, 0, 0], [0, -608, 0, 0, 0, 56, 0, 0], [0, 0, 0, 0, 123, 0, -87, -123], [0, 0, -313, 0, -80, 0, 0, 0], [48, 0, 0, 52, 0, 0, 0, 0], [0, 0, -355, 0, 0, 0, -355, 64], [0, 516, 0, 0, 0, 8, 0, 0], [0, 0, 0, 0, -6, 0, 894, 6], [0, -696, 0, 0, 0, 7, 0, 0], [0, 0, 0, 0, 8, 0, 884, -8], [-518, 0, 0, -36, 0, 0, 0, 0], [0, 0, -99, 0, 81, 0, 0, 0], [-484, 484, 0, -48, 0, -48, 0, 0], [0, 0, -74, 0, 0, 0, -74, 14], [-66, 66, 0, 88, 0, 88, 0, 0], [0, 0, 0, 0, -129, 0, -21, 129], [-724, 0, 0, 78, 0, 0, 0, 0], [0, 0, 696, 0, 0, 0, 696, -12], [0, 284, 0, 0, 0, -14, 0, 0], [0, 0, 0, 0, -90, 0, 402, 90], [240, -240, 0, -14, 0, -14, 0, 0], [0, 0, -243, 0, 0, 0, -243, 129], [-1262, 0, 0, 0, 0, 0, 0, 0], [0, 0, 703, 0, -43, 0, 0, 0], [0, 0, 238, 0, 0, 0, 238, 110], [0, 0, 0, 0, -8, 0, 73, 8], [1268, 0, 0, 36, 0, 0, 0, 0], [0, 0, 684, 0, 0, 0, 684, 60], [0, 0, -81, 0, -30, 0, 0, 0], [0, 0, 36, 0, -72, 0, 0, 0], [0, 112, 0, 0, 0, 78, 0, 0], [0, 842, 0, 0, 0, -36, 0, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_882_s_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_882_3_s_i();". // 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_882_3_s_i(:prec:=8) chi := MakeCharacter_882_s(); 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_882_3_s_i();". // 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_882_3_s_i( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_882_s(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign))); Vf := Kernel([<5,R![8503056, 0, -338256, 0, 10540, 0, -116, 0, 1]>,<11,R![2176782336, 0, -21648384, 0, 168640, 0, -464, 0, 1]>,<13,R![-48, -16, 1]>],Snew); return Vf; end function;