// Make newform 384.4.d.e in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_384_d();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_384_d_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_384_4_d_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_384_4_d_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 := [9, 0, 7, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 4, 0, 1], [0, 40, 0, 4], [28, 0, 8, 0]]; Rf_basisdens := [1, 3, 3, 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_384_d();" function MakeCharacter_384_d() N := 384; order := 2; char_gens := [127, 133, 257]; v := [2, 1, 2]; // 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_384_d_Hecke();" function MakeCharacter_384_d_Hecke(Kf) N := 384; order := 2; char_gens := [127, 133, 257]; 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 := 4; raw_aps := [[0, 0, 0, 0], [0, 3, 0, 0], [0, -4, 1, 0], [8, 0, 0, -1], [0, -4, -4, 0], [0, -36, -2, 0], [-18, 0, 0, -4], [0, -68, 4, 0], [128, 0, 0, 2], [0, -76, 9, 0], [40, 0, 0, 13], [0, 68, -4, 0], [218, 0, 0, -20], [0, -356, 4, 0], [112, 0, 0, -14], [0, 172, 15, 0], [0, -324, 0, 0], [0, -324, 0, 0], [0, -228, -48, 0], [1024, 0, 0, -2], [-330, 0, 0, 48], [-248, 0, 0, -59], [0, -388, -28, 0], [-266, 0, 0, 8], [-610, 0, 0, -88], [0, -612, -77, 0], [856, 0, 0, 73], [0, -244, 8, 0], [0, -1364, 42, 0], [114, 0, 0, 56], [-2152, 0, 0, 35], [0, 1308, 48, 0], [1594, 0, 0, 68], [0, 284, -112, 0], [0, -212, 53, 0], [-312, 0, 0, -69], [0, -1508, -140, 0], [0, 1676, -52, 0], [-528, 0, 0, -60], [0, 2980, 23, 0], [0, 1292, -112, 0], [0, 308, -234, 0], [-1568, 0, 0, -20], [-914, 0, 0, 272], [0, -980, 131, 0], [-1064, 0, 0, -119], [0, 76, -32, 0], [1720, 0, 0, -161], [0, 1628, -100, 0], [0, 516, 50, 0], [102, 0, 0, 8], [864, 0, 0, 360], [-4162, 0, 0, -184], [0, 3660, -84, 0], [642, 0, 0, 160], [1168, 0, 0, -416], [0, 2180, -227, 0], [4248, 0, 0, -81], [0, 1476, 286, 0], [-3434, 0, 0, 224], [0, 1708, -272, 0], [0, -2452, -233, 0], [0, -3876, -240, 0], [-2480, 0, 0, -284], [5018, 0, 0, -240], [0, 8116, 93, 0], [0, -10100, -56, 0], [5410, 0, 0, -80], [0, 4044, 12, 0], [0, -244, 224, 0], [-6270, 0, 0, -136], [-1152, 0, 0, 486], [3144, 0, 0, 525], [0, 6548, -328, 0], [0, 4540, 364, 0], [6368, 0, 0, 356], [0, -3412, 237, 0], [0, 8636, -184, 0], [3694, 0, 0, 468], [2026, 0, 0, 424], [0, 3340, -836, 0], [0, 7236, 430, 0], [-1200, 0, 0, -426], [2414, 0, 0, 144], [-2472, 0, 0, -483], [0, 4188, 588, 0], [-8610, 0, 0, -308], [506, 0, 0, 56], [0, -4492, 355, 0], [2968, 0, 0, -713], [0, 1836, -180, 0], [-8240, 0, 0, 58], [-3416, 0, 0, -5], [0, 2076, 312, 0], [0, 2188, 208, 0], [5472, 0, 0, -882], [0, 484, 425, 0], [-16550, 0, 0, -292], [0, -12052, -100, 0], [0, 3372, -46, 0], [0, 4188, 804, 0], [0, -3804, 395, 0], [0, 588, 228, 0], [-502, 0, 0, 1068], [0, 21580, 232, 0], [12946, 0, 0, 560], [0, -7732, -424, 0], [2002, 0, 0, -424], [7200, 0, 0, -162], [4710, 0, 0, -800], [-22952, 0, 0, 331], [0, 7380, 576, 0], [17574, 0, 0, -80], [0, 4364, 80, 0], [-5176, 0, 0, 467], [1022, 0, 0, -1124], [0, -1956, -1020, 0], [6240, 0, 0, -222], [0, 1940, 495, 0], [0, -11700, -360, 0], [0, -10108, 1052, 0], [11550, 0, 0, 1184], [0, -4884, -775, 0], [0, -6772, -76, 0], [0, 10252, 460, 0], [0, -13116, -359, 0], [0, 2068, -1638, 0], [-13872, 0, 0, 1590], [17704, 0, 0, -737], [0, 3164, 966, 0], [0, -5428, -1576, 0], [29072, 0, 0, 344], [2152, 0, 0, 2233], [0, -27532, -134, 0], [810, 0, 0, -1948], [-3218, 0, 0, 384], [0, -12036, 1127, 0], [0, -8148, -228, 0], [0, 15732, -113, 0], [2842, 0, 0, -1500], [0, 17164, -188, 0], [0, 3052, 1715, 0], [22952, 0, 0, 1127], [0, -4660, -88, 0], [0, -19412, 1434, 0], [5856, 0, 0, 834], [0, 8180, 548, 0], [-10454, 0, 0, -468], [0, 1244, -1780, 0], [35104, 0, 0, 796], [0, 10748, -232, 0], [15506, 0, 0, -240], [0, -12868, 1388, 0], [-3024, 0, 0, 1728], [0, -2372, 220, 0], [3552, 0, 0, 1608], [9192, 0, 0, -897], [-4034, 0, 0, -1500], [-9210, 0, 0, -608], [0, -1356, -469, 0], [0, 21084, -912, 0], [-25622, 0, 0, -292], [-23112, 0, 0, 2133], [0, 24268, -1292, 0], [51310, 0, 0, -508], [-16528, 0, 0, 608], [-10360, 0, 0, 17], [0, -19036, -1720, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_384_d_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_384_4_d_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_384_4_d_e(:prec:=4) chi := MakeCharacter_384_d(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 4)); 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_384_4_d_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_384_4_d_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_384_d(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<5,R![36864, 0, 448, 0, 1]>,<7,R![-144, -16, 1]>],Snew); return Vf; end function;