// Make newform 126.4.g.g in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_126_g();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_126_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_126_4_g_g();". // 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_126_4_g_g();". // 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 := [112896, 336, 337, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [112896, -337, 337, -1], [673, 0, 0, 1]]; Rf_basisdens := [1, 1, 113232, 337]; 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_126_g();" function MakeCharacter_126_g() N := 126; order := 3; char_gens := [29, 73]; v := [3, 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_126_g_Hecke();" function MakeCharacter_126_g_Hecke(Kf) N := 126; order := 3; char_gens := [29, 73]; char_values := [[1, 0, 0, 0], [0, 0, -1, 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, 2, 0], [0, 0, 0, 0], [0, 1, 2, 0], [1, 0, -3, 1], [33, 1, -34, 1], [20, 0, 0, 1], [-48, 4, 44, 4], [0, 3, -23, 0], [0, -4, 76, 0], [-33, 0, 0, -11], [259, 2, -261, 2], [0, -9, 1, 0], [216, 0, 0, -6], [-46, 0, 0, -15], [0, 12, 282, 0], [-123, 3, 120, 3], [9, -25, 16, -25], [0, 4, -114, 0], [-338, -11, 349, -11], [-246, 0, 0, 20], [478, -35, -443, -35], [0, -8, 267, 0], [123, 0, 0, -25], [0, -42, 408, 0], [959, 0, 0, 35], [342, 20, -362, 20], [0, -21, 1237, 0], [0, -45, 132, 0], [274, 49, -323, 49], [840, 0, 0, 82], [-1393, 0, 0, -68], [0, 7, 86, 0], [-1710, 50, 1660, 50], [-850, 0, 0, 101], [0, 6, 468, 0], [-437, -59, 496, -59], [628, 18, -646, 18], [0, -132, -1028, 0], [-1266, 0, 0, -8], [0, 70, 212, 0], [1380, -114, -1266, -114], [-1264, 0, 0, -143], [0, 78, -414, 0], [811, -222, -589, -222], [1410, 0, 0, 170], [2908, -36, -2872, -36], [2834, 0, 0, -102], [-4489, 0, 0, 9], [3135, -57, -3078, -57], [0, -211, -289, 0], [0, 62, 1882, 0], [-3024, 0, 0, 54], [-71, -39, 110, -39], [-9, 0, 0, -75], [0, -48, -2322, 0], [5070, 102, -5172, 102], [1845, -55, -1790, -55], [0, 247, -2256, 0], [2728, 255, -2983, 255], [-2406, 0, 0, 38], [-2858, -17, 2875, -17], [2007, 0, 0, 9], [3920, 0, 0, -175], [-1152, 86, 1066, 86], [0, 38, -2935, 0], [0, -63, 3408, 0], [0, -147, -1385, 0], [-1039, 0, 0, -44], [-3228, 168, 3060, 168], [5594, 0, 0, -232], [-1464, -162, 1626, -162], [0, -218, -1660, 0], [-5231, 354, 4877, 354], [0, -157, 9437, 0], [-4864, 0, 0, -307], [0, 302, -1328, 0], [-11934, 68, 11866, 68], [0, 319, 2469, 0], [0, -264, 7776, 0], [-143, -366, 509, -366], [-1728, 0, 0, 174], [6530, 0, 0, -53], [2604, -246, -2358, -246], [-4114, 0, 0, 201], [0, -41, -8640, 0], [0, -91, -4544, 0], [10956, 0, 0, -38], [0, 518, 3347, 0], [7482, 0, 0, 274], [80, 0, 0, 797], [0, 502, 6134, 0], [-12342, -218, 12560, -218], [-1019, -282, 1301, -282], [-2637, 0, 0, 633], [0, 101, 2045, 0], [2916, 0, 0, -606], [0, 423, -138, 0], [-13506, -426, 13932, -426], [0, -675, 4807, 0], [0, 583, -7041, 0], [4664, 0, 0, 364], [17163, -301, -16862, -301], [20805, -143, -20662, -143], [0, -690, -7890, 0], [-18626, 287, 18339, 287], [7249, 544, -7793, 544], [-7455, 0, 0, 623], [0, 510, 7506, 0], [5178, -1014, -4164, -1014], [19115, 0, 0, -348], [0, 802, 6627, 0], [3976, -538, -3438, -538], [78, 0, 0, -472], [526, -1259, 733, -1259], [-4213, 0, 0, 645], [30570, 82, -30652, 82], [-13654, 0, 0, -591], [-366, 420, -54, 420], [0, -399, -12006, 0], [-12936, 0, 0, -688], [-12158, 211, 11947, 211], [10397, 0, 0, -798], [0, 1131, -4212, 0], [-18789, 21, 18768, 21], [0, -627, 3715, 0], [-3561, 0, 0, 493], [0, -744, 10186, 0], [0, 264, -3750, 0], [-2479, 0, 0, -496], [0, 823, 9651, 0], [14146, 191, -14337, 191], [-8046, 0, 0, 1128], [0, -356, -21951, 0], [-10690, 0, 0, 724], [0, 80, 3124, 0], [17045, 0, 0, -910], [5886, 1616, -7502, 1616], [3490, -562, -2928, -562], [-24975, 0, 0, -813], [6480, -958, -5522, -958], [5318, 0, 0, 674], [0, 1127, 3310, 0], [20272, -608, -19664, -608], [-20061, 0, 0, 627], [8308, 703, -9011, 703], [5208, 0, 0, -1876], [-30058, 0, 0, -567], [-10626, -1374, 12000, -1374], [0, -486, 13312, 0], [0, 196, -19930, 0], [0, 458, 7862, 0], [-13626, 0, 0, 1782], [-19384, 0, 0, -963], [0, 846, 12210, 0], [-6320, -637, 6957, -637], [-37674, 0, 0, -450], [0, 1613, -18493, 0], [0, 384, -46422, 0], [13355, 0, 0, 204], [10431, 977, -11408, 977], [0, 1822, -19342, 0], [-12234, 0, 0, 962], [-21475, 0, 0, -804], [0, 891, -26472, 0], [13386, -444, -12942, -444], [-5388, 1788, 3600, 1788], [8695, -2394, -6301, -2394], [18802, -873, -17929, -873]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_126_g_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_126_4_g_g();". // 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_126_4_g_g(:prec:=4) chi := MakeCharacter_126_g(); 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_126_4_g_g();". // 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_126_4_g_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_126_g(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<5,R![108900, 1650, 355, -5, 1]>],Snew); return Vf; end function;