// Make newform 405.4.e.s in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_405_e();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_405_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_405_4_e_s();". // 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_405_4_e_s();". // 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 := [243, 0, 223, 0, 29, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0], [9, 13, 1, 1, 0, 0], [-36, 79, 0, 20, 0, 1], [9, 0, 1, 0, 0, 0], [-207, 29, -144, -20, -9, -1], [207, 29, 144, -20, 9, -1]]; Rf_basisdens := [1, 4, 72, 2, 72, 72]; 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_405_e();" function MakeCharacter_405_e() N := 405; order := 3; char_gens := [326, 82]; v := [2, 3]; // 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_405_e_Hecke();" function MakeCharacter_405_e_Hecke(Kf) N := 405; order := 3; char_gens := [326, 82]; char_values := [[-1, 0, -1, 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 := 4; raw_aps := [[0, -1, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0], [5, 0, 5, 0, 0, 0], [0, -5, -10, 5, 3, 0], [0, -7, 17, 7, 5, 0], [20, 13, 20, 0, 0, 13], [14, 0, 0, 8, -8, 8], [-5, 0, 0, -10, -14, 14], [22, 15, 22, 0, 0, 19], [0, -65, 95, 65, -13, 0], [-211, 5, -211, 0, 0, 1], [-156, 0, 0, -54, 2, -2], [-59, 2, -59, 0, 0, -62], [0, -28, -288, 28, -8, 0], [0, -67, 56, 67, -31, 0], [186, 0, 0, 53, -21, 21], [-159, 58, -159, 0, 0, 90], [0, 122, 6, -122, 58, 0], [-38, -154, -38, 0, 0, 18], [177, 0, 0, 79, 69, -69], [-226, 0, 0, 32, 12, -12], [0, 82, -184, -82, -54, 0], [0, 210, 648, -210, -30, 0], [-297, 0, 0, -39, 3, -3], [0, 336, 112, -336, 32, 0], [0, 393, 329, -393, 5, 0], [402, -153, 402, 0, 0, 143], [1870, 0, 0, -12, -16, 16], [631, 0, 0, 15, 133, -133], [646, 412, 646, 0, 0, 136], [-954, 0, 0, -319, 227, -227], [-723, -252, -723, 0, 0, 60], [0, -378, -1108, 378, 38, 0], [43, -522, 43, 0, 0, 78], [-254, 32, -254, 0, 0, -152], [0, -391, 1131, 391, -59, 0], [1932, -351, 1932, 0, 0, -139], [-170, 0, 0, -666, -206, 206], [-74, 570, -74, 0, 0, 46], [0, -507, 794, 507, -211, 0], [-477, 0, 0, 344, -360, 360], [-981, 0, 0, 561, 83, -83], [0, 97, 669, -97, 213, 0], [-40, -708, -40, 0, 0, -488], [392, 0, 0, -243, 175, -175], [144, 0, 0, -160, 200, -200], [1697, 702, 1697, 0, 0, 574], [0, -60, 1004, 60, 360, 0], [0, 622, -1694, -622, 538, 0], [-502, 878, -502, 0, 0, 718], [-688, 0, 0, -24, -260, 260], [-1808, 322, -1808, 0, 0, 682], [0, -349, 2713, 349, -457, 0], [2628, 0, 0, 465, 291, -291], [1026, -24, 1026, 0, 0, 876], [0, 1147, 1498, -1147, -581, 0], [-1963, 0, 0, -1561, 301, -301], [-1976, 0, 0, -1162, 682, -682], [0, -11, -3860, 11, 5, 0], [0, 855, 2686, -855, -701, 0], [946, 986, 946, 0, 0, -102], [-2960, -429, -2960, 0, 0, 715], [-2012, 0, 0, 2132, -452, 452], [-815, -441, -815, 0, 0, 283], [0, -168, 1696, 168, 272, 0], [0, 1351, -6340, -1351, 203, 0], [0, -557, -549, 557, 463, 0], [-2612, 1558, -2612, 0, 0, -270], [-5806, -1124, -5806, 0, 0, -604], [0, -1621, -787, 1621, 951, 0], [0, 474, 7952, -474, 566, 0], [-2075, 0, 0, 27, -871, 871], [0, -458, -1706, 458, -178, 0], [-6442, -956, -6442, 0, 0, 604], [9888, 0, 0, 113, -517, 517], [-3464, -843, -3464, 0, 0, 37], [0, -702, 3886, 702, 82, 0], [-5910, 0, 0, 1364, 236, -236], [-8090, -1155, -8090, 0, 0, 169], [-2230, 3519, -2230, 0, 0, 1363], [7276, 1580, 7276, 0, 0, 916], [0, -793, 4341, 793, 1291, 0], [1339, 0, 0, -861, 1121, -1121], [-5998, 0, 0, -2230, -618, 618], [0, 1877, 1587, -1877, -1151, 0], [0, -1434, 4552, 1434, 958, 0], [-3687, 0, 0, 590, 1218, -1218], [0, 344, 5120, -344, 1752, 0], [0, -1275, -475, 1275, -1399, 0], [5054, -1681, 5054, 0, 0, 595], [-106, 0, 0, 1384, -2084, 2084], [0, -873, -5107, 873, -1933, 0], [2864, 0, 0, 1047, 1737, -1737], [4445, -358, 4445, 0, 0, -622], [2689, 2668, 2689, 0, 0, -108], [-13308, 0, 0, -1772, 1656, -1656], [-1030, 3112, -1030, 0, 0, -1240], [-7926, 0, 0, -2549, 153, -153], [5596, 0, 0, 182, 1318, -1318], [5775, 0, 0, 1903, 485, -485], [0, 188, 6608, -188, 0, 0], [1144, 0, 0, 1953, -2797, 2797], [-6148, -2328, -6148, 0, 0, 1940], [0, 342, -7739, -342, -1970, 0], [-2873, -4437, -2873, 0, 0, -1689], [738, 0, 0, -1110, -2914, 2914], [0, 4422, 7364, -4422, -478, 0], [-5694, 0, 0, 4372, -768, 768], [5353, 335, 5353, 0, 0, -1781], [0, 1788, 6793, -1788, -2476, 0], [8304, -3032, 8304, 0, 0, -1696], [1720, 0, 0, 4637, 127, -127], [1400, -1362, 1400, 0, 0, 1874], [0, 5331, -11708, -5331, 287, 0], [14107, 0, 0, -2629, 889, -889], [0, 4989, -10081, -4989, 1361, 0], [10384, -6712, 10384, 0, 0, -684], [-9970, 0, 0, -1330, -1502, 1502], [-3064, 698, -3064, 0, 0, -2338], [0, 2777, 6024, -2777, 1581, 0], [-101, 991, -101, 0, 0, 3411], [0, -200, -8474, 200, -820, 0], [0, -6117, -3666, 6117, 1251, 0], [-4932, 0, 0, 4926, 426, -426], [0, -2779, -5612, 2779, 2657, 0], [2447, 0, 0, 1837, -1625, 1625], [0, 1436, -9758, -1436, 1196, 0], [4575, 0, 0, 7053, 231, -231], [0, 3963, -2380, -3963, 2235, 0], [-2870, -3492, -2870, 0, 0, -4056], [9157, 0, 0, -7447, 2995, -2995], [-22434, -3554, -22434, 0, 0, -1158], [2232, 5470, 2232, 0, 0, -1090], [6228, 0, 0, -4627, -385, 385], [17519, -1422, 17519, 0, 0, 1682], [-11145, -315, -11145, 0, 0, -1351], [-15940, 0, 0, -1986, 4918, -4918], [-20114, -2804, -20114, 0, 0, 2532], [10000, 5102, 10000, 0, 0, -1262], [16647, 0, 0, -2054, 1158, -1158], [7395, 0, 0, 4699, -271, 271], [0, 3213, 8995, -3213, -239, 0], [4110, -1182, 4110, 0, 0, 3662], [15868, 0, 0, -3400, 3356, -3356], [-7421, 0, 0, 2299, -1439, 1439], [0, -107, -9991, 107, 161, 0], [0, -5204, -5266, 5204, 996, 0], [0, -5734, -7392, 5734, 3978, 0], [-7779, -6191, -7779, 0, 0, 581], [-4830, 0, 0, 3523, -4659, 4659], [-28772, -5815, -28772, 0, 0, 69], [5919, 0, 0, 521, -5229, 5229], [9902, 0, 0, -4128, -1452, 1452], [-6264, 6477, -6264, 0, 0, -3771], [0, -5972, -11234, 5972, -3472, 0], [0, 7325, -7589, -7325, 313, 0], [-1323, 0, 0, 4229, -3337, 3337], [0, 2707, -20503, -2707, -265, 0], [-538, 0, 0, -38, 3450, -3450], [20962, -6988, 20962, 0, 0, -1316], [0, 4734, 29902, -4734, -470, 0], [-14336, 0, 0, -9952, -2212, 2212], [1738, -7398, 1738, 0, 0, 3162], [21121, 0, 0, 3252, 3956, -3956], [5272, -4286, 5272, 0, 0, -746], [0, 1998, 2766, -1998, 5622, 0], [5595, 0, 0, 8259, -5431, 5431], [0, 3253, 23418, -3253, 5569, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_405_e_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_405_4_e_s();". // 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_405_4_e_s(:prec:=6) chi := MakeCharacter_405_e(); 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_405_4_e_s();". // 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_405_4_e_s( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_405_e(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<2,R![324, 252, 214, 22, 15, 1, 1]>,<7,R![260499600, -10458720, 823404, -16080, 1273, -25, 1]>],Snew); return Vf; end function;