// Make newform 1350.4.c.y in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1350_c();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1350_c_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_1350_4_c_y();". // 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_1350_4_c_y();". // 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 := [2704, 0, 105, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 53, 0, 1], [0, 209, 0, 1], [158, 0, 3, 0]]; Rf_basisdens := [1, 52, 52, 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_1350_c();" function MakeCharacter_1350_c() N := 1350; order := 2; char_gens := [1001, 1027]; v := [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_1350_c_Hecke();" function MakeCharacter_1350_c_Hecke(Kf) N := 1350; order := 2; char_gens := [1001, 1027]; char_values := [[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, -2, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -7, -1, 0], [8, 0, 0, 2], [0, 4, 1, 0], [0, -2, 2, 0], [-22, 0, 0, 1], [0, 68, 4, 0], [-16, 0, 0, 2], [73, 0, 0, 2], [0, 121, 12, 0], [12, 0, 0, 6], [0, 7, -14, 0], [0, -154, 10, 0], [0, 430, 14, 0], [8, 0, 0, 8], [320, 0, 0, -27], [0, 622, 15, 0], [-60, 0, 0, -36], [0, 701, 12, 0], [259, 0, 0, 0], [0, 2, -62, 0], [780, 0, 0, -24], [0, 525, -23, 0], [1148, 0, 0, 14], [0, 755, 12, 0], [0, -932, -40, 0], [-309, 0, 0, -4], [0, -364, 16, 0], [0, 88, -36, 0], [1012, 0, 0, 10], [0, -1014, 54, 0], [109, 0, 0, 48], [1260, 0, 0, -18], [1394, 0, 0, 39], [0, 509, 20, 0], [0, 1542, -85, 0], [0, -3794, -10, 0], [0, 1732, 104, 0], [3092, 0, 0, 8], [-2057, 0, 0, -115], [3652, 0, 0, -38], [0, 1853, 105, 0], [0, -1366, 118, 0], [1005, 0, 0, 53], [-945, 0, 0, 115], [0, -3141, -58, 0], [0, -2496, -72, 0], [2099, 0, 0, -152], [0, 1902, -114, 0], [2632, 0, 0, 94], [-2165, 0, 0, -43], [1460, 0, 0, -70], [0, -6062, 38, 0], [0, 5026, 62, 0], [1564, 0, 0, 160], [-3631, 0, 0, 192], [0, 1563, -128, 0], [3524, 0, 0, -202], [0, -195, -178, 0], [0, -1908, -228, 0], [0, -1013, 294, 0], [-7112, 0, 0, -110], [0, -5808, -25, 0], [0, -478, -350, 0], [-1165, 0, 0, -420], [0, 1212, 139, 0], [0, -2928, 168, 0], [4741, 0, 0, -360], [0, 6408, 204, 0], [-4436, 0, 0, -158], [0, 4654, 249, 0], [0, -267, 125, 0], [-577, 0, 0, -521], [0, -4042, 334, 0], [-4192, 0, 0, -58], [0, -3459, 16, 0], [-12468, 0, 0, 72], [3845, 0, 0, 391], [4952, 0, 0, -400], [1853, 0, 0, 456], [2340, 0, 0, -18], [0, 6139, -224, 0], [8863, 0, 0, 402], [0, -2380, -560, 0], [-6464, 0, 0, 154], [0, 1822, -132, 0], [9540, 0, 0, -156], [0, 5243, -48, 0], [0, 7008, -72, 0], [-1900, 0, 0, 668], [0, -3214, -751, 0], [-17584, 0, 0, 98], [-2579, 0, 0, -258], [0, -898, 910, 0], [6876, 0, 0, 348], [-2580, 0, 0, -732], [0, 12941, 231, 0], [1848, 0, 0, -353], [0, 14065, 468, 0], [0, 6846, -594, 0], [0, -13532, -256, 0], [-9524, 0, 0, 64], [3126, 0, 0, -533], [0, 691, 663, 0], [0, 19278, -174, 0], [0, -5972, -520, 0], [-1836, 0, 0, 624], [-1761, 0, 0, 292], [0, 5965, -576, 0], [0, -3661, 324, 0], [0, -1752, -624, 0], [-10910, 0, 0, -159], [20773, 0, 0, -127], [8652, 0, 0, -12], [0, 4532, -300, 0], [0, 15040, 596, 0], [0, -7468, 808, 0], [-17140, 0, 0, -610], [-1561, 0, 0, 480], [0, 791, -996, 0], [0, 21862, 350, 0], [0, 6758, -134, 0], [80, 0, 0, -948], [-10192, 0, 0, 1028], [-5824, 0, 0, 265], [5352, 0, 0, 498], [0, 10105, -618, 0], [0, -11722, -300, 0], [3664, 0, 0, -744], [0, -17412, -456, 0], [4517, 0, 0, 96], [0, -13379, 405, 0], [-6128, 0, 0, 1030], [5739, 0, 0, 269], [0, 4766, -242, 0], [0, 9271, 1173, 0], [0, -2572, 1768, 0], [14220, 0, 0, 312], [16795, 0, 0, 254], [5180, 0, 0, -226], [0, 9238, 1585, 0], [0, 1526, -878, 0], [30775, 0, 0, 60], [-7644, 0, 0, 420], [0, 3512, -1341, 0], [0, -4468, -1448, 0], [-21095, 0, 0, -1248], [0, -2458, -722, 0], [0, 15323, 1067, 0], [3912, 0, 0, 330], [0, -18273, -787, 0], [0, -10622, 338, 0], [0, -15881, 768, 0], [20364, 0, 0, -1410], [-4919, 0, 0, -978], [20000, 0, 0, 644], [0, -44485, 359, 0], [11152, 0, 0, -146], [0, 3734, 538, 0], [0, 10202, 598, 0], [0, 22459, -960, 0], [8992, 0, 0, -356], [0, -33492, -576, 0], [0, -11938, 202, 0], [41438, 0, 0, -75], [0, -40970, -324, 0], [15388, 0, 0, 1533], [0, -7462, -74, 0], [20912, 0, 0, 416], [-29953, 0, 0, 396], [25532, 0, 0, -244], [0, -11033, 1627, 0], [-25923, 0, 0, 1457], [1544, 0, 0, 704], [10375, 0, 0, 242], [39640, 0, 0, -914], [0, -14294, -191, 0], [-10719, 0, 0, -1684], [0, 3138, -941, 0], [1480, 0, 0, 586], [0, -30931, 1356, 0], [0, -25304, 548, 0], [0, 27152, 328, 0], [30908, 0, 0, 446], [0, -39908, -1485, 0], [0, 8382, -1273, 0], [-26884, 0, 0, 553], [-14620, 0, 0, 1448], [0, 6927, 2075, 0], [0, 22970, -326, 0], [48426, 0, 0, -1259], [28944, 0, 0, 2196], [0, -45600, 252, 0], [0, 22330, 446, 0], [-17101, 0, 0, -1128], [0, -9727, 792, 0], [0, -5924, 980, 0], [0, 21202, 914, 0], [38720, 0, 0, -1828], [57809, 0, 0, 144], [0, 35719, 1224, 0], [-32813, 0, 0, -600], [22320, 0, 0, 606], [0, -32378, -1174, 0], [9086, 0, 0, 1189], [0, 49138, 530, 0], [7380, 0, 0, 2154], [3265, 0, 0, -247], [0, -36840, 949, 0], [-9772, 0, 0, -3154], [0, -6046, -1647, 0], [0, -57714, 714, 0], [7396, 0, 0, 850], [-29521, 0, 0, -2460], [0, 24257, 410, 0], [-31328, 0, 0, -1202], [0, -31796, 248, 0], [0, 43152, -1488, 0], [11728, 0, 0, 311], [-66467, 0, 0, 1086], [4120, 0, 0, 2242], [0, 83840, 348, 0], [0, 60204, 516, 0], [-15657, 0, 0, 2525], [0, -19908, -2016, 0], [-34524, 0, 0, 882], [0, -17442, -2303, 0], [-70988, 0, 0, -980], [0, -18077, 2371, 0], [-51104, 0, 0, 108], [34575, 0, 0, -1370], [77500, 0, 0, -686], [0, -32605, 1692, 0], [0, 10456, -676, 0], [-2333, 0, 0, -2772], [0, -11520, 948, 0], [8448, 0, 0, 3876], [-3488, 0, 0, 1432], [0, -45230, 3050, 0], [44453, 0, 0, -1575], [0, 35945, -633, 0], [11731, 0, 0, -2415], [0, 20422, 1478, 0], [18000, 0, 0, -4086], [0, 59139, 1069, 0], [70220, 0, 0, -28], [-34633, 0, 0, 2950], [0, -30656, -2752, 0], [0, -64694, -1044, 0], [-2920, 0, 0, 1544], [0, 43874, -590, 0], [25321, 0, 0, -921], [0, -31348, 2788, 0], [-23452, 0, 0, 1502], [-85312, 0, 0, -1761], [0, 77668, 1260, 0], [0, 40684, -748, 0], [0, -59774, 2100, 0], [0, -76953, -1111, 0], [0, 35598, 2058, 0], [25459, 0, 0, -1440], [0, 54565, -1547, 0], [0, 103114, -982, 0], [40216, 0, 0, -1884], [-22804, 0, 0, 1472], [-55968, 0, 0, 30], [0, 48872, -1296, 0], [0, -3468, -1500, 0], [-80856, 0, 0, 841], [0, -98363, 180, 0], [0, -33082, 3600, 0], [5527, 0, 0, 939], [0, 52095, -1664, 0], [0, 22631, 1449, 0], [0, 54892, 1028, 0], [59453, 0, 0, 1492], [-32849, 0, 0, 1532], [-63760, 0, 0, -850], [0, -6952, -740, 0], [-72124, 0, 0, -612], [0, -15274, -4766, 0], [-64939, 0, 0, 975], [0, 778, -5055, 0], [-2816, 0, 0, 226], [0, 26992, 229, 0], [0, 95612, 1108, 0], [54397, 0, 0, -3444], [-58520, 0, 0, 58], [-106740, 0, 0, 2352], [0, 118912, 620, 0], [0, 72256, -184, 0], [100, 0, 0, -2996], [0, -1333, 3600, 0], [-140236, 0, 0, -112], [-15218, 0, 0, -1255], [0, 3990, 378, 0], [24756, 0, 0, 3612], [0, -26441, -5700, 0], [0, -114229, -384, 0], [0, -27520, -3824, 0], [-33585, 0, 0, -2590]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1350_c_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1350_4_c_y();". // 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_1350_4_c_y(:prec:=4) chi := MakeCharacter_1350_c(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 4)); 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_1350_4_c_y();". // 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_1350_4_c_y( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1350_c(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<7,R![183184, 0, 1025, 0, 1]>,<11,R![-1800, -18, 1]>],Snew); return Vf; end function;