// Make newform 1728.3.q.g in Magma, downloaded from the LMFDB on 19 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1728_q();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1728_q_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_1728_3_q_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_1728_3_q_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 := [9, -3, -2, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [-9, 16, 2, 1], [-9, -2, 2, 1], [12, 8, 1, -4]]; Rf_basisdens := [1, 6, 6, 3]; 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_1728_q();" function MakeCharacter_1728_q() N := 1728; order := 6; char_gens := [703, 325, 1217]; v := [6, 6, 5]; // 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_1728_q_Hecke();" function MakeCharacter_1728_q_Hecke(Kf) N := 1728; order := 6; char_gens := [703, 325, 1217]; char_values := [[1, 0, 0, 0], [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 := 3; raw_aps := [[0, 0, 0, 0], [0, 0, 0, 0], [3, -1, 1, 0], [-1, 2, 0, -1], [-7, 0, 5, -2], [1, -1, 3, 2], [7, 1, 15, -1], [-1, -1, -1, -1], [-33, -1, -17, 0], [-7, 0, 14, 7], [3, -3, 5, 6], [18, 4, 4, 4], [6, -8, -1, 0], [23, 0, 23, 0], [15, 0, -12, 3], [16, -8, 24, 8], [42, -8, 17, 0], [-19, -6, -22, 3], [6, -6, 61, 12], [-14, -2, -30, 2], [23, 3, 3, 3], [39, 10, 44, -5], [-15, 0, 12, -3], [-56, -8, -120, 8], [-99, 4, -97, -2], [73, 0, -74, -1], [3, -3, 101, 6], [-5, -11, -21, 11], [-80, 6, 6, 6], [99, -21, 39, 0], [20, -18, -18, -18], [21, 29, 25, 0], [-65, 0, 67, 2], [2, -2, -43, 4], [-39, -23, -31, 0], [-173, -18, -182, 9], [21, -21, -101, 42], [-126, 4, 4, 4], [9, 21, 15, 0], [-87, 0, 66, -21], [-68, -20, -156, 20], [-108, -14, -14, -14], [-1, 0, 38, 37], [0, 0, -49, 0], [56, -64, 48, 64], [-134, -8, -8, -8], [-17, 17, -249, -34], [19, 30, 34, -15], [-135, 0, 153, 18], [15, -15, 175, 30], [155, -19, 291, 19], [135, -9, 63, 0], [65, -64, 33, 32], [-77, 61, -93, -61], [-54, -12, -33, 0], [-49, 0, 26, -23], [-92, -44, -228, 44], [56, 42, 42, 42], [-307, 42, -286, -21], [-137, 0, 76, -61], [3, -3, 367, 6], [123, 7, 65, 0], [145, -3, -3, -3], [429, -47, 191, 0], [-295, 8, -291, -4], [-23, 0, 28, 5], [177, -82, 136, 41], [-38, 38, -29, -76], [-234, -84, -159, 0], [13, -86, -30, 43], [223, 0, -257, -34], [-270, -18, -558, 18], [-113, 66, -80, -33], [1, -1, -141, 2], [-225, 31, 31, 31], [105, 85, 95, 0], [73, 0, -110, -37], [-194, -24, -24, -24], [-12, 10, -1, 0], [-38, 38, -173, -76], [-243, 69, -87, 0], [97, 118, 156, -59], [-422, 22, -822, -22], [47, 27, 27, 27], [19, -114, -38, 57], [-231, 0, 255, 24], [33, 87, 153, -87], [-107, 12, -101, -6], [137, 0, -148, -11], [-19, 19, 393, -38], [-185, 25, -345, -25], [-81, 0, -18, -99], [-218, -68, -68, -68], [-264, 10, -127, 0], [-54, 54, -107, -108], [120, -24, 216, 24], [669, 5, 337, 0], [13, 43, 69, -43], [-590, -12, -12, -12], [-210, -56, -56, -56], [191, 72, 227, -36], [-46, 86, -6, -86], [1134, -12, 561, 0], [287, 0, -151, 136], [86, -86, -19, 172], [-121, 3, 3, 3], [-215, 0, 61, -154], [340, 28, 708, -28], [417, -67, 175, 0], [349, 132, 415, -66], [-29, 29, -587, -58], [154, -36, -36, -36], [852, 10, 431, 0], [-517, 60, -487, -30], [534, -52, -52, -52], [-113, 0, -17, -130], [0, 0, 577, 0], [378, -90, 666, 90], [915, 79, 497, 0], [193, 0, -8, 185], [15, -15, -1073, 30], [-297, -158, -376, 79], [505, 0, -524, -19], [-39, -57, -135, 57], [-679, 54, -652, -27], [232, -80, 384, 80], [-27, 50, -2, -25], [54, -198, -90, 198], [-581, -90, -626, 45], [-105, 105, 223, -210], [-575, -3, -3, -3], [117, -87, 15, 0], [-35, 35, -167, -70], [790, 48, 48, 48], [1395, 27, 711, 0], [-43, 43, 705, -86], [-156, -156, -468, 156], [-69, 69, -593, -138], [-585, 75, -255, 0], [-551, -17, -1119, 17], [223, -81, -81, -81], [313, 0, -374, -61], [-7, 7, 273, -14], [284, 92, 660, -92], [-466, -40, -40, -40], [-273, 0, 204, -69], [-1171, -6, -1174, 3], [-297, 0, 252, -45], [-4, 4, 521, -8], [226, -50, 402, 50], [5, -5, -757, 10], [-532, 68, -996, -68], [515, 59, 59, 59], [-759, -11, -385, 0], [-129, -136, -197, 68], [-33, 0, 48, 15], [-1252, -42, -42, -42], [279, 0, -180, 99], [-498, 116, 116, 116], [177, -47, 65, 0], [-135, 0, 261, 126], [171, -243, 99, 243], [129, -129, -223, 258], [108, -324, -108, 324], [-456, -154, -305, 0], [623, 0, -598, 25], [248, -150, -150, -150], [917, -30, 902, 15], [410, 120, 120, 120], [237, 149, 193, 0], [473, 0, -469, 4], [111, -111, 367, 222], [-1029, -29, -529, 0], [113, -256, -15, 128], [65, -65, -159, 130], [-69, -13, -41, 0], [-61, -60, -91, 30], [198, 18, 414, -18], [478, 100, 100, 100], [-247, -162, -328, 81], [551, -262, 420, 131], [217, 0, -137, 80], [-33, 33, -497, -66], [33, -345, -279, 345], [-183, 181, -1, 0], [-887, 0, 820, -67], [292, -30, -30, -30], [481, -50, 456, 25], [20, -20, -1833, 40], [322, 46, 690, -46], [817, -23, -23, -23], [-183, 0, -21, -204], [-1791, 13, 13, 13], [-647, 0, 520, -127], [569, -73, 1065, 73], [-1572, -94, -833, 0], [170, -170, 627, 340], [1077, 98, 1126, -49], [-97, 0, 293, 196], [-116, 4, -228, -4], [783, -237, 273, 0], [555, 70, 590, -35], [25, -25, -525, 50], [517, 300, 667, -150], [691, -35, 1347, 35], [-364, -124, -852, 124], [-290, -140, -140, -140], [-330, -244, -287, 0], [-433, 0, 695, 262], [-172, 172, 905, -344], [-423, -31, -31, -31], [-2055, 73, -991, 0], [-1481, 138, -1412, -69], [-95, 0, 316, 221], [531, -213, 159, 0], [157, -492, -89, 246], [39, -39, -787, 78], [975, 0, -1059, -84], [596, 188, 1380, -188], [141, 85, 113, 0], [-47, 47, 51, -94], [69, -69, 857, 138], [-570, -296, -433, 0], [-490, 60, 60, 60], [-576, 162, -207, 0], [-1047, 62, -1016, -31], [-537, 0, 84, -453], [-478, 302, -654, -302], [-1057, 50, -1032, -25], [929, 0, -868, 61], [-85, 85, 791, -170], [-1226, -144, -144, -144], [141, -141, -727, 282], [78, -176, -49, 0], [-1727, -2, -1728, 1], [319, 0, 262, 581], [-43, 43, -591, -86], [6, -174, -162, 174], [-1188, 102, -543, 0], [152, -328, -24, 328], [217, 0, -353, -136], [226, -108, -108, -108], [-87, 87, 353, -174], [1480, 78, 78, 78], [-132, 386, 127, 0], [1039, 0, -1124, -85], [-1304, -134, -134, -134], [1704, -166, 769, 0], [135, -135, 1063, 270], [-780, -276, -1836, 276], [147, -147, 151, 294], [735, 105, 1575, -105], [75, 467, 271, 0], [-641, -54, -668, 27], [-807, 0, 444, -363], [1097, -265, 1929, 265], [-968, -210, -210, -210], [257, 126, 320, -63], [-198, -306, -702, 306], [851, -57, -57, -57], [1347, -2, 1346, 1], [-231, 0, -69, -300], [-111, 111, -605, -222], [-294, 172, 172, 172], [-1461, 355, -553, 0], [-1741, 324, -1579, -162], [68, 20, 156, -20], [-273, 0, 231, -42], [24, -24, -95, 48], [-9, 555, 273, 0], [-235, 235, 1067, -470], [2118, 64, 64, 64], [1321, 312, 1477, -156], [9, -9, 2993, 18], [-216, 216, 575, -432], [224, -270, -270, -270], [453, -115, 169, 0], [1505, -42, 1484, 21], [1127, -45, -45, -45], [-311, 0, 307, -4], [-1155, -383, -769, 0], [39, -39, 2381, 78], [79, 0, -518, -439], [1685, -126, 1622, 63], [252, -252, 985, 504], [1308, -204, 2412, 204], [-2162, 4, 4, 4], [-1137, 115, -511, 0], [-509, 174, -422, -87], [831, 57, 1719, -57], [-615, 0, 114, -501], [-485, -203, -1173, 203], [930, 380, 655, 0], [-1278, 384, -447, 0], [2129, -234, 2012, 117], [2805, -179, 1313, 0], [-165, -506, -418, 253], [1273, 0, -1394, -121], [-336, -120, -792, 120], [57, -394, -140, 197], [-160, 160, 399, -320], [1008, -216, 1800, 216], [1490, 96, 96, 96]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1728_q_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1728_3_q_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_1728_3_q_g(:prec:=4) chi := MakeCharacter_1728_q(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 3)); 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_1728_3_q_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_1728_3_q_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1728_q(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign))); Vf := Kernel([<5,R![324, 162, 9, -9, 1]>,<7,R![5476, -74, 75, 1, 1]>],Snew); return Vf; end function;