// Make newform 392.3.k.l in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_392_k();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_392_k_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_392_3_k_l();". // 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_392_3_k_l();". // 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 := [4, -20, 60, -116, 164, -166, 155, -163, 86, 3, -4, -3, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-114476732, 298885728, -675410750, 492489836, -266302456, 906643427, -324275527, -280693408, 33927641, 36949178, -128409, -2848753], [-26684466, 111915639, -270088448, 416913921, -267919561, 376525973, -469374709, 91567337, 28087343, 3252536, -6494043, 908144], [13204764, -46482276, 48635510, -177078348, 215038026, -122846749, 201471321, -175954938, 1459025, 11867940, 4608465, -2016555], [16276324, -40145506, 87465566, -46480180, 1783409, -103354841, 13990590, 80688819, -6680758, -8410783, -880921, 894552], [-32192540, 70705150, -68273940, 4804172, 130000577, -2408183, 91183726, -310515677, 24171184, 28320593, 5070151, -3736366], [29235872, -154926652, 426711138, -667643040, 889048300, -701452097, 912321093, -742433540, 5607853, 48705482, 20159299, -8673537], [49862132, -236999870, 526622432, -836256216, 981317799, -797312818, 927115697, -751559539, 8785933, 48834261, 20455402, -8864783], [15160294, -43781960, 160381312, -254949459, 278083421, -243549767, 282488251, -234558069, 3410783, 15510516, 6284192, -2781796], [-55637788, 194264578, -603936480, 975668172, -1267262003, 959373046, -1071722261, 1263522623, -55362465, -95322325, -28755150, 15221487], [65016998, -292014698, 678019552, -1067833785, 1141965083, -1002633735, 1189138614, -752734228, -12733522, 41996941, 23461665, -8729167], [6095104, -51802364, 142102466, -311221520, 291501740, -225704689, 386668629, -240378660, -8114019, 13528514, 7514003, -2679889]]; Rf_basisdens := [1, 20513668, 5128417, 10256834, 2930524, 10256834, 20513668, 20513668, 5128417, 20513668, 10256834, 2930524]; 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_392_k();" function MakeCharacter_392_k() N := 392; order := 6; char_gens := [295, 197, 297]; v := [3, 3, 4]; // 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_392_k_Hecke();" function MakeCharacter_392_k_Hecke(Kf) N := 392; order := 6; char_gens := [295, 197, 297]; char_values := [[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 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 := 3; raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, -1, 0, 1, 1, 0, 0, -1, 0, 0], [0, 0, -1, 0, -1, 0, 0, 0, 1, 0, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [6, 0, 0, 0, 0, 0, -3, 6, 0, 3, 0, 0], [0, 2, -2, -5, 0, -5, 0, 0, -5, 0, -5, -2], [-5, 0, 0, 1, 0, -1, 2, -5, 0, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 16, -4, 1, 4, 0], [0, 0, 0, 0, -7, 0, 0, 0, -5, 0, -5, -7], [0, 8, -8, 3, 0, 3, 0, 0, 3, 0, 3, -2], [0, 0, 0, -6, -9, -6, 0, 0, 0, 0, 0, 0], [0, 0, 9, 0, -1, 0, 0, 0, -4, 0, -4, -1], [26, 0, 0, 1, 0, -1, -18, 0, 1, 0, -1, 0], [-20, 0, 0, -10, 0, 10, -10, 0, -10, 0, 10, 0], [0, 0, 16, 0, -5, 0, 0, 0, -2, 0, -2, -5], [0, 3, 0, 20, 7, 20, 0, 0, 0, 0, 0, 0], [12, 0, 0, -5, 0, 5, 9, 12, 0, -9, 0, 0], [0, 0, 15, 0, 25, 0, 0, 0, -10, 0, -10, 25], [70, 0, 0, -3, 0, 3, 1, 70, 0, -1, 0, 0], [0, -6, 6, -22, 0, -22, 0, 0, -22, 0, -22, 8], [-15, 0, 0, 12, 0, -12, 18, -15, 0, -18, 0, 0], [0, 0, 16, 0, 25, 0, 0, 0, 5, 0, 5, 25], [50, 0, 0, 14, 0, -14, 12, 0, 14, 0, -14, 0], [0, 0, 0, 0, 0, 0, 0, 13, 36, 4, -36, 0], [0, 0, 0, -15, 0, 15, 8, 0, -15, 0, 15, 0], [0, -5, 0, 21, -1, 21, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, -5, 0, 0, 0, -31, 0, -31, -5], [0, 0, 0, 0, 0, 0, 0, 0, -8, 35, 8, 0], [0, 19, 0, 1, -11, 1, 0, 0, 0, 0, 0, 0], [-60, 0, 0, 5, 0, -5, 20, 0, 5, 0, -5, 0], [0, 46, -46, -10, 0, -10, 0, 0, -10, 0, -10, -28], [0, 0, 0, 0, 0, 0, 0, -14, -45, -25, 45, 0], [65, 0, 0, -9, 0, 9, 42, 65, 0, -42, 0, 0], [-22, 0, 0, 0, 0, 0, -40, 0, 0, 0, 0, 0], [0, 0, 37, 0, 77, 0, 0, 0, -63, 0, -63, 77], [0, -20, 0, 6, 49, 6, 0, 0, 0, 0, 0, 0], [0, 11, 0, 26, 37, 26, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 40, -25, -13, 25, 0], [0, -22, 22, -65, 0, -65, 0, 0, -65, 0, -65, 48], [0, 0, 49, 0, 5, 0, 0, 0, -22, 0, -22, 5], [-94, 0, 0, -14, 0, 14, 53, -94, 0, -53, 0, 0], [0, 6, -6, 34, 0, 34, 0, 0, 34, 0, 34, -76], [0, 0, 20, 0, -35, 0, 0, 0, -40, 0, -40, -35], [55, 0, 0, 71, 0, -71, -60, 55, 0, 60, 0, 0], [0, -88, 88, 1, 0, 1, 0, 0, 1, 0, 1, -2], [0, -34, 0, -84, -81, -84, 0, 0, 0, 0, 0, 0], [154, 0, 0, -10, 0, 10, 0, 0, -10, 0, 10, 0], [0, -46, 46, -36, 0, -36, 0, 0, -36, 0, -36, -32], [100, 0, 0, -4, 0, 4, 21, 100, 0, -21, 0, 0], [0, 0, -47, 0, 59, 0, 0, 0, 9, 0, 9, 59], [0, 0, 0, 0, 0, 0, 0, -55, 1, -40, -1, 0], [0, -42, 42, -61, 0, -61, 0, 0, -61, 0, -61, 44], [79, 0, 0, 72, 0, -72, 64, 79, 0, -64, 0, 0], [6, 0, 0, 17, 0, -17, 48, 0, 17, 0, -17, 0], [0, 0, 0, 0, 0, 0, 0, 5, -59, 34, 59, 0], [0, -18, 0, 81, -15, 81, 0, 0, 0, 0, 0, 0], [0, -29, 0, 64, -19, 64, 0, 0, 0, 0, 0, 0], [0, 0, -18, 0, -21, 0, 0, 0, -1, 0, -1, -21], [0, -69, 0, 40, -9, 40, 0, 0, 0, 0, 0, 0], [-76, 0, 0, 49, 0, -49, 8, 0, 49, 0, -49, 0], [30, 0, 0, -2, 0, 2, -9, 30, 0, 9, 0, 0], [0, -20, 20, 16, 0, 16, 0, 0, 16, 0, 16, 64], [-60, 0, 0, 40, 0, -40, -74, 0, 40, 0, -40, 0], [0, 24, 0, 71, -21, 71, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -205, 56, -102, -56, 0], [0, 0, 3, 0, -25, 0, 0, 0, 106, 0, 106, -25], [0, 0, 0, 0, 0, 0, 0, 134, 7, 17, -7, 0], [-50, 0, 0, -135, 0, 135, -62, 0, -135, 0, 135, 0], [180, 0, 0, 71, 0, -71, 75, 180, 0, -75, 0, 0], [0, 24, -24, -45, 0, -45, 0, 0, -45, 0, -45, -50], [-395, 0, 0, -75, 0, 75, -52, -395, 0, 52, 0, 0], [0, 0, 26, 0, -75, 0, 0, 0, 130, 0, 130, -75], [0, 162, 0, 15, 43, 15, 0, 0, 0, 0, 0, 0], [0, 0, -65, 0, 23, 0, 0, 0, -26, 0, -26, 23], [-26, 0, 0, 87, 0, -87, -4, 0, 87, 0, -87, 0], [0, 0, 130, 0, -9, 0, 0, 0, 84, 0, 84, -9], [0, 37, 0, 63, 37, 63, 0, 0, 0, 0, 0, 0], [0, 0, -131, 0, -73, 0, 0, 0, -29, 0, -29, -73], [0, 0, 0, 0, 0, 0, 0, 119, -3, -110, 3, 0], [-33, 0, 0, -33, 0, 33, -118, -33, 0, 118, 0, 0], [-284, 0, 0, -140, 0, 140, -62, 0, -140, 0, 140, 0], [0, 26, -26, 5, 0, 5, 0, 0, 5, 0, 5, 50], [0, -62, 0, -114, -11, -114, 0, 0, 0, 0, 0, 0], [-20, 0, 0, -85, 0, 85, -176, 0, -85, 0, 85, 0], [0, 0, 54, 0, 91, 0, 0, 0, 66, 0, 66, 91], [0, 0, 0, 0, 0, 0, 0, 330, 18, -31, -18, 0], [-126, 0, 0, -133, 0, 133, 186, 0, -133, 0, 133, 0], [0, 0, 0, 0, 0, 0, 0, 265, -150, 58, 150, 0], [0, 144, -144, 153, 0, 153, 0, 0, 153, 0, 153, -42], [0, -28, 28, -132, 0, -132, 0, 0, -132, 0, -132, 180], [0, 0, 0, 0, 0, 0, 0, -280, 167, 141, -167, 0], [0, -212, 0, -159, -101, -159, 0, 0, 0, 0, 0, 0], [0, 82, 0, -63, 29, -63, 0, 0, 0, 0, 0, 0], [-138, 0, 0, 91, 0, -91, 196, 0, 91, 0, -91, 0], [0, 0, 0, 0, 0, 0, 0, 92, 179, 11, -179, 0], [0, 214, -214, -16, 0, -16, 0, 0, -16, 0, -16, -72], [0, 0, 167, 0, 51, 0, 0, 0, -29, 0, -29, 51], [179, 0, 0, -58, 0, 58, 120, 179, 0, -120, 0, 0], [0, 0, 0, 0, 0, 0, 0, 480, 50, 77, -50, 0], [0, 0, 165, 0, 181, 0, 0, 0, -309, 0, -309, 181], [510, 0, 0, 115, 0, -115, -236, 0, 115, 0, -115, 0], [0, -77, 0, -225, -183, -225, 0, 0, 0, 0, 0, 0], [150, 0, 0, 179, 0, -179, 39, 150, 0, -39, 0, 0], [0, 0, 0, 0, 0, 0, 0, -431, 52, 198, -52, 0], [-282, 0, 0, 32, 0, -32, -103, -282, 0, 103, 0, 0], [-305, 0, 0, -6, 0, 6, -196, -305, 0, 196, 0, 0], [700, 0, 0, -44, 0, 44, -26, 0, -44, 0, 44, 0], [0, 0, 0, 0, 0, 0, 0, 195, -129, 92, 129, 0], [0, -104, 0, 189, 101, 189, 0, 0, 0, 0, 0, 0], [-212, 0, 0, -53, 0, 53, 36, 0, -53, 0, 53, 0], [0, 0, -182, 0, 95, 0, 0, 0, 58, 0, 58, 95], [0, 223, 0, -29, 81, -29, 0, 0, 0, 0, 0, 0], [-350, 0, 0, -3, 0, 3, -42, 0, -3, 0, 3, 0], [-486, 0, 0, -109, 0, 109, 199, -486, 0, -199, 0, 0], [0, 114, -114, 146, 0, 146, 0, 0, 146, 0, 146, -24], [121, 0, 0, -52, 0, 52, -52, 121, 0, 52, 0, 0], [180, 0, 0, -14, 0, 14, 102, 0, -14, 0, 14, 0], [0, -32, 0, -59, -153, -59, 0, 0, 0, 0, 0, 0], [0, 0, -77, 0, -85, 0, 0, 0, 42, 0, 42, -85], [482, 0, 0, -52, 0, 52, 176, 0, -52, 0, 52, 0], [0, 103, 0, 312, -7, 312, 0, 0, 0, 0, 0, 0], [70, 0, 0, -111, 0, 111, -26, 0, -111, 0, 111, 0], [0, 0, 181, 0, -5, 0, 0, 0, 83, 0, 83, -5], [-680, 0, 0, -46, 0, 46, 179, -680, 0, -179, 0, 0], [0, 0, 0, 0, 0, 0, 0, -362, -67, -265, 67, 0], [0, -224, 224, 44, 0, 44, 0, 0, 44, 0, 44, 84], [0, 0, -211, 0, 79, 0, 0, 0, -56, 0, -56, 79], [0, 0, 158, 0, 123, 0, 0, 0, 283, 0, 283, 123], [0, 30, -30, -102, 0, -102, 0, 0, -102, 0, -102, 56], [0, 0, -163, 0, -247, 0, 0, 0, 156, 0, 156, -247], [-44, 0, 0, -244, 0, 244, 215, -44, 0, -215, 0, 0], [0, 134, -134, 144, 0, 144, 0, 0, 144, 0, 144, -324], [0, 0, -20, 0, -235, 0, 0, 0, 150, 0, 150, -235], [0, 84, -84, 43, 0, 43, 0, 0, 43, 0, 43, 14], [0, 0, 0, 0, 0, 0, 0, -339, 81, 214, -81, 0], [606, 0, 0, 19, 0, -19, 162, 0, 19, 0, -19, 0], [0, -305, 0, -279, -133, -279, 0, 0, 0, 0, 0, 0], [720, 0, 0, -111, 0, 111, -167, 720, 0, 167, 0, 0], [0, -128, 128, 209, 0, 209, 0, 0, 209, 0, 209, -262], [71, 0, 0, 66, 0, -66, -10, 71, 0, 10, 0, 0], [-644, 0, 0, 38, 0, -38, 186, 0, 38, 0, -38, 0], [0, 0, 207, 0, -31, 0, 0, 0, -171, 0, -171, -31], [0, -252, 0, -523, -211, -523, 0, 0, 0, 0, 0, 0], [-920, 0, 0, -122, 0, 122, 174, 0, -122, 0, 122, 0], [0, -39, 0, 3, -193, 3, 0, 0, 0, 0, 0, 0], [0, -164, 164, -140, 0, -140, 0, 0, -140, 0, -140, -40], [0, 62, -62, -215, 0, -215, 0, 0, -215, 0, -215, 238], [-675, 0, 0, -46, 0, 46, -158, -675, 0, 158, 0, 0], [0, 0, 0, 0, 0, 0, 0, 718, -60, 75, 60, 0], [0, 0, 44, 0, 109, 0, 0, 0, -259, 0, -259, 109], [0, 0, 169, 0, -47, 0, 0, 0, -65, 0, -65, -47], [-444, 0, 0, -48, 0, 48, 222, 0, -48, 0, 48, 0], [290, 0, 0, -98, 0, 98, -100, 0, -98, 0, 98, 0], [0, 0, 128, 0, -53, 0, 0, 0, -124, 0, -124, -53], [180, 0, 0, -245, 0, 245, -145, 180, 0, 145, 0, 0], [0, -436, 436, -62, 0, -62, 0, 0, -62, 0, -62, 228], [0, 0, 206, 0, 141, 0, 0, 0, -74, 0, -74, 141], [0, 0, 0, 0, 0, 0, 0, -109, 248, -46, -248, 0], [-470, 0, 0, -172, 0, 172, 384, 0, -172, 0, 172, 0], [0, 213, 0, 298, 477, 298, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -610, 103, -307, -103, 0], [320, 0, 0, 203, 0, -203, 204, 0, 203, 0, -203, 0], [0, 2, -2, 210, 0, 210, 0, 0, 210, 0, 210, -4], [0, 0, 0, 0, 0, 0, 0, 466, -175, 295, 175, 0], [-1085, 0, 0, -157, 0, 157, 152, -1085, 0, -152, 0, 0], [0, -224, 0, 244, 219, 244, 0, 0, 0, 0, 0, 0], [0, 316, 0, 227, 313, 227, 0, 0, 0, 0, 0, 0], [0, 251, 0, 234, 285, 234, 0, 0, 0, 0, 0, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_392_k_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_392_3_k_l();". // 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_392_3_k_l(:prec:=12) chi := MakeCharacter_392_k(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 3)); 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_392_3_k_l();". // 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_392_3_k_l( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_392_k(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign))); Vf := Kernel([<3,R![1225, -665, 466, -13, 28, -3, 1]>,<5,R![117649, 0, -64141, 0, 25022, 0, -4737, 0, 654, 0, -29, 0, 1]>],Snew); return Vf; end function;