// Make newform 2100.2.bi.l in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2100_bi();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2100_bi_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_2100_2_bi_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_2100_2_bi_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 := [6561, -6561, 6561, -4374, 2592, -972, 0, 153, -167, 51, 0, -36, 32, -18, 9, -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, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [6561, -6561, 4374, -2592, 972, 0, -153, 167, -51, 0, 36, -32, 18, -9, 3, -1], [20670066, -24411051, 19216764, -9118332, 3207744, -263352, -1296236, 938494, -339726, -105324, 151832, -121504, 76824, -29220, 6454, 1153], [-54877662, 58844394, -39695913, 15893982, -4226967, -681303, 3585486, -1965202, 437691, 407982, -359337, 258388, -149400, 45069, -4206, -5395], [52235766, -81111456, 70727904, -40658976, 17467020, -4108284, -2590688, 3390626, -1670856, 72792, 430760, -438368, 304524, -149151, 46258, -5968], [-108434376, 151849728, -124258779, 69652251, -28159650, 5733957, 5956964, -6035418, 2652570, 73005, -848087, 786063, -534429, 241188, -71512, 6954], [185919057, -202887261, 135325242, -52760484, 13815873, 4191318, -12576744, 7041979, -1318053, -1588383, 1306899, -887410, 498699, -143820, 951, 23263], [276613947, -293889789, 211208067, -85116744, 26531901, 3420909, -17596443, 10069919, -2572404, -2035269, 1838910, -1365137, 767331, -242181, 32013, 24626], [47911338, -82166319, 76141134, -45346419, 19818666, -5344728, -2213422, 3602156, -1928373, 217959, 431323, -457733, 329391, -170922, 56618, -9427], [-33668865, -13079718, 41955408, -44623872, 25189920, -11666124, 4108284, 1677584, -2393970, 1366488, -72792, -215912, 247392, -197100, 95439, -28354], [60732990, -100513305, 87490611, -52309665, 23354802, -6075699, -2877419, 4183034, -2127084, 241278, 491762, -532664, 389808, -194877, 65950, -11554], [220725162, -195970509, 107884710, -28293624, -166968, 10498437, -15338910, 5622202, 221379, -2239380, 1323084, -769078, 367722, -13698, -50757, 35773], [33668865, 29541267, -55368522, 56817612, -32506164, 12817755, -4213662, -2550716, 2777847, -1637460, 42684, 321290, -315135, 242262, -110493, 35881], [-49687182, 13861206, 12957975, -25329105, 15176142, -8119533, 4215935, 476724, -1456956, 1097229, -176321, -49284, 108063, -114183, 62870, -21897], [-23491053, 8130132, 2059155, -7566768, 5458719, -3083533, 1822111, 101459, -481167, 424168, -93712, -302, 29256, -40300, 21901, -8452]]; Rf_basisdens := [1, 1, 2187, 609687, 1829061, 1829061, 1829061, 5487183, 5487183, 1829061, 5487183, 1829061, 5487183, 5487183, 1829061, 609687]; 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_2100_bi();" function MakeCharacter_2100_bi() N := 2100; order := 6; char_gens := [1051, 701, 1177, 1501]; v := [6, 3, 6, 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_2100_bi_Hecke();" function MakeCharacter_2100_bi_Hecke(Kf) N := 2100; order := 6; char_gens := [1051, 701, 1177, 1501]; char_values := [[1, 0, 0, 0, 0, 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], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 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 := 2; raw_aps := [[0, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, -1, 1], [1, -1, 1, 1, 0, -1, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0], [0, -1, 0, 1, -1, 0, 1, 1, -1, 0, 0, 0, 0, 0, 1, 1], [-2, 1, 0, 0, 1, 1, -1, -2, 0, -1, 1, 0, 0, 0, 0, 0], [0, 1, 0, 0, -1, 1, 0, -1, 0, 0, 0, 0, 0, -2, 0, 0], [0, 2, -1, 1, -1, 0, -1, -1, 0, -1, 0, 0, -2, -2, 0, 0], [0, -1, -2, -2, 1, -1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 6, 6, -2, -1, 2, 0, 0, -2, -1, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 2, 0, -1, 1, -1, 0, 1, 0, 0, 0, 0], [0, -1, 0, 0, 1, -1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 2, -1, 0, -1, 1, -2, -1, 2, -2, 0, -1, -4, -2, 1, -1], [0, 2, 0, -1, 0, -1, -1, 0, 1, -1, 0, 0, -2, 0, 1, -1], [0, 1, 0, 1, -3, 2, 0, -1, -1, -1, 0, 0, -2, -4, 1, 1], [2, 2, 0, 0, -1, 2, -2, -1, 0, -2, -1, 0, 0, 0, 0, 0], [-3, -3, 1, 1, 1, -3, -2, -1, 0, -1, 3, 0, 0, 0, 0, 0], [0, 0, -1, -3, 3, -2, -1, 0, 2, 2, 0, -2, 2, 2, -2, -2], [-3, -1, 3, 3, 1, -1, 0, 1, 0, 1, -3, 0, 0, 0, 0, 0], [0, -2, -2, -2, 3, -2, -1, 2, 0, 1, 1, 0, 0, 0, 0, 0], [0, -3, -3, 1, 0, 1, 0, 1, -1, 1, 0, -1, 0, 0, 0, 0], [0, 0, -1, 0, 1, -1, 2, 1, 2, 2, 0, -1, 4, 2, 1, -1], [1, -1, 0, 0, 1, -1, -1, 2, 0, -2, -2, 0, 0, 0, 0, 0], [0, -2, 1, -2, 2, -1, 1, 2, -1, 1, 0, 2, 2, 4, 1, -1], [6, 0, 0, 0, -2, 0, -1, 1, 0, -1, -3, 0, 0, 0, 0, 0], [0, -1, -3, 2, -3, 2, 1, 0, -2, 1, 0, -1, 0, -4, 1, 1], [-5, 0, 1, 1, -2, 0, 2, 2, 0, 0, 5, 0, 0, 0, 0, 0], [0, 0, 0, 4, -2, 0, 2, 1, -2, -1, 0, 2, 0, 0, 2, 2], [-5, 1, 1, 1, 0, 1, 0, -3, 0, -3, 0, 0, 0, 0, 0, 0], [0, -1, 0, -2, 1, -1, -1, 0, -2, 1, 0, 0, 0, 0, -2, 0], [0, 0, -2, 0, 0, 2, -1, 0, 2, -1, 0, 0, -2, 0, 2, -2], [-1, -5, 7, 7, -2, -5, 2, 2, 0, -2, 2, 0, 0, 0, 0, 0], [0, -2, 3, -4, 1, -5, 1, 4, -2, 1, 0, -1, 0, 4, 1, 1], [-1, 3, -2, -2, 1, 3, -3, -1, 0, -2, 1, 0, 0, 0, 0, 0], [3, -1, 4, 4, 1, -1, 4, 1, 0, -3, 3, 0, 0, 0, 0, 0], [0, 5, 0, 0, -5, 5, 0, -5, 0, 0, 1, 0, 0, 0, 0, 0], [0, 4, 6, -4, 0, -6, 0, 0, -1, 0, 0, -1, 0, 0, -2, 2], [0, 3, 0, 2, -3, 3, -3, -2, 2, -5, 0, 0, -8, -4, 2, 0], [0, 4, 0, -2, 0, -2, 0, 0, 2, 0, 0, 0, 0, 0, 2, -2], [-6, 0, -1, -1, 1, 0, -1, 3, 0, -3, 12, 0, 0, 0, 0, 0], [0, 3, 3, -2, -2, 0, 0, -2, 0, 0, 0, -1, 0, -4, -1, 1], [-6, 6, -4, -4, 2, 6, -1, -2, 0, 1, 6, 0, 0, 0, 0, 0], [0, 1, 1, -1, -2, 3, -2, -2, -2, -2, 0, 2, -4, -4, 0, 0], [3, 2, 4, 4, -2, 2, 1, -2, 0, -3, 3, 0, 0, 0, 0, 0], [1, 2, 1, 1, -1, 2, -1, -5, 0, -5, 0, 0, 0, 0, 0, 0], [3, -5, 2, 2, 3, -5, -3, -2, 0, 2, -6, 0, 0, 0, 0, 0], [0, 1, -2, 2, 1, -1, -1, -1, 2, 0, 0, -2, 0, 0, -2, 0], [-2, -2, 0, 0, -2, -2, 3, 5, 0, 3, 1, 0, 0, 0, 0, 0], [0, 5, 5, -6, -2, 0, 0, -2, 0, 0, 0, 1, 0, -4, 1, -1], [0, 2, -3, 3, 1, -4, 1, 1, -2, 1, 0, 2, 2, 2, 0, 0], [-3, 1, 3, 3, -1, 1, 3, -1, 0, -4, -3, 0, 0, 0, 0, 0], [0, -3, -4, 2, 1, 3, -1, 1, 1, -1, 0, 1, -2, 2, 2, -2], [0, -4, 1, 2, 0, 3, 3, 1, 0, 1, 0, 1, 4, 2, 1, 1], [0, -2, 1, 1, 0, 0, 2, 0, -2, 2, 0, 0, 4, 0, -2, 2], [0, 2, 4, -5, 2, -1, -2, -2, 3, 0, 0, -2, 0, 0, -3, -1], [4, -6, 0, 0, -1, -6, -3, -2, 0, -3, -2, 0, 0, 0, 0, 0], [-9, -1, 2, 2, -3, -1, 0, 3, 0, -3, 9, 0, 0, 0, 0, 0], [0, 3, -2, 6, -3, -1, 1, 0, -1, -2, 0, 1, -2, -2, 2, 2], [-1, -3, -8, -8, 3, -3, 0, 3, 0, 3, -1, 0, 0, 0, 0, 0], [0, -3, 1, 3, 2, -1, -2, 1, 0, -3, 0, 0, -4, 4, -2, 2], [2, 9, -4, -4, -5, 9, 5, 1, 0, -1, -4, 0, 0, 0, 0, 0], [0, 0, -2, 4, 0, -2, 0, 0, 2, 0, 0, -4, 0, 0, -2, 2], [12, 4, 0, 0, 0, 4, 1, 1, 0, 1, -6, 0, 0, 0, 0, 0], [0, -1, -2, 4, 2, 1, -1, -1, 2, -1, 0, 0, 0, 2, -2, 0], [0, 2, -4, -4, 0, 2, 2, 2, 0, -2, 7, 0, 0, 0, 0, 0], [16, -6, -6, -6, 0, -6, 0, 3, 0, 3, 0, 0, 0, 0, 0, 0], [0, -5, -2, -1, 1, 0, 0, 1, -3, 3, 0, -2, 2, 0, -3, 1], [4, -7, 3, 3, 4, -7, -4, -3, 0, 3, -8, 0, 0, 0, 0, 0], [0, -2, -3, 0, 2, 1, 1, 2, -3, 1, 0, 6, 2, 4, 3, -3], [0, 7, 11, -8, 3, -4, -2, -3, 4, -2, 0, 1, 0, 2, -3, -1], [12, 1, -6, -6, -1, 1, -3, -1, 0, 2, 12, 0, 0, 0, 0, 0], [0, 4, -6, -6, -1, 4, -2, -3, 0, 2, 1, 0, 0, 0, 0, 0], [1, 6, 7, 7, 1, 6, 1, -5, 0, -5, 0, 0, 0, 0, 0, 0], [0, 2, -2, -6, 2, -4, -5, -1, -2, 1, 0, -2, -4, -2, -4, -2], [0, 1, 3, -4, 1, -8, 1, 1, -5, 4, 0, -2, 4, 0, -5, 3], [8, -2, 0, 0, 5, -2, 1, -4, 0, 1, -4, 0, 0, 0, 0, 0], [0, 1, -1, -4, -2, 2, 1, 1, -2, 1, 0, 3, 0, -2, 5, -3], [7, 4, -12, -12, -4, 4, -2, -4, 0, -2, 7, 0, 0, 0, 0, 0], [0, -7, -7, 9, 0, 9, 0, -1, 1, -1, 0, 1, 0, 0, 0, 0], [-9, -1, 4, 4, 5, -1, 5, 2, 0, 2, 0, 0, 0, 0, 0, 0], [0, 5, 0, -6, 1, -7, 2, 1, 0, 5, 0, -2, 6, 0, 0, -2], [6, -4, 5, 5, -1, -4, 1, -1, 0, 1, -12, 0, 0, 0, 0, 0], [18, -4, 0, 0, 5, -4, 3, -2, 0, 3, -9, 0, 0, 0, 0, 0], [0, 5, 8, -2, -1, -3, -1, -4, 2, -1, 0, -4, 0, -4, -6, 4], [0, 5, -5, 5, 0, -5, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0], [0, 1, 0, 0, -1, 1, 4, 3, 0, -4, 7, 0, 0, 0, 0, 0], [0, -4, -6, -2, 0, 0, 0, 3, -2, 3, 0, -2, 0, 0, 2, -2], [13, -2, -1, -1, 1, -2, 1, 6, 0, 6, 0, 0, 0, 0, 0, 0], [0, -3, -2, -6, 3, -1, -3, 0, -2, 3, 0, -2, 0, 0, -4, -2], [0, -4, 3, -4, 4, -3, 2, 4, -1, 2, 0, 2, 4, 8, 1, -1], [-14, 8, -3, -3, -2, 8, 3, 2, 0, 1, 14, 0, 0, 0, 0, 0], [0, -2, 2, 6, -4, 2, 4, 2, -2, -2, 0, 2, 0, 0, 4, 4], [0, 4, -8, -8, 0, 4, 2, 2, 0, -2, 7, 0, 0, 0, 0, 0], [0, -6, -7, 11, -1, 12, 1, -3, 2, -1, 0, 2, 2, -2, 0, 0], [0, -4, -4, -8, 6, -4, 0, 2, 0, 8, 0, -4, 8, 4, -4, -4], [0, 4, 4, -4, -4, 4, -2, -4, 0, -2, 0, 0, -4, -8, 0, 0], [-4, -1, 0, 0, 4, -1, -1, -5, 0, -1, 2, 0, 0, 0, 0, 0], [0, -5, 4, 4, 3, -5, -2, 1, 0, 2, 7, 0, 0, 0, 0, 0], [-2, -7, -7, -7, 0, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -6, 2, 10, -2, 10, 5, -2, 0, -1, 0, 4, 6, 0, 0, 4], [0, 0, 8, -5, 0, -3, 0, 0, 3, 0, 0, -6, 0, 0, -3, 3], [0, 8, 7, -8, -1, 1, -1, -4, 2, -1, 0, 3, 0, -4, 1, -3], [11, 5, 0, 0, -5, 5, 4, 5, 0, -1, -11, 0, 0, 0, 0, 0], [10, -3, -6, -6, 3, -3, -1, 3, 0, 4, 10, 0, 0, 0, 0, 0], [0, 5, -1, -3, -4, 3, 4, -3, 0, 5, 0, 0, 8, -8, 2, -2], [0, -2, 4, 4, 0, -2, 6, 2, -4, 2, 0, 4, 8, 4, 0, 4], [0, 9, 7, 6, -3, 2, 0, -3, 5, -9, 0, 6, -6, 0, 5, 1], [-3, -9, 11, 11, -2, -9, 2, -3, 0, 3, 6, 0, 0, 0, 0, 0], [18, -1, 0, 0, -1, -1, 1, 2, 0, 1, -9, 0, 0, 0, 0, 0], [-8, 12, -1, -1, -10, 12, 9, 10, 0, -1, 8, 0, 0, 0, 0, 0], [0, 4, 0, -4, -2, 4, -6, -5, 4, -3, 0, -4, -8, -8, -2, -2], [14, -9, -7, -7, 9, -9, 2, 9, 0, 7, 14, 0, 0, 0, 0, 0], [13, 0, -5, -5, -5, 0, -5, 1, 0, 1, 0, 0, 0, 0, 0, 0], [0, -1, -2, 4, -1, 7, -3, -1, 4, -6, 0, 2, -8, 0, 4, -2], [5, -6, 14, 14, -8, -6, 8, 2, 0, -2, -10, 0, 0, 0, 0, 0], [0, 4, -2, -1, 0, 3, -3, -4, 1, -1, 0, 2, -2, -4, -1, -3], [0, -4, -1, -6, 1, -3, 2, 7, -4, 2, 0, 3, 0, 6, 7, -3], [0, 2, -5, 1, 5, -8, 1, 2, 5, 4, 0, -5, 6, 6, -2, -2], [-5, -1, -17, -17, 1, -1, -6, 1, 0, 7, -5, 0, 0, 0, 0, 0], [5, -3, 0, 0, 3, -3, 3, 4, 0, 4, 0, 0, 0, 0, 0, 0], [0, 2, 2, 0, 1, -6, 2, 1, -4, 2, 0, 2, 4, 2, -2, 2], [0, 0, 0, -7, 2, -9, -5, 2, -5, 1, 0, -4, -6, 0, -5, 1], [-14, 3, 0, 0, 6, 3, 2, -4, 0, 2, 7, 0, 0, 0, 0, 0], [0, 3, -3, 7, -2, -3, 2, 1, -3, -1, 0, 3, 0, 0, 2, 2], [0, 0, -8, -8, 4, 0, -9, -5, 0, 9, -9, 0, 0, 0, 0, 0], [0, 1, 5, 8, -5, 4, 2, -1, -2, -6, 0, 5, -4, -2, 3, 5], [5, 13, -3, -3, -10, 13, 10, 2, 0, -2, -10, 0, 0, 0, 0, 0], [-22, 6, 0, 0, -3, 6, -2, 1, 0, -2, 11, 0, 0, 0, 0, 0], [-1, -10, 8, 8, -6, -10, 8, 6, 0, 2, 1, 0, 0, 0, 0, 0], [0, 3, -3, 3, 0, -3, 0, 0, 2, 0, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3, 3, 0, -3, -13, 0, 0, 0, 0, 0], [15, 0, 0, 0, 0, 0, 0, -7, 0, -7, 0, 0, 0, 0, 0, 0], [0, 1, -1, -4, 4, -8, 2, 2, -2, 6, 0, -1, 8, 4, -3, -1], [6, -10, 18, 18, -8, -10, 8, 7, 0, -7, -12, 0, 0, 0, 0, 0], [0, 2, 0, 0, -2, 2, -1, -2, 0, -1, 0, 0, -2, -4, 0, 0], [3, 7, 23, 23, -7, 7, 3, -7, 0, -10, 3, 0, 0, 0, 0, 0], [0, 3, 1, -9, 2, -7, -2, 4, 0, 0, 0, 0, -4, 4, 4, -4], [0, -4, -3, -5, 2, -4, -1, 2, -4, 5, 0, -4, 2, 0, -4, 0], [-15, 7, -13, -13, 6, 7, -6, -5, 0, 5, 30, 0, 0, 0, 0, 0], [0, 1, 7, 4, 3, -6, -2, -3, 4, -2, 0, -7, 0, 2, -11, 7], [-19, -2, -1, -1, 4, -2, -7, -4, 0, -3, 19, 0, 0, 0, 0, 0], [0, -2, 2, 2, -2, 2, 2, 1, 8, -1, 0, -8, 0, 0, 2, 2], [1, 0, -2, -2, 0, 0, -1, 0, 0, 1, 1, 0, 0, 0, 0, 0], [0, -6, -8, 8, -2, 10, 2, -2, 0, 2, 0, 0, 4, -4, 0, 0], [-2, -3, -2, -2, 5, -3, -5, -2, 0, 2, 4, 0, 0, 0, 0, 0], [0, -2, -4, 4, -6, 6, 3, 2, -8, -1, 0, 8, -2, -4, 8, 0], [24, 2, 0, 0, -6, 2, -11, -5, 0, -11, -12, 0, 0, 0, 0, 0], [0, -4, -3, -2, 5, -1, 0, 5, 0, 0, 0, 5, 0, 10, 5, -5], [0, -13, 6, 6, 10, -13, -9, 1, 0, 9, -9, 0, 0, 0, 0, 0], [0, -5, 3, 9, 0, 1, 0, -2, -2, -2, 0, -2, 0, 0, -8, 8], [-17, 13, 11, 11, -2, 13, -2, -12, 0, -12, 0, 0, 0, 0, 0, 0], [0, -3, 8, 12, -7, 9, 4, -1, -4, -8, 0, 8, -4, -2, 4, 8], [-35, 2, 2, 2, -6, 2, 2, 6, 0, -4, 35, 0, 0, 0, 0, 0], [0, 7, -3, -9, 2, 1, -10, -7, 2, -1, 0, -2, -8, -8, -6, -6], [0, 1, -4, -4, 1, 1, -1, 0, 0, 1, 2, 0, 0, 0, 0, 0], [0, 11, 5, 6, -4, -8, 1, -1, -4, -5, 0, 5, -4, -2, 1, 5], [10, -4, 1, 1, 3, -4, -3, 1, 0, -1, -20, 0, 0, 0, 0, 0], [0, 1, 10, -10, -3, 3, 0, -1, -2, -1, 0, 2, -2, -4, 2, 0], [0, -6, -6, 4, -8, 0, 4, 4, -8, 4, 0, -6, 0, -8, 2, 6], [0, 6, 0, 4, -8, 6, -4, -5, -1, -7, 0, 1, -12, -12, 2, 2], [-5, -6, -2, -2, 4, -6, 4, 13, 0, 13, 0, 0, 0, 0, 0, 0], [0, -9, -1, -4, 6, -2, 6, 4, -2, 10, 0, -1, 16, 8, -3, -1], [0, -6, -9, -4, 2, 3, -2, 2, 1, 4, 0, -4, 0, 0, 1, -5], [0, 11, 2, -5, -9, 12, -6, -11, -1, -5, 0, 4, -10, -20, 1, -3], [16, 7, -7, -7, 7, 7, 0, -7, 0, 7, -16, 0, 0, 0, 0, 0], [11, -2, -2, -2, 2, -2, 1, 2, 0, 1, 11, 0, 0, 0, 0, 0], [-2, 13, 5, 5, -8, 13, -8, -7, 0, -7, 0, 0, 0, 0, 0, 0], [0, -4, 2, -4, 5, -8, 4, 3, -8, 8, 0, 2, 12, 6, -6, 2], [0, 9, 2, -8, 1, -11, -3, 1, 0, 0, 0, -2, -4, 0, 0, -2], [5, 1, 8, 8, -9, 1, 9, -6, 0, 6, -10, 0, 0, 0, 0, 0], [0, -3, 3, -4, 5, -6, 0, 5, 0, 0, 0, 1, 0, 10, 1, -1], [-5, -4, -3, -3, 10, -4, -5, -10, 0, 5, 5, 0, 0, 0, 0, 0], [1, 3, 8, 8, -3, 3, 7, -3, 0, -10, 1, 0, 0, 0, 0, 0], [0, -1, -8, 4, 1, 11, -1, -1, 6, -3, 0, 6, -2, 2, 8, -8], [-4, -3, 1, 1, 4, -3, 4, 7, 0, 7, 0, 0, 0, 0, 0, 0], [0, -13, -4, 3, 1, 6, 6, 1, -5, 9, 0, -2, 14, 0, -5, 3], [-11, -3, 0, 0, 3, -3, -3, 4, 0, -4, 22, 0, 0, 0, 0, 0], [0, -10, 0, 0, 10, -10, 10, 0, 0, 10, 0, 0, 0, 0, 0, 0], [0, 3, 10, 10, -8, 3, 6, -2, 0, -6, -18, 0, 0, 0, 0, 0], [0, 12, 11, 1, 1, 2, -1, -6, 8, -8, 0, 8, -2, 2, 2, -2], [6, 12, 0, 0, -12, 12, -12, -12, 0, -12, 0, 0, 0, 0, 0, 0], [0, 4, -5, -4, 2, -3, -2, 0, 6, 2, 0, -5, 0, 0, 1, -5], [0, -1, 6, 11, -3, 8, 7, -3, 1, -2, 0, 6, 8, 0, 1, 5], [0, 1, 4, 1, -3, -2, 0, -1, 3, -1, 0, -8, -2, -4, -3, 5], [-5, -23, 13, 13, -3, -23, 7, 3, 0, 4, 5, 0, 0, 0, 0, 0], [18, -8, -6, -6, 8, -8, 5, 8, 0, 3, 18, 0, 0, 0, 0, 0], [0, 11, 2, 2, -12, 11, 4, -8, 0, -4, -8, 0, 0, 0, 0, 0], [0, 6, -4, 7, -6, 3, -3, -6, 3, -3, 0, -6, -6, -12, -3, 3], [26, -12, 0, 0, -1, -12, 10, 11, 0, 10, -13, 0, 0, 0, 0, 0], [0, 5, -4, 12, -5, -3, 3, 1, -3, -3, 0, 3, -2, -2, 4, 4], [0, 4, -6, -6, -1, 4, -5, -6, 0, 5, -21, 0, 0, 0, 0, 0], [0, -7, -2, 0, 3, 3, 6, 3, 4, 6, 0, -2, 12, 6, 2, -2], [0, 10, -2, -12, 2, -12, -2, 2, 2, 4, 0, -4, 0, 0, 2, -6], [0, 3, -7, 9, -5, 3, -1, -3, 0, -2, 0, -2, -4, -8, 0, 2], [9, 4, 1, 1, -6, 4, 5, 6, 0, -1, -9, 0, 0, 0, 0, 0], [0, -2, 28, 28, -12, -2, 14, 2, 0, -14, -22, 0, 0, 0, 0, 0], [0, 0, -3, 1, 1, 4, -1, 0, 3, -2, 0, 3, -2, 2, 4, -4], [0, 5, 4, 6, -4, -1, 1, -1, -2, -5, 0, 4, -4, -2, 2, 4], [0, 6, 3, 4, -2, 3, 4, -2, 5, -2, 0, 4, 4, 0, 5, -1], [1, -10, -4, -4, 14, -10, -14, -5, 0, 5, -2, 0, 0, 0, 0, 0], [16, -7, 0, 0, 10, -7, 9, -1, 0, 9, -8, 0, 0, 0, 0, 0], [2, -6, 7, 7, 6, -6, -6, 6, 0, 12, 2, 0, 0, 0, 0, 0], [0, -7, -10, 4, -3, 7, 3, 0, -3, 6, 0, -3, 6, -6, 0, 0], [0, 2, 7, -2, -6, 1, 0, -2, 1, -2, 0, -6, -4, -8, -1, 5], [22, 0, 0, 0, -2, 0, 1, 3, 0, 1, -11, 0, 0, 0, 0, 0], [0, -5, -9, -2, 0, 4, 2, 6, -4, 2, 0, 7, 0, 4, 11, -7], [0, -2, -1, -3, 5, -4, 1, 2, 4, 4, 0, -4, 6, 6, -2, -2], [-7, -3, 16, 16, 3, -3, -4, 3, 0, 7, -7, 0, 0, 0, 0, 0], [0, -14, 2, 2, 13, -14, 0, 13, 0, 0, -11, 0, 0, 0, 0, 0], [0, 5, 1, 5, -2, 9, 2, -6, 5, -2, 0, 5, 4, -4, 2, -2], [19, -3, 6, 6, 9, -3, 9, -1, 0, -1, 0, 0, 0, 0, 0, 0], [0, 1, -9, -8, 3, 2, -6, -1, 10, 2, 0, -9, -4, -2, 1, -9], [0, -11, 0, -1, 9, -8, 6, 11, -3, 5, 0, 4, 10, 20, 3, -1], [26, -10, 0, 0, -5, -10, 13, 18, 0, 13, -13, 0, 0, 0, 0, 0], [-31, 18, -12, -12, 6, 18, 0, -6, 0, 6, 31, 0, 0, 0, 0, 0], [0, 3, 6, -8, 3, -1, -3, -3, 4, 0, 0, -2, 0, 0, -4, -2], [0, 0, 6, -8, 7, -6, -1, 4, 2, -1, 0, 4, 0, 12, 2, -4], [0, 4, -2, -2, 0, 0, -4, -3, 8, -1, 0, -8, -4, -4, -2, -2], [0, 0, 10, 10, -5, 0, -1, -6, 0, 1, -6, 0, 0, 0, 0, 0], [-19, 12, -15, -15, 3, 12, -3, -4, 0, 4, 38, 0, 0, 0, 0, 0], [0, -4, 6, -2, -4, -10, 3, 5, -6, 3, 0, -10, 0, -2, -4, 10], [0, -5, 8, 8, 1, -5, 9, 10, 0, -9, 2, 0, 0, 0, 0, 0], [0, 2, 1, -3, -3, -2, 3, -1, -3, 5, 0, -3, 6, -6, -2, 2], [15, 2, 7, 7, 5, 2, 5, 5, 0, 5, 0, 0, 0, 0, 0, 0], [0, 2, -8, -4, 3, 0, 0, 1, 12, 4, 0, -8, 4, 2, 4, -8], [0, -5, -3, -8, 3, -8, 0, 3, -7, 9, 0, -6, 6, 0, -7, 1], [-4, 21, 0, 0, -2, 21, -7, -5, 0, -7, 2, 0, 0, 0, 0, 0], [0, -6, 8, -14, 4, -14, 3, 13, -6, 3, 0, 0, 0, 14, 6, 0], [-29, 12, -2, -2, -8, 12, 1, 8, 0, -7, 29, 0, 0, 0, 0, 0], [8, -1, -14, -14, 1, -1, -5, 1, 0, 6, 8, 0, 0, 0, 0, 0], [6, -14, -8, -8, 6, -14, 6, 3, 0, 3, 0, 0, 0, 0, 0, 0], [0, -14, -16, 0, 2, 14, 0, 2, 4, 6, 0, -4, 4, 0, 4, -8], [-1, -8, 8, 8, 0, -8, 0, -7, 0, 7, 2, 0, 0, 0, 0, 0], [0, 3, -11, 6, 3, 2, -3, -3, 1, 0, 0, 4, 0, 0, -1, -5], [-10, 13, 0, 0, -9, 13, -1, 8, 0, -1, 5, 0, 0, 0, 0, 0], [0, 6, 1, 6, 3, 5, -4, -9, 8, -4, 0, 1, 0, -2, -7, -1], [0, -9, 5, 3, 0, 1, 8, 6, -6, 2, 0, 6, 8, 8, 4, 4], [0, -4, -3, -5, 3, -6, -3, 6, -2, 0, 0, -2, -6, 6, 2, -2], [0, -11, -6, 9, -1, 16, -4, -1, 3, -7, 0, 2, -10, 0, 3, -1], [-24, 2, 0, 0, 4, 2, 9, 5, 0, 9, 12, 0, 0, 0, 0, 0], [9, 9, 15, 15, -9, 9, 5, -9, 0, -14, 9, 0, 0, 0, 0, 0], [0, -4, 14, 14, -3, -4, 0, -3, 0, 0, -7, 0, 0, 0, 0, 0], [0, -1, 4, 0, -1, -5, 1, 0, -4, 2, 0, -4, 2, -2, -6, 6], [0, -12, 1, -4, 5, 1, 4, 3, -6, 8, 0, 1, 12, 6, -5, 1], [0, 10, -24, -24, 14, 10, -14, -11, 0, 11, 0, 0, 0, 0, 0, 0], [0, -4, -11, 10, 8, -7, 1, 4, 3, 3, 0, -2, 6, 12, -3, -1], [-35, 4, 2, 2, -8, 4, 8, 8, 0, 0, 35, 0, 0, 0, 0, 0], [0, -10, 6, 6, -2, 2, 10, 7, -5, 1, 0, 5, 8, 8, 6, 6], [31, -12, -1, -1, 11, -12, 11, 9, 0, 9, 0, 0, 0, 0, 0, 0], [0, 4, 4, 2, -3, -2, -3, -2, -6, -5, 0, 4, -8, -4, -2, 4], [0, 10, 8, 2, -2, -4, 0, -2, 2, -6, 0, 4, -4, 0, 2, 2], [19, -3, 4, 4, -1, -3, 1, -9, 0, 9, -38, 0, 0, 0, 0, 0], [0, 1, -7, 4, 1, 2, -1, -1, -1, 0, 0, 4, 0, 0, 1, -3], [0, -1, 0, 2, 0, -1, 0, 0, 0, 0, 0, -2, 0, 0, -2, 2], [36, -14, 7, 7, 0, -14, 1, 0, 0, 1, -36, 0, 0, 0, 0, 0], [11, 14, 1, 1, -14, 14, -8, -14, 0, -6, 11, 0, 0, 0, 0, 0], [0, -3, 8, 6, -1, -5, 1, -2, -5, 0, 0, -5, 2, -2, -12, 12], [44, 2, 0, 0, -2, 2, -3, -1, 0, -3, -22, 0, 0, 0, 0, 0], [-17, -11, 4, 4, 3, -11, 7, -3, 0, 10, 17, 0, 0, 0, 0, 0], [0, 9, -4, -8, 1, 1, -11, -8, -2, -2, 0, 2, -10, -10, -6, -6], [11, 10, 18, 18, -10, 10, -1, -10, 0, -9, 11, 0, 0, 0, 0, 0], [10, 3, 15, 15, -18, 3, 18, 8, 0, -8, -20, 0, 0, 0, 0, 0], [0, 5, 13, -19, 9, -3, -6, -5, 8, 1, 0, -2, 2, 4, -8, -6], [46, -10, 0, 0, 10, -10, 4, -6, 0, 4, -23, 0, 0, 0, 0, 0], [0, 1, -4, -12, 11, -7, -5, -1, -1, 7, 0, 1, 6, 6, -8, -8], [0, -6, -10, 4, -6, 8, 6, -2, -5, 10, 0, -5, 12, -12, -2, 2], [40, 0, 3, 3, 3, 0, 3, -2, 0, -2, 0, 0, 0, 0, 0, 0], [0, 6, 11, -3, 0, -14, 4, 0, -8, 4, 0, 0, 8, 0, -8, 8], [-26, 17, 0, 0, 6, 17, -12, -18, 0, -12, 13, 0, 0, 0, 0, 0], [6, -14, 7, 7, 0, -14, -2, 0, 0, -2, -6, 0, 0, 0, 0, 0], [-5, -9, 4, 4, 9, -9, 6, 9, 0, 3, -5, 0, 0, 0, 0, 0], [0, 11, -28, -28, 3, 11, -3, 0, 0, 3, 27, 0, 0, 0, 0, 0], [31, 5, 6, 6, -11, 5, 11, 0, 0, 0, -62, 0, 0, 0, 0, 0], [-6, 27, 0, 0, -8, 27, -19, -11, 0, -19, 3, 0, 0, 0, 0, 0], [0, 2, 12, -8, 5, -10, -1, 2, 2, -1, 0, -2, 0, 8, -4, 2], [4, -1, -4, -4, 9, -1, 7, -9, 0, 16, -4, 0, 0, 0, 0, 0], [0, -2, 24, 24, -10, -2, 10, 0, 0, -10, 11, 0, 0, 0, 0, 0], [0, 3, -2, 0, -1, 1, -2, -1, 4, -2, 0, -2, -4, -2, 2, -2], [0, -3, 9, -6, -3, 0, 3, 3, -3, 0, 0, 0, 0, 0, 3, 3], [-5, 4, 1, 1, -6, 4, 5, 6, 0, -1, 5, 0, 0, 0, 0, 0], [0, 0, 12, 10, 4, -2, -4, -3, 3, -11, 0, 3, -8, 8, -8, 8], [11, -15, 3, 3, 12, -15, -12, -7, 0, 7, -22, 0, 0, 0, 0, 0], [-2, -14, 0, 0, -4, -14, 11, 15, 0, 11, 1, 0, 0, 0, 0, 0], [0, 11, -3, 4, -10, 14, -1, -13, 2, -1, 0, 1, 0, -22, -1, -1], [5, -6, 3, 3, 0, -6, 3, 0, 0, 3, -5, 0, 0, 0, 0, 0], [0, -11, -2, -6, 17, -15, 9, 11, -3, 15, 0, 3, 26, 26, -4, -4], [5, -17, -1, -1, 17, -17, 6, 17, 0, 11, 5, 0, 0, 0, 0, 0], [0, 7, 12, 0, -1, -5, 1, -4, 0, -2, 0, 0, 2, -2, -6, 6], [0, -1, -1, 4, -1, 6, -5, -1, 3, -8, 0, 2, -12, 0, 3, -1], [0, -4, -5, 7, 4, -6, 2, 4, 2, 2, 0, -4, 4, 8, -2, 2], [0, 5, -2, 2, -6, 7, 0, -6, 0, 0, 0, 0, 0, -12, 0, 0], [0, 6, 7, -13, 5, -14, -5, 6, 1, -4, 0, 1, -10, 10, 4, -4], [-3, -8, -8, -8, 0, -8, 0, -4, 0, -4, 0, 0, 0, 0, 0, 0], [0, 4, -6, 2, -8, 12, -1, -4, -8, -3, 0, 12, -6, -12, 8, -4], [-28, 11, 0, 0, 10, 11, -11, -21, 0, -11, 14, 0, 0, 0, 0, 0], [0, 6, 9, 7, -1, 4, 1, -7, 4, -5, 0, 4, 2, -2, -4, 4], [0, -6, -3, -4, 7, -5, 8, 5, 2, 12, 0, -3, 20, 10, -1, -3], [1, 17, -19, -19, 2, 17, -2, 6, 0, -6, -2, 0, 0, 0, 0, 0], [-8, 5, 0, 0, 4, 5, -13, -17, 0, -13, 4, 0, 0, 0, 0, 0], [0, 0, 2, -4, 5, -2, -1, 2, 2, -1, 0, 4, 0, 8, 2, -4], [-3, 11, -5, -5, -1, 11, -3, 1, 0, -4, 3, 0, 0, 0, 0, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2100_bi_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2100_2_bi_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_2100_2_bi_l(:prec:=16) chi := MakeCharacter_2100_bi(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 2)); 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_2100_2_bi_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_2100_2_bi_l( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2100_bi(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![1475789056, 0, -559605872, 0, 144546913, 0, -19966919, 0, 1984747, 0, -101180, 0, 3715, 0, -74, 0, 1]>,<13,R![25, 0, 78, 0, 71, 0, 21, 0, 1]>,<19,R![6889, 27888, 40869, 13104, 596, -351, -12, 9, 1]>,<37,R![12837889, 1311378, 631993, -29376, 16836, -315, 148, -3, 1]>],Snew); return Vf; end function;