// Make newform 1050.2.bc.e in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1050_bc();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1050_bc_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_1050_2_bc_e();". // 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_1050_2_bc_e();". // 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 := [1679616, 0, -1306368, 0, 672624, 0, -194544, 0, 40705, 0, -5404, 0, 519, 0, -28, 0, 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], [0, 1559363223744, 0, -1048992541632, 0, 320563651380, 0, -54184962811, 0, 5822728240, 0, -340572477, 0, 12398488, 0, -2107], [1559363223744, 0, -1048992541632, 0, 320563651380, 0, -54184962811, 0, 5822728240, 0, -340572477, 0, 12398488, 0, -2107, 0], [-87800666688, 0, 39264450624, 0, -15792050700, 0, 2897647075, 0, -398280184, 0, 26810709, 0, -1267888, 0, 11299, 0], [100335652560, 0, -122817017232, 0, 32313724443, 0, -7670501839, 0, 1048998769, 0, -109750641, 0, 6070207, 0, -236299, 0], [0, -348009515532, 0, 123446110131, 0, -45329453883, 0, 6704863382, 0, -823985897, 0, 32000682, 0, -908495, 0, -90523], [0, 94833934272, 0, -35554785300, 0, 5273830425, 0, -479450177, 0, -14543173, 0, 5317761, 0, -458203, 0, 17575], [-667398916800, 0, 163015417944, 0, -31002566364, 0, -7211681519, 0, 1771996688, 0, -307019433, 0, 19357688, 0, -961463, 0], [-14125104000, 0, 9299398752, 0, -3406609584, 0, 718144501, 0, -98147980, 0, 9009267, 0, -483532, 0, 15973, 0], [-242518237920, 0, 322801951680, 0, -134626595796, 0, 29388081667, 0, -4086660592, 0, 339648981, 0, -17467576, 0, 443203, 0], [0, -63434342592, 0, 1083154212, 0, 9236878637, 0, -3994950813, 0, 733266583, 0, -85575203, 0, 5115945, 0, -198541], [0, 7949027664, 0, -2301711228, 0, 642448512, 0, -12141473, 0, -4730788, 0, 1857273, 0, -127492, 0, 7603], [0, 1746136865088, 0, -2905816074768, 0, 1107271381356, 0, -328567062775, 0, 51247418824, 0, -6005325921, 0, 351463840, 0, -14621479], [0, -14125104000, 0, 9299398752, 0, -3406609584, 0, 718144501, 0, -98147980, 0, 9009267, 0, -483532, 0, 15973], [1746136865088, 0, -2905816074768, 0, 1107271381356, 0, -328567062775, 0, 51247418824, 0, -6005325921, 0, 351463840, 0, -14621479, 0]]; Rf_basisdens := [1, 1, 1091452828032, 181908804672, 12993486048, 136431603504, 204647405256, 38980458144, 272863207008, 3389605056, 90954402336, 90954402336, 1917071712, 3274358484096, 3389605056, 545726414016]; 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_1050_bc();" function MakeCharacter_1050_bc() N := 1050; order := 12; char_gens := [701, 127, 451]; v := [12, 3, 2]; // 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_1050_bc_Hecke();" function MakeCharacter_1050_bc_Hecke(Kf) N := 1050; order := 12; char_gens := [701, 127, 451]; char_values := [[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, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 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, 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, -1, 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, 1, 0, 0, 0, 1, 0, 1, 0, 0], [0, 0, 0, 1, -1, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 0], [0, -1, 0, 0, 0, 0, -1, 1, 0, 0, 0, 1, -1, 0, -1, 0], [-1, 0, 0, 1, 1, -1, 0, 0, -1, 3, 1, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0], [-1, 0, 0, -1, 2, 2, 0, 0, 1, -2, 1, 0, 0, 0, 0, -1], [1, 0, 0, -1, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 1], [0, -1, 6, 0, 0, 0, 0, 2, 0, 0, 0, 2, -1, -2, -2, 0], [1, 0, 0, 0, -1, 0, 0, 0, 0, -6, -1, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 7, 1, 0, 0, 0, 1, 1, -6, 0, 0], [0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, -1, 0], [-6, 0, 0, -2, 2, -4, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0], [-2, 0, 0, -1, 0, -1, 0, 0, 5, 0, 0, 0, 0, 0, 0, 1], [0, -2, 0, 0, 0, 0, 1, 3, 0, 0, 0, -3, 3, 1, 2, 0], [6, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1], [0, -1, 0, 0, 0, 0, -11, 1, 0, 0, 0, -4, 1, 6, 1, 0], [1, 0, 0, 0, 1, 1, 0, 0, -6, -1, -1, 0, 0, 0, 0, -1], [0, -1, 6, 0, 0, 0, -1, -1, 0, 0, 0, 5, -1, 0, -2, 0], [-1, 0, 0, -1, 1, 9, 0, 0, -4, 9, 1, 0, 0, 0, 0, -2], [0, -2, 0, 0, 0, 0, -3, 4, 0, 0, 0, 4, 2, 5, -2, 0], [-2, 0, 0, 3, 3, -1, 0, 0, 2, 1, 2, 0, 0, 0, 0, -2], [0, -2, -1, 0, 0, 0, 3, -1, 0, 0, 0, -1, 3, 0, 1, 0], [0, 0, 0, 0, 0, 0, 8, -2, 0, 0, 0, -2, 2, -12, 2, 0], [5, 0, 0, 0, 0, -5, 0, 0, 6, -6, 1, 0, 0, 0, 0, -1], [0, 1, 0, 0, 0, 0, 11, -2, 0, 0, 0, -2, -1, -12, 1, 0], [0, 1, 0, 0, 0, 0, -4, 2, 0, 0, 0, 2, 1, -5, 3, 0], [-11, 0, 0, 0, 1, -5, 0, 0, 1, -1, -1, 0, 0, 0, 0, -1], [0, 3, 0, 0, 0, 0, -15, -3, 0, 0, 0, -9, -3, 6, -3, 0], [3, 0, 0, 0, -3, 0, 0, 0, 2, -1, 3, 0, 0, 0, 0, 0], [10, 0, 0, -2, -2, 4, 0, 0, -10, 2, 2, 0, 0, 0, 0, 4], [6, 0, 0, -2, 2, 8, 0, 0, -1, -1, 1, 0, 0, 0, 0, 1], [0, -1, -12, 0, 0, 0, -1, 1, 0, 0, 0, -5, -1, 1, -1, 0], [0, -1, -1, 0, 0, 0, 7, 2, 0, 0, 0, 0, 1, 0, 2, 0], [0, 2, 0, 0, 0, 0, 4, -4, 0, 0, 0, -4, -2, -6, 2, 0], [0, 0, -6, 0, 0, 0, -2, 2, 0, 0, 0, 2, -2, 0, -2, 0], [8, 0, 0, -4, -4, -4, 0, 0, -8, 4, -2, 0, 0, 0, 0, 2], [-5, 0, 0, -3, 2, -6, 0, 0, 3, 0, -1, 0, 0, 0, 0, -3], [-1, 0, 0, 1, 1, 5, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 1, -1, 0, 0, 0, -1, 1, 6, 1, 0], [0, -2, 6, 0, 0, 0, 4, -2, 0, 0, 0, 4, -2, 6, -4, 0], [-2, 0, 0, 3, -3, -5, 0, 0, -3, 4, 2, 0, 0, 0, 0, 2], [6, 0, 0, -4, -5, 0, 0, 0, -4, 0, 1, 0, 0, 0, 0, 4], [0, -1, 5, 0, 0, 0, -1, -1, 0, 0, 0, 4, -1, 0, -2, 0], [0, 4, -12, 0, 0, 0, 2, -2, 0, 0, 0, -8, 2, 6, 4, 0], [3, 0, 0, -1, -3, -7, 0, 0, -7, 11, -3, 0, 0, 0, 0, 2], [0, 4, -6, 0, 0, 0, 8, -2, 0, 0, 0, -14, 2, 0, -2, 0], [5, 0, 0, 2, -1, 8, 0, 0, -2, 4, 1, 0, 0, 0, 0, 2], [4, 0, 0, -4, -4, 0, 0, 0, -5, 1, 1, 0, 0, 0, 0, -1], [-4, 0, 0, 0, 4, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, -4, -6, 0, 0, 0, -6, -4, 0, -6, 0], [0, -1, 0, 0, 0, 0, -1, 1, 0, 0, 0, 1, -1, 18, -1, 0], [-20, 0, 0, 3, -3, -23, 0, 0, -8, -1, 2, 0, 0, 0, 0, 2], [-2, 0, 0, -1, -2, -2, 0, 0, 9, 2, 2, 0, 0, 0, 0, 3], [0, 5, 0, 0, 0, 0, -13, -3, 0, 0, 0, -9, -3, 5, -5, 0], [13, 0, 0, -2, -1, 0, 0, 0, -2, 0, -1, 0, 0, 0, 0, 2], [0, 0, 0, 0, 0, 0, -23, 1, 0, 0, 0, 0, 1, 12, 0, 0], [0, 3, 0, 0, 0, 0, 3, 3, 0, 0, 0, 3, 3, 0, 6, 0], [0, 0, 0, 0, 0, 0, -4, -4, 0, 0, 0, -4, 0, 4, 4, 0], [-3, 0, 0, 0, 0, 3, 0, 0, -9, 9, -3, 0, 0, 0, 0, 3], [0, 3, -2, 0, 0, 0, 5, -2, 0, 0, 0, -2, -1, -12, 2, 0], [0, 4, -6, 0, 0, 0, -7, 3, 0, 0, 0, 3, -7, 0, -3, 0], [3, 0, 0, 1, -3, 2, 0, 0, 2, -7, -3, 0, 0, 0, 0, 4], [0, 1, 6, 0, 0, 0, 1, -2, 0, 0, 0, 4, 1, 0, -1, 0], [0, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 5, 0], [11, 0, 0, -1, 4, 0, 0, 0, 11, -6, -5, 0, 0, 0, 0, 1], [0, 8, 0, 0, 0, 0, -13, -1, 0, 0, 0, -1, -1, 6, -8, 0], [14, 0, 0, 2, 2, 8, 0, 0, 4, -2, -2, 0, 0, 0, 0, -4], [0, -2, -24, 0, 0, 0, 0, 0, 0, 0, 0, -12, 0, 4, -2, 0], [0, 1, -5, 0, 0, 0, 13, 0, 0, 0, 0, -10, 1, 0, 0, 0], [10, 0, 0, -1, 3, 26, 0, 0, 1, 5, 2, 0, 0, 0, 0, -1], [0, 1, -18, 0, 0, 0, 9, -4, 0, 0, 0, -4, 3, -12, 4, 0], [-2, 0, 0, 5, 5, -3, 0, 0, 2, 3, 8, 0, 0, 0, 0, -8], [0, -4, 12, 0, 0, 0, -4, 2, 0, 0, 0, 26, -2, 0, 2, 0], [-2, 0, 0, 2, 2, 10, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0], [4, 0, 0, -4, 4, 8, 0, 0, -1, -13, -4, 0, 0, 0, 0, -4], [-1, 0, 0, -5, 0, 0, 0, 0, 7, -6, -5, 0, 0, 0, 0, 5], [1, 0, 0, 5, 5, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, -5], [-7, 0, 0, -4, 4, -3, 0, 0, -7, -9, 1, 0, 0, 0, 0, 1], [0, 2, 2, 0, 0, 0, 8, -1, 0, 0, 0, 1, 2, 6, 1, 0], [1, 0, 0, 0, -1, -13, 0, 0, 6, -13, -1, 0, 0, 0, 0, 1], [0, -2, 0, 0, 0, 0, 3, -1, 0, 0, 0, -1, -3, 0, -1, 0], [4, 0, 0, -1, 3, 14, 0, 0, 1, 4, 2, 0, 0, 0, 0, -1], [0, -5, 18, 0, 0, 0, -1, -1, 0, 0, 0, -1, 6, 14, 1, 0], [0, 0, 0, 2, -2, -4, 0, 0, -2, 10, 0, 0, 0, 0, 0, 2], [0, 5, -7, 0, 0, 0, -5, -9, 0, 0, 0, -2, -5, 0, 4, 0], [0, -6, -6, 0, 0, 0, -11, 7, 0, 0, 0, -5, 1, 0, 7, 0], [-19, 0, 0, -1, 1, -18, 0, 0, 5, -6, -5, 0, 0, 0, 0, -5], [0, -3, 0, 0, 0, 0, 25, 1, 0, 0, 0, 1, 1, -12, 3, 0], [27, 0, 0, 1, 4, 0, 0, 0, 13, -6, -3, 0, 0, 0, 0, -1], [-21, 0, 0, -5, 3, -9, 0, 0, -1, -3, -3, 0, 0, 0, 0, 2], [0, 0, 18, 0, 0, 0, 6, -6, 0, 0, 0, 12, 0, 6, -6, 0], [1, 0, 0, 4, -1, 3, 0, 0, -5, 9, -1, 0, 0, 0, 0, 5], [-5, 0, 0, -3, -3, 8, 0, 0, 5, -8, -1, 0, 0, 0, 0, 1], [0, 2, 2, 0, 0, 0, -1, 5, 0, 0, 0, 5, -7, -12, -5, 0], [-3, 0, 0, 1, 3, 12, 0, 0, 14, -25, 3, 0, 0, 0, 0, -2], [0, -3, 6, 0, 0, 0, -15, -1, 0, 0, 0, 5, -3, -12, -4, 0], [0, -6, 0, 0, 0, 0, -6, 6, 0, 0, 0, 0, 6, 6, 6, 0], [0, -9, 0, 0, 0, 0, 14, 2, 0, 0, 0, 8, 2, -6, 9, 0], [15, 0, 0, -1, 3, 9, 0, 0, 2, -3, -3, 0, 0, 0, 0, -2], [-4, 0, 0, 2, -2, -6, 0, 0, -5, -5, -1, 0, 0, 0, 0, -1], [0, 6, -24, 0, 0, 0, 2, -2, 0, 0, 0, -14, 2, -10, 6, 0], [0, 0, 6, 0, 0, 0, 24, 4, 0, 0, 0, -2, 0, -24, -4, 0], [0, 5, -12, 0, 0, 0, -2, -3, 0, 0, 0, -3, -2, 0, 3, 0], [-1, 0, 0, 1, 1, 0, 0, 0, -23, 24, -5, 0, 0, 0, 0, 5], [-11, 0, 0, 1, -3, -28, 0, 0, -1, 8, -2, 0, 0, 0, 0, 1], [0, -2, 12, 0, 0, 0, -10, -3, 0, 0, 0, 21, -5, 0, -3, 0], [0, -1, -12, 0, 0, 0, -6, 6, 0, 0, 0, 0, -6, 6, -1, 0], [0, -4, 6, 0, 0, 0, 8, 3, 0, 0, 0, 9, -4, 12, -1, 0], [8, 0, 0, -3, 3, 11, 0, 0, 14, 7, -2, 0, 0, 0, 0, -2], [8, 0, 0, 5, 8, 0, 0, 0, 5, 0, -3, 0, 0, 0, 0, -5], [5, 0, 0, 3, -3, 2, 0, 0, -1, 4, 1, 0, 0, 0, 0, 1], [0, -2, 23, 0, 0, 0, 4, 5, 0, 0, 0, 28, -2, 6, 3, 0], [0, -2, 24, 0, 0, 0, -6, 6, 0, 0, 0, 18, -6, 12, -2, 0], [0, 0, -6, 0, 0, 0, 0, 6, 0, 0, 0, -6, 6, 0, 6, 0], [5, 0, 0, -7, -4, 2, 0, 0, 7, -8, -11, 0, 0, 0, 0, -7], [6, 0, 0, 6, 6, -12, 0, 0, -11, 17, 1, 0, 0, 0, 0, -1], [0, 0, -11, 0, 0, 0, 0, 2, 0, 0, 0, 13, 0, 0, -2, 0], [0, 4, -12, 0, 0, 0, -1, 1, 0, 0, 0, -23, 5, 0, 1, 0], [0, -4, 0, 0, 0, 0, 2, -2, 0, 0, 0, -2, 2, -12, -4, 0], [20, 0, 0, 1, -6, 7, 0, 0, -5, 6, 6, 0, 0, 0, 0, 5], [8, 0, 0, 2, -2, 0, 0, 0, 14, -6, 4, 0, 0, 0, 0, -2], [-16, 0, 0, 1, 8, -4, 0, 0, 20, -8, -8, 0, 0, 0, 0, -9], [0, 0, 0, 1, 0, 10, 0, 0, 17, -34, 0, 0, 0, 0, 0, 1], [0, 7, -12, 0, 0, 0, -8, -3, 0, 0, 0, 9, -7, 1, -4, 0], [0, -5, 25, 0, 0, 0, 1, 4, 0, 0, 0, 4, 1, 0, -4, 0], [-5, 0, 0, -2, -2, 7, 0, 0, -14, 12, 5, 0, 0, 0, 0, -5], [0, -6, 0, 0, 0, 0, 18, -7, 0, 0, 0, -7, 6, -12, 13, 0], [-1, 0, 0, 3, 1, 0, 0, 0, 10, -19, 1, 0, 0, 0, 0, 2], [0, -1, -12, 0, 0, 0, 10, -2, 0, 0, 0, -14, -1, 11, -3, 0], [20, 0, 0, -1, -4, 8, 0, 0, -5, 4, 4, 0, 0, 0, 0, 5], [8, 0, 0, 4, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, -4], [0, 3, 0, 0, 0, 0, -12, 0, 0, 0, 0, 6, 0, 6, -3, 0], [0, 3, -24, 0, 0, 0, 5, -5, 0, 0, 0, -17, 5, 17, 3, 0], [0, -5, 0, 0, 0, 0, 23, -3, 0, 0, 0, -3, 5, -18, 8, 0], [8, 0, 0, -4, -4, -4, 0, 0, -14, 10, -2, 0, 0, 0, 0, 2], [-1, 0, 0, 5, -3, -8, 0, 0, -5, 4, 2, 0, 0, 0, 0, 5], [3, 0, 0, -4, -3, 5, 0, 0, -11, 19, -3, 0, 0, 0, 0, -1], [0, -3, -22, 0, 0, 0, 1, -1, 0, 0, 0, -12, 1, 0, -3, 0], [0, 7, -18, 0, 0, 0, 1, -5, 0, 0, 0, -23, 7, -6, 2, 0], [21, 0, 0, 5, -5, 16, 0, 0, -4, 7, 3, 0, 0, 0, 0, 3], [9, 0, 0, 6, 3, 0, 0, 0, 6, 0, 3, 0, 0, 0, 0, -6], [0, -3, 4, 0, 0, 0, -9, -5, 0, 0, 0, -1, -3, -6, -8, 0], [0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -18, -4, 0], [1, 0, 0, -2, -1, -27, 0, 0, 1, -3, -1, 0, 0, 0, 0, -1], [0, -7, 6, 0, 0, 0, -9, 4, 0, 0, 0, 16, -3, 0, 4, 0], [0, 9, 12, 0, 0, 0, -11, -3, 0, 0, 0, -3, -6, 10, 3, 0], [1, 0, 0, 0, -1, 0, 0, 0, 0, 6, -1, 0, 0, 0, 0, 0], [0, 0, 11, 0, 0, 0, -6, 7, 0, 0, 0, -4, 0, 6, -7, 0], [0, -2, 0, 0, 0, 0, -22, -2, 0, 0, 0, -2, -4, 0, -2, 0], [0, -1, 0, 0, 0, 0, 23, 1, 0, 0, 0, 25, 1, -11, 1, 0], [8, 0, 0, -1, -5, 0, 0, 0, -1, 0, 4, 0, 0, 0, 0, 1], [-33, 0, 0, 7, -9, -21, 0, 0, -24, 9, 9, 0, 0, 0, 0, 2], [2, 0, 0, -8, -2, 14, 0, 0, 6, -14, -2, 0, 0, 0, 0, -6], [0, 4, -12, 0, 0, 0, 6, -6, 0, 0, 0, 6, -4, -10, 2, 0], [8, 0, 0, -9, -9, 1, 0, 0, -14, 5, -8, 0, 0, 0, 0, 8], [0, -2, -12, 0, 0, 0, 12, -4, 0, 0, 0, -4, 6, -12, 4, 0], [0, -2, -12, 0, 0, 0, -10, 6, 0, 0, 0, 18, 2, 12, -4, 0], [0, 5, -24, 0, 0, 0, -2, -9, 0, 0, 0, -33, 5, -7, -4, 0], [15, 0, 0, 2, 3, 9, 0, 0, 23, -3, -3, 0, 0, 0, 0, -5], [0, 3, 0, 0, 0, 0, -33, 3, 0, 0, 0, 15, 3, 18, -3, 0], [0, 6, 0, 0, 0, 0, -8, -8, 0, 0, 0, 22, -8, 0, -6, 0], [-3, 0, 0, -5, 5, 2, 0, 0, 7, 12, 5, 0, 0, 0, 0, 5], [0, -2, 0, 0, 0, 0, 7, -7, 0, 0, 0, -7, 7, -17, -2, 0], [-13, 0, 0, 4, 3, -20, 0, 0, -4, 18, 7, 0, 0, 0, 0, 4], [0, 4, 18, 0, 0, 0, 8, -6, 0, 0, 0, -6, 2, -12, 6, 0], [-2, 0, 0, -4, -4, 6, 0, 0, -34, 30, 2, 0, 0, 0, 0, -2], [-6, 0, 0, -2, 7, 2, 0, 0, 2, -22, 5, 0, 0, 0, 0, -2], [-6, 0, 0, -4, 6, 14, 0, 0, 4, -2, 6, 0, 0, 0, 0, -10], [0, -4, 14, 0, 0, 0, -4, 4, 0, 0, 0, 11, -4, -12, -4, 0], [30, 0, 0, 7, -7, 23, 0, 0, -1, 18, 6, 0, 0, 0, 0, 6], [-6, 0, 0, -7, -7, 0, 0, 0, 17, -12, 0, 0, 0, 0, 0, 7], [15, 0, 0, -7, -2, 0, 0, 0, -19, 6, -5, 0, 0, 0, 0, 7], [30, 0, 0, -2, 2, 32, 0, 0, -18, -8, 6, 0, 0, 0, 0, 6], [0, 4, 1, 0, 0, 0, -8, 1, 0, 0, 0, 2, 4, -12, 5, 0], [1, 0, 0, -6, -1, -7, 0, 0, 11, -23, -1, 0, 0, 0, 0, -5], [0, 2, 24, 0, 0, 0, -22, 2, 0, 0, 0, 2, -4, 36, -2, 0], [-12, 0, 0, 0, 0, -24, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0], [0, -2, -12, 0, 0, 0, 2, -8, 0, 0, 0, 4, 2, 0, 10, 0], [0, 6, 0, 0, 0, 0, 22, -2, 0, 0, 0, -2, 4, 0, -2, 0], [0, -3, 24, 0, 0, 0, -1, 1, 0, 0, 0, 13, -1, -12, -3, 0], [18, 0, 0, 0, 0, 18, 0, 0, -24, -24, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3, -1, 0, 0, 0, 11, -1, 1, 0, 0], [0, -4, 0, 0, 0, 0, 30, 6, 0, 0, 0, -14, 6, -12, 4, 0], [-14, 0, 0, -5, -2, -8, 0, 0, 15, 2, 2, 0, 0, 0, 0, 7], [-6, 0, 0, 4, 4, 2, 0, 0, 42, -38, 0, 0, 0, 0, 0, 0], [0, -5, -5, 0, 0, 0, -1, 0, 0, 0, 0, 0, 5, 12, 0, 0], [0, -6, 6, 0, 0, 0, 0, 10, 0, 0, 0, 4, 6, 6, -4, 0], [-1, 0, 0, 3, 1, -12, 0, 0, -8, 17, 1, 0, 0, 0, 0, 2], [5, 0, 0, 2, -7, -1, 0, 0, -17, 7, 7, 0, 0, 0, 0, 5], [0, 5, 0, 0, 0, 0, -10, 2, 0, 0, 0, 20, 2, 6, -5, 0], [0, -12, 0, 0, 0, 0, 42, 6, 0, 0, 0, 24, 6, -18, 12, 0], [4, 0, 0, -6, 6, 10, 0, 0, 23, 7, -5, 0, 0, 0, 0, -5], [0, -2, 24, 0, 0, 0, -12, 2, 0, 0, 0, 50, 0, 0, 2, 0], [0, 6, -12, 0, 0, 0, 0, 0, 0, 0, 0, 12, -6, -6, -6, 0], [0, 0, -12, 0, 0, 0, -12, -6, 0, 0, 0, -6, 6, 36, 6, 0], [-18, 0, 0, 6, 6, 12, 0, 0, -12, 18, -6, 0, 0, 0, 0, 6], [0, 0, 0, -5, 0, 0, 0, 0, 5, -22, -5, 0, 0, 0, 0, -5], [0, -2, 6, 0, 0, 0, 1, 7, 0, 0, 0, 19, 5, 0, 7, 0], [35, 0, 0, -5, 5, 40, 0, 0, 5, -10, -5, 0, 0, 0, 0, -5], [-26, 0, 0, 2, 0, 0, 0, 0, 14, -6, 2, 0, 0, 0, 0, -2], [0, 8, 24, 0, 0, 0, 8, -8, 0, 0, 0, 4, 8, -24, 8, 0], [1, 0, 0, 0, -1, -25, 0, 0, -2, 3, -1, 0, 0, 0, 0, 1], [0, 0, -6, 0, 0, 0, -3, 9, 0, 0, 0, -3, 9, 0, 9, 0], [-30, 0, 0, -3, 3, -54, 0, 0, 3, -6, 0, 0, 0, 0, 0, -3], [-8, 0, 0, -3, -3, 11, 0, 0, -3, 0, 9, 0, 0, 0, 0, -9], [0, -8, 12, 0, 0, 0, 5, 4, 0, 0, 0, 4, 4, -2, -4, 0], [9, 0, 0, 7, -4, 10, 0, 0, -7, -4, 3, 0, 0, 0, 0, 7], [0, -3, -5, 0, 0, 0, -9, 1, 0, 0, 0, 6, 3, 12, 2, 0], [0, 2, -18, 0, 0, 0, -1, 3, 0, 0, 0, -33, 5, 0, 3, 0], [1, 0, 0, 7, -7, -6, 0, 0, -17, 0, 5, 0, 0, 0, 0, 5], [-8, 0, 0, 2, -6, -7, 0, 0, 20, 6, 6, 0, 0, 0, 0, 4], [0, 5, 0, 0, 0, 0, 14, 0, 0, 0, 0, 12, 0, -7, -5, 0], [-6, 0, 0, 6, 6, 0, 0, 0, 0, 6, -6, 0, 0, 0, 0, 6], [0, 6, -24, 0, 0, 0, 4, 2, 0, 0, 0, 2, -8, -24, -2, 0], [0, 5, -18, 0, 0, 0, -5, -1, 0, 0, 0, 17, -5, 0, -4, 0], [5, 0, 0, -6, -5, -13, 0, 0, 7, -19, -5, 0, 0, 0, 0, -1], [-3, 0, 0, 15, 12, 0, 0, 0, 1, 7, 3, 0, 0, 0, 0, -15], [3, 0, 0, -7, 3, 3, 0, 0, 8, -3, -3, 0, 0, 0, 0, 4], [0, 1, -12, 0, 0, 0, -6, 5, 0, 0, 0, -19, 6, 0, 5, 0], [0, -6, 6, 0, 0, 0, 6, 6, 0, 0, 0, 0, 6, 0, 0, 0], [0, 0, 0, -3, -3, -6, 0, 0, 3, 17, -6, 0, 0, 0, 0, -3], [0, -3, 6, 0, 0, 0, -24, 9, 0, 0, 0, 9, -6, 36, -9, 0], [-9, 0, 0, 13, 13, -4, 0, 0, -3, 16, 3, 0, 0, 0, 0, -3], [0, -15, 0, 0, 0, 0, 17, 6, 0, 0, 0, 6, -9, 0, 6, 0], [2, 0, 0, -6, -2, 10, 0, 0, 10, -22, -2, 0, 0, 0, 0, -4], [0, -3, -2, 0, 0, 0, 9, -9, 0, 0, 0, -10, 9, -12, -3, 0], [20, 0, 0, -11, 11, 31, 0, 0, 14, -1, -2, 0, 0, 0, 0, -2], [11, 0, 0, 4, -7, 0, 0, 0, -20, 12, 11, 0, 0, 0, 0, -4], [23, 0, 0, 5, -5, 18, 0, 0, 5, 24, 7, 0, 0, 0, 0, 7], [0, 0, 1, 0, 0, 0, -6, -1, 0, 0, 0, 0, 0, -6, -1, 0], [0, -4, 12, 0, 0, 0, -3, 3, 0, 0, 0, 9, -3, 42, -4, 0], [2, 0, 0, 1, -2, -10, 0, 0, -34, 66, -2, 0, 0, 0, 0, 3], [0, -2, -6, 0, 0, 0, 11, 1, 0, 0, 0, -11, -1, 0, 1, 0], [-14, 0, 0, 6, 2, -24, 0, 0, -6, -12, 8, 0, 0, 0, 0, 6], [-12, 0, 0, -3, -3, -30, 0, 0, 3, -6, -6, 0, 0, 0, 0, -3], [0, -9, 12, 0, 0, 0, -10, 10, 0, 0, 0, 16, -10, 0, -9, 0], [-4, 0, 0, 0, 8, 2, 0, 0, 20, -8, -8, 0, 0, 0, 0, -8], [0, 5, 0, 0, 0, 0, 17, -7, 0, 0, 0, 18, -7, -12, -5, 0], [-27, 0, 0, -9, 9, -9, 0, 0, 1, -9, -9, 0, 0, 0, 0, 0], [0, -1, -12, 0, 0, 0, -19, 6, 0, 0, 0, -6, -1, -18, 5, 0], [0, 0, 0, 9, 0, -12, 0, 0, -3, 6, 0, 0, 0, 0, 0, 9], [0, 2, 12, 0, 0, 0, -4, 11, 0, 0, 0, -1, -2, 2, -13, 0], [8, 0, 0, 5, 5, -13, 0, 0, -2, 7, 4, 0, 0, 0, 0, -4], [-9, 0, 0, -10, -10, 19, 0, 0, -2, -8, -3, 0, 0, 0, 0, 3], [0, 12, 12, 0, 0, 0, -42, -7, 0, 0, 0, -19, -12, 30, -5, 0], [0, -5, 0, 0, 0, 0, 18, 10, 0, 0, 0, 10, -5, 23, 5, 0], [24, 0, 0, -12, 0, 12, 0, 0, -12, 0, 0, 0, 0, 0, 0, 12], [0, -6, 0, 0, 0, 0, 7, -5, 0, 0, 0, 31, -5, -6, 6, 0], [-5, 0, 0, 2, 3, 0, 0, 0, 16, -7, -1, 0, 0, 0, 0, -2], [0, 4, 0, 0, 0, 0, 46, -2, 0, 0, 0, 4, -2, -24, -4, 0], [-17, 0, 0, -4, 7, -5, 0, 0, -15, -7, -7, 0, 0, 0, 0, -3], [24, 0, 0, -4, 4, 28, 0, 0, -2, -2, 2, 0, 0, 0, 0, 2], [0, -6, 12, 0, 0, 0, -9, 9, 0, 0, 0, 15, -9, 5, -6, 0], [0, -2, 12, 0, 0, 0, 2, 4, 0, 0, 0, -8, 2, 0, -2, 0], [0, -11, 6, 0, 0, 0, 0, 13, 0, 0, 0, 25, 2, 0, 13, 0], [0, 9, -14, 0, 0, 0, 13, -13, 0, 0, 0, -20, 13, -12, 9, 0], [0, 4, 12, 0, 0, 0, -8, -2, 0, 0, 0, 10, 4, -12, 2, 0], [11, 0, 0, -5, 5, 16, 0, 0, -12, -15, 1, 0, 0, 0, 0, 1], [0, -6, -2, 0, 0, 0, -6, 12, 0, 0, 0, 10, -6, 0, 6, 0], [0, -9, 36, 0, 0, 0, -3, 3, 0, 0, 0, 21, -3, -24, -9, 0], [-1, 0, 0, -5, 1, -13, 0, 0, 13, -25, 1, 0, 0, 0, 0, -6], [4, 0, 0, 4, -2, 4, 0, 0, -4, 14, 2, 0, 0, 0, 0, 4], [6, 0, 0, 12, -6, 0, 0, 0, -12, 18, 6, 0, 0, 0, 0, 12], [0, 0, 6, 0, 0, 0, -12, 8, 0, 0, 0, 2, 0, 12, -8, 0], [0, 9, 0, 0, 0, 0, 6, -6, 0, 0, 0, -6, 6, -6, 9, 0], [-52, 0, 0, -5, -2, -27, 0, 0, -13, 2, 2, 0, 0, 0, 0, 7], [0, 2, 0, 0, 0, 0, 16, 4, 0, 0, 0, -8, 4, -6, -2, 0], [0, 2, 0, 0, 0, 0, -6, -6, 0, 0, 0, -49, -6, 0, -2, 0], [-23, 0, 0, 6, 1, -11, 0, 0, 6, -1, -1, 0, 0, 0, 0, -7], [0, -3, 6, 0, 0, 0, 20, -2, 0, 0, 0, -8, 3, -17, 5, 0], [0, 6, -19, 0, 0, 0, -3, -3, 0, 0, 0, -3, -3, 0, 3, 0], [0, -4, -12, 0, 0, 0, 26, -4, 0, 0, 0, -4, 8, -36, 4, 0], [8, 0, 0, 1, 1, -9, 0, 0, 38, -37, -2, 0, 0, 0, 0, 2], [0, 0, 0, 6, 0, 25, 0, 0, 6, -12, 0, 0, 0, 0, 0, 6], [-14, 0, 0, 4, -2, -8, 0, 0, -22, 2, 2, 0, 0, 0, 0, -2], [0, -3, 0, 0, 0, 0, 63, 3, 0, 0, 0, 27, 3, -30, 3, 0], [-13, 0, 0, 1, -1, -14, 0, 0, 6, -5, -6, 0, 0, 0, 0, -6], [0, 5, -6, 0, 0, 0, -47, -2, 0, 0, 0, 4, -5, 42, -3, 0], [10, 0, 0, -4, 7, 34, 0, 0, 4, 22, 3, 0, 0, 0, 0, -4], [0, 7, 12, 0, 0, 0, -8, 2, 0, 0, 0, 26, 9, 0, 2, 0], [-7, 0, 0, 7, 7, -25, 0, 0, -12, 31, 7, 0, 0, 0, 0, 0], [0, -4, -10, 0, 0, 0, 2, -2, 0, 0, 0, -7, 2, -12, -4, 0], [0, 15, 6, 0, 0, 0, 3, -9, 0, 0, 0, -3, 15, -12, 6, 0], [19, 0, 0, 5, -5, 14, 0, 0, 26, 41, 5, 0, 0, 0, 0, 5], [-18, 0, 0, -5, 7, 0, 0, 0, -5, 0, -12, 0, 0, 0, 0, 5], [36, 0, 0, 12, -12, 24, 0, 0, 0, 36, 12, 0, 0, 0, 0, 12], [0, -3, 0, 0, 0, 0, 4, -4, 0, 0, 0, -4, 4, 42, -3, 0], [0, -3, -6, 0, 0, 0, -4, 5, 0, 0, 0, -7, 2, 0, 5, 0], [-13, 0, 0, -3, -2, -30, 0, 0, 3, -6, -5, 0, 0, 0, 0, -3], [0, -5, 11, 0, 0, 0, -25, 2, 0, 0, 0, -9, 5, 30, 3, 0], [4, 0, 0, 9, -9, -5, 0, 0, -14, -1, 2, 0, 0, 0, 0, 2], [-4, 0, 0, -8, 8, 2, 0, 0, -24, -8, -8, 0, 0, 0, 0, 0], [0, 17, -6, 0, 0, 0, 17, -19, 0, 0, 0, -25, 17, 0, -2, 0], [11, 0, 0, -16, -11, -15, 0, 0, -7, 3, -11, 0, 0, 0, 0, -5], [0, 0, -12, 0, 0, 0, 13, -3, 0, 0, 0, 9, 0, -13, 3, 0], [0, 1, -2, 0, 0, 0, 17, -12, 0, 0, 0, -12, 11, -12, 12, 0], [0, 10, -12, 0, 0, 0, 18, -16, 0, 0, 0, -16, 6, -24, 16, 0], [-8, 0, 0, 5, 5, 3, 0, 0, 1, 4, -4, 0, 0, 0, 0, 4]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1050_bc_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1050_2_bc_e();". // 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_1050_2_bc_e(:prec:=16) chi := MakeCharacter_1050_bc(); 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_1050_2_bc_e();". // 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_1050_2_bc_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1050_bc(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![20736, -3456, 4320, 48, 580, -4, 30, 2, 1]>,<13,R![4096, 0, 0, 0, 11618176, 0, 0, 0, 283569, 0, 0, 0, 1714, 0, 0, 0, 1]>,<17,R![1679616, 0, -3359232, 0, 2642544, 0, -806112, 0, 12481, 0, 29856, 0, 3383, 0, 96, 0, 1]>],Snew); return Vf; end function;