
// Make newform 1050.2.j.c in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1050_j();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1050_j_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_j_c();".
// 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_j_c();".
// 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, 0, 60, 0, 185, 0, 196, 0, 86, 0, 16, 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], [10, 22, 53, 35, 47, 12, 13, 1, 1, 0, 0, 0], [10, -22, 53, -35, 47, -12, 13, -1, 1, 0, 0, 0], [24, -84, 82, -152, 127, -151, 71, -73, 15, -15, 1, -1], [24, 84, 82, 152, 127, 151, 71, 73, 15, 15, 1, 1], [-28, -32, 4, -298, 103, -421, 69, -217, 15, -45, 1, -3], [-28, 32, 4, 298, 103, 421, 69, 217, 15, 45, 1, 3], [0, 46, 0, 243, 0, 321, 0, 157, 0, 31, 0, 2], [-12, 140, -192, 588, -327, 633, -191, 271, -43, 49, -3, 3], [48, -92, 342, -412, 491, -515, 241, -243, 47, -47, 3, -3], [12, 140, 192, 588, 327, 633, 191, 271, 43, 49, 3, 3], [48, 92, 342, 412, 491, 515, 241, 243, 47, 47, 3, 3]];
    Rf_basisdens := [1, 8, 8, 16, 16, 16, 16, 8, 16, 16, 16, 16];
    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_j();"
function MakeCharacter_1050_j()
    N := 1050;
    order := 4;
    char_gens := [701, 127, 451];
    v := [2, 1, 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_1050_j_Hecke();"
function MakeCharacter_1050_j_Hecke(Kf)
    N := 1050;
    order := 4;
    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, -1, 0, 0, 0, 0], [1, 0, 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<X, Y> := 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<T> := 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, -1, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, -1, 0, 0, 0, 1, 0, 1, 0], [0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1], [-4, -2, 0, -1, 0, -2, 1, -4, 2, 1, -1, 0], [0, 0, 0, 1, -1, -1, 1, 0, -1, 0, -1, 0], [1, 0, -2, 0, -1, 1, 0, -1, 1, 0, 0, 1], [2, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -2], [-4, 0, 0, 0, 0, 1, 1, 0, 2, 3, -2, 3], [4, 0, 2, 0, 3, -1, 0, -4, -1, 0, 0, -3], [0, -2, 2, 0, 0, 0, 0, -2, 0, -2, 0, 2], [-4, -4, 0, -3, 1, -3, 1, -4, 3, 3, -1, 1], [2, 4, 0, 3, -1, 1, -3, 2, -1, -3, 3, -1], [0, 0, -2, 2, -2, 0, -2, 0, 0, 2, -2, 2], [4, 2, 2, 3, 3, 1, 1, 0, 0, 1, 0, 1], [2, 2, 2, 3, 3, 0, 0, 0, 1, -1, -1, -1], [-2, 0, -4, 1, -1, 3, -1, 2, 3, 1, -1, 1], [0, -1, 1, -1, 1, -3, 3, -6, 1, -1, 1, 1], [6, 6, 0, 2, 0, 4, -4, 6, -4, -2, 4, 0], [0, 5, -5, 1, -1, 4, -4, 6, 1, 0, 1, 0], [4, 0, 4, -1, 2, -1, 1, -4, -1, -1, 1, -2], [10, 4, 4, 1, 1, -3, -3, 0, -5, -3, 5, -3], [-2, 0, -2, 2, -2, -2, -2, 2, -2, 2, -2, 2], [0, 8, -8, 2, -2, 3, -3, 6, -2, -2, -2, 2], [0, 8, 0, 4, -2, 4, 0, 0, -4, -4, 0, -2], [-6, 0, 0, -1, -1, 1, -1, -6, -1, 1, 1, -1], [0, -2, 2, -3, 3, -1, 1, -6, 1, 3, 1, -3], [4, 0, 4, -1, -1, -1, -3, -4, -1, -1, -3, 1], [-1, 0, -2, -1, -2, 4, 1, 1, 4, -1, 1, 2], [0, -6, 6, -2, 2, -2, 2, -4, 1, 4, 1, -4], [-1, -8, 0, -2, 1, -3, 4, -1, 3, 2, -4, 1], [0, -4, 4, 0, 0, 1, -1, 0, -1, 1, -1, -1], [-2, -6, -6, -4, -4, -2, -2, 0, 4, 2, -4, 2], [2, 2, 2, 3, 3, 1, 1, 0, 1, -3, -1, -3], [-2, 0, -2, -1, -2, 3, 3, 2, 3, -1, 3, 2], [0, -8, 0, -1, 3, -3, 5, 0, 3, 1, -5, 3], [0, 0, 0, -1, -1, -1, -5, 0, 1, 1, 5, -1], [6, 0, -2, 0, 2, -4, 1, -6, -4, 0, 1, -2], [0, 2, 2, -2, -2, 1, 1, 0, 0, 1, 0, 1], [-2, 4, 4, -4, -4, 0, 0, 0, -3, 0, 3, 0], [0, 3, -3, -3, 3, 1, -1, -6, 3, 3, 3, -3], [2, 8, 0, 3, -2, 0, -3, 2, 0, -3, 3, -2], [-6, -2, 0, 0, 0, -6, 4, -6, 6, 0, -4, 0], [0, -4, 4, -1, 1, -2, 2, 0, 1, -4, 1, 4], [-4, 4, 4, -4, -4, 4, 4, 0, 2, 2, -2, 2], [-2, -8, 0, -5, -1, -3, 3, -2, 3, 5, -3, -1], [4, 12, 0, -4, 3, 1, -1, 4, -1, 4, 1, 3], [0, -6, 6, -1, 1, -3, 3, -10, -2, -1, -2, 1], [3, 0, -4, 1, 4, -2, 1, -3, -2, 1, 1, -4], [12, -6, -6, 1, 1, 0, 0, 0, 3, 2, -3, 2], [18, 6, 6, 4, 4, 2, 2, 0, -4, -8, 4, -8], [0, 2, -2, 1, -1, 6, -6, 4, -3, -3, -3, 3], [2, 2, 0, 1, 0, 6, -3, 2, -6, -1, 3, 0], [-2, 0, 6, -1, -3, 3, 3, 2, 3, -1, 3, 3], [-10, -2, -2, -3, -3, 0, 0, 0, 7, 3, -7, 3], [-8, -8, -8, -2, -2, -5, -5, 0, 4, -1, -4, -1], [2, 0, 6, -4, 1, -3, 2, -2, -3, -4, 2, -1], [0, -4, 4, 1, -1, -3, 3, 14, -1, 1, -1, -1], [4, 20, 0, 3, -2, -1, -1, 4, 1, -3, 1, -2], [6, 0, 10, -4, 2, -4, 5, -6, -4, -4, 5, -2], [0, 0, 0, 1, 4, 0, -2, 0, 0, 1, -2, -4], [0, 2, -2, -2, 2, -8, 8, -8, 4, 2, 4, -2], [-10, -6, 0, -4, 2, 2, 0, -10, -2, 4, 0, 2], [12, 2, 0, 6, 0, 6, 2, 12, -6, -6, -2, 0], [8, -8, -8, -2, -2, 2, 2, 0, -2, 0, 2, 0], [6, 0, 8, -2, 7, 1, 8, -6, 1, -2, 8, -7], [2, 4, 0, 6, -2, 4, 2, 2, -4, -6, -2, -2], [0, 0, 0, -2, 2, 1, -1, 2, -2, 0, -2, 0], [4, 0, 6, 4, 9, -1, 0, -4, -1, 4, 0, -9], [2, 8, 8, 2, 2, 5, 5, 0, -6, -9, 6, -9], [4, 0, 24, 3, 1, -1, 3, -4, -1, 3, 3, -1], [0, 2, 0, -5, 2, 2, 5, 0, -2, 5, -5, 2], [0, -6, 6, 0, 0, -6, 6, -12, 4, 4, 4, -4], [-10, 0, -12, 5, -5, 5, -1, 10, 5, 5, -1, 5], [10, 4, 4, 6, 6, -2, -2, 0, -4, -2, 4, -2], [2, 0, 2, 5, 2, -3, 5, -2, -3, 5, 5, -2], [0, 5, -5, -2, 2, -1, 1, 0, -6, -5, -6, 5], [0, 8, -8, 6, -6, 4, -4, 6, -2, -2, -2, 2], [4, -10, -10, 0, 0, 1, 1, 0, 2, -4, -2, -4], [-6, 2, 2, 3, 3, 7, 7, 0, -5, 1, 5, 1], [0, -16, 16, -7, 7, -4, 4, -14, -1, 4, -1, -4], [-6, -2, 0, -2, -2, 0, 4, -6, 0, 2, -4, -2], [0, -2, 2, -1, 1, 4, -4, -12, -1, -2, -1, 2], [-10, 0, -16, -2, -12, 0, -2, 10, 0, -2, -2, 12], [-30, -8, -8, -7, -7, -3, -3, 0, 5, 3, -5, 3], [-2, 0, -18, 3, -8, -2, -1, 2, -2, 3, -1, 8], [0, -10, 10, 0, 0, 2, -2, 10, 1, 0, 1, 0], [2, 2, 0, -1, 7, -1, -3, 2, 1, 1, 3, 7], [-12, -16, 0, -4, 5, -2, 8, -12, 2, 4, -8, 5], [0, 8, 8, 2, 2, 0, 0, 0, 0, 0, 0, 0], [3, 0, 6, -3, -4, -2, 3, -3, -2, -3, 3, 4], [0, -8, 8, 1, -1, -4, 4, 4, -5, -8, -5, 8], [0, -14, 14, -6, 6, -8, 8, -4, 0, 6, 0, -6], [-10, 0, 12, 1, 1, -5, 1, 10, -5, 1, 1, -1], [-14, 0, 0, 5, 5, 5, 5, 0, 2, 7, -2, 7], [0, 14, -14, 5, -5, -1, 1, -2, -3, -1, -3, 1], [-4, 4, 0, 6, 1, -1, 1, -4, 1, -6, -1, 1], [6, 8, 8, 2, 2, 0, 0, 0, 0, 6, 0, 6], [-8, 0, -8, 5, -5, 7, 1, 8, 7, 5, 1, 5], [2, -2, 0, 8, 4, 4, 2, 2, -4, -8, -2, 4], [4, 0, -24, -1, -4, 4, 0, -4, 4, -1, 0, 4], [8, 13, 13, -3, -3, -6, -6, 0, -11, -6, 11, -6], [24, 18, 18, 2, 2, -2, -2, 0, -14, -8, 14, -8], [-6, 0, -30, -2, -6, 4, -2, 6, 4, -2, -2, 6], [4, 0, 0, 0, -1, 4, 4, 4, -4, 0, -4, -1], [-16, 0, -6, -2, -11, -1, -2, 16, -1, -2, -2, 11], [-6, -1, -1, 1, 1, -7, -7, 0, -1, -1, 1, -1], [-6, -10, -10, 0, 0, -4, -4, 0, 0, 4, 0, 4], [-2, 0, -16, 3, -5, 3, -11, 2, 3, 3, -11, 5], [6, 10, 0, 1, 8, 4, 1, 6, -4, -1, -1, 8], [0, 0, 0, -7, -1, 5, 5, 0, -5, 7, -5, -1], [0, -12, 12, 2, -2, -5, 5, 0, 5, 3, 5, -3], [18, -5, -5, 0, 0, -1, -1, 0, 0, -1, 0, -1], [0, -15, 15, -4, 4, -9, 9, 0, 4, -5, 4, 5], [16, 12, 0, 10, 5, 7, 1, 16, -7, -10, -1, 5], [-6, 8, 0, 7, 1, -3, -1, -6, 3, -7, 1, 1], [0, 0, 6, -8, 10, 0, 6, 0, 0, -8, 6, -10], [-4, -10, -10, 0, 0, 9, 9, 0, 0, 1, 0, 1], [-22, 4, 4, -4, -4, -2, -2, 0, 13, 2, -13, 2], [-11, -8, 0, 0, -4, -4, 4, -11, 4, 0, -4, -4], [2, -10, 0, -12, 4, -3, 2, 2, 3, 12, -2, 4], [-2, 0, 16, -2, 8, -2, 4, 2, -2, -2, 4, -8], [-16, -8, -8, -2, -2, 1, 1, 0, 1, 3, -1, 3], [0, -8, 8, 8, -8, 4, -4, -2, 0, 2, 0, -2], [0, 0, 0, -3, 3, -11, 11, -18, 1, 5, 1, -5], [-12, 6, 6, 4, 4, 0, 0, 0, -2, 4, 2, 4], [-16, 0, -12, -5, -1, 9, -7, 16, 9, -5, -7, 1], [-14, 10, 0, -8, -3, -7, 1, -14, 7, 8, -1, -3], [0, 24, -24, 8, -8, 10, -10, 4, -6, -6, -6, 6], [3, 0, 14, 3, 0, -2, -3, -3, -2, 3, -3, 0], [14, 10, 10, 5, 5, 3, 3, 0, 3, -1, -3, -1], [2, 0, -6, 8, 1, -3, 0, -2, -3, 8, 0, -1], [0, -8, 8, -10, 10, 0, 0, 10, 2, 2, 2, -2], [0, 22, -22, 2, -2, 4, -4, 2, 2, 2, 2, -2], [-16, 0, -10, 4, -2, -1, 8, 16, -1, 4, 8, 2], [16, 0, 12, -5, 4, -9, 5, -16, -9, -5, 5, -4], [24, 10, 0, 12, -4, 8, -11, 24, -8, -12, 11, -4], [-4, -3, -3, 3, 3, 2, 2, 0, -5, -2, 5, -2], [16, 8, 8, 12, 12, 1, 1, 0, -9, -9, 9, -9], [0, -8, 8, -16, 16, -6, 6, -2, 0, 2, 0, -2], [-18, -14, 0, -15, -3, -7, 7, -18, 7, 15, -7, -3], [0, 8, 0, 1, 3, 3, -5, 0, -3, -1, 5, 3], [0, 10, -10, 1, -1, 1, -1, 26, -4, -5, -4, 5], [-8, 0, 0, -10, -10, -4, -4, 0, 6, -2, -6, -2], [10, 10, 0, 0, -5, 7, -5, 10, -7, 0, 5, -5], [-12, -10, 0, -11, -2, -12, 3, -12, 12, 11, -3, -2], [0, 0, 0, -1, 1, 1, -1, 16, -3, -10, -3, 10], [-6, 0, 6, -2, 0, -6, -4, 6, -6, -2, -4, 0], [-2, 0, -22, -6, -1, 3, -6, 2, 3, -6, -6, 1], [0, -2, 2, -3, 3, -5, 5, -6, -1, 3, -1, -3], [-6, 4, 0, -5, -3, 3, -1, -6, -3, 5, 1, -3], [14, 12, 0, 7, 3, -5, -1, 14, 5, -7, 1, 3], [18, 0, 32, 3, 13, -7, 17, -18, -7, 3, 17, -13], [0, 6, -6, -6, 6, 7, -7, -4, 8, 1, 8, -1], [0, -9, 9, -17, 17, -6, 6, -30, 3, 6, 3, -6], [22, 0, 0, 0, 0, -4, -4, 0, -2, 0, 2, 0], [-4, 0, -10, 4, 6, 6, 2, 4, 6, 4, 2, -6], [0, 26, -26, 2, -2, 4, -4, -6, 1, 6, 1, -6], [2, -28, 0, -2, 2, -6, 12, 2, 6, 2, -12, 2], [-13, 0, -4, 7, 4, 2, -1, 13, 2, 7, -1, -4], [3, 0, 14, -9, -4, -2, -7, -3, -2, -9, -7, 4], [0, -2, 2, -5, 5, -8, 8, -28, -1, 3, -1, -3], [-3, -24, 0, -12, -1, -9, 2, -3, 9, 12, -2, -1], [6, 0, -8, -9, 3, -9, -1, -6, -9, -9, -1, -3], [-14, -25, -25, -10, -10, 5, 5, 0, 10, 9, -10, 9], [4, 0, -10, 3, 0, 4, -10, -4, 4, 3, -10, 0], [0, 15, -15, -1, 1, -8, 8, -8, -3, 0, -3, 0], [0, 0, 26, -8, 2, -5, 4, 0, -5, -8, 4, -2], [-8, -18, -18, -2, -2, 3, 3, 0, 6, 13, -6, 13], [-10, 2, 2, 3, 3, 10, 10, 0, 5, 5, -5, 5], [0, 17, -17, 11, -11, 7, -7, 38, 5, -3, 5, 3], [25, 12, 0, 12, 0, 8, 0, 25, -8, -12, 0, 0], [0, 4, -4, 3, -3, 10, -10, -20, 5, 4, 5, -4], [2, -20, -20, 0, 0, 0, 0, 0, 0, 8, 0, 8], [-8, -10, -10, -10, -10, -14, -14, 0, 4, 4, -4, 4], [0, -6, 6, 6, -6, 4, -4, -2, 8, 4, 8, -4], [-14, -12, 0, -7, 1, -3, -9, -14, 3, 7, 9, 1], [0, -12, 12, 2, -2, 3, -3, -2, -10, -2, -10, 2], [1, 0, 2, -3, 4, 6, -1, -1, 6, -3, -1, -4], [0, -10, 10, -4, 4, 4, -4, -12, -1, 10, -1, -10], [8, 26, 0, 11, 2, 8, -7, 8, -8, -11, 7, 2], [-10, 30, 0, 5, -2, 6, -5, -10, -6, -5, 5, -2], [-5, 0, -18, -3, -2, 0, -9, 5, 0, -3, -9, 2], [-2, 2, 2, -2, -2, 3, 3, 0, 4, -6, -4, -6], [14, 0, 18, -14, 5, -11, 4, -14, -11, -14, 4, -5], [-24, -8, 0, -6, 3, -4, 6, -24, 4, 6, -6, 3], [0, 8, -8, -6, 6, -2, 2, -8, 2, -2, 2, 2], [0, 4, -4, -6, 6, 2, -2, -16, 10, 4, 10, -4], [-4, -34, 0, -10, -2, -8, 6, -4, 8, 10, -6, -2], [12, 0, 28, -5, -3, -13, 11, -12, -13, -5, 11, 3], [-16, -12, -12, 2, 2, -2, -2, 0, 4, 4, -4, 4], [0, 10, -10, -3, 3, -7, 7, 2, -10, -5, -10, 5], [-20, -20, 0, -8, 0, 0, 0, -20, 0, 8, 0, 0], [-4, 0, -14, 6, -11, -9, -2, 4, -9, 6, -2, 11], [16, 27, 27, 8, 8, 7, 7, 0, -16, -11, 16, -11], [4, -6, 0, 9, -2, 4, -1, 4, -4, -9, 1, -2], [20, 30, 0, 13, -4, 0, -5, 20, 0, -13, 5, -4], [12, 0, -8, 11, 9, -3, -3, -12, -3, 11, -3, -9], [-6, -22, -22, 2, 2, 4, 4, 0, 20, 6, -20, 6], [-16, -10, -10, 0, 0, 9, 9, 0, 0, 1, 0, 1], [4, 0, -22, 7, -12, 4, -4, -4, 4, 7, -4, 12], [0, -4, 4, -8, 8, -4, 4, -10, -2, 6, -2, -6], [4, 2, 2, 8, 8, 1, 1, 0, -2, -12, 2, -12], [-2, 2, 0, 0, 8, 9, -2, -2, -9, 0, 2, 8], [0, 12, -12, 9, -9, 4, -4, 8, 3, 2, 3, -2], [4, 0, 4, 11, -5, -1, 3, -4, -1, 11, 3, 5], [48, 3, 3, 7, 7, 0, 0, 0, -13, -12, 13, -12], [8, 0, 0, 5, 5, -3, -3, 0, -7, -4, 7, -4], [-18, 0, -14, -1, -14, 12, -9, 18, 12, -1, -9, 14], [0, -2, 2, -6, 6, -4, 4, -10, -1, -2, -1, 2], [21, 10, 0, 10, -5, 21, -4, 21, -21, -10, 4, -5], [32, 28, 0, 24, -3, 7, -3, 32, -7, -24, 3, -3], [-4, 8, 8, -8, -8, -2, -2, 0, -2, 4, 2, 4], [18, -12, -12, 2, 2, 0, 0, 0, -4, 4, 4, 4], [-10, 0, -22, 5, 1, 15, -15, 10, 15, 5, -15, -1], [0, 10, -10, 8, -8, 0, 0, 10, 12, -10, 12, 10], [-30, -6, 0, -17, 1, -11, 5, -30, 11, 17, -5, 1], [16, 0, 30, -8, 2, -14, 8, -16, -14, -8, 8, -2], [34, 2, 2, 3, 3, 1, 1, 0, -9, -3, 9, -3], [0, -6, 6, -1, 1, 6, -6, -12, 1, 14, 1, -14], [-60, 0, 0, -10, -10, 4, 4, 0, 14, 12, -14, 12], [-3, -6, 0, -4, 9, 7, 2, -3, -7, 4, -2, 9], [-4, -28, 0, -12, -3, -5, 11, -4, 5, 12, -11, -3], [0, -8, 8, -13, 13, -11, 11, -22, -3, 7, -3, -7], [10, 0, 2, -4, 1, 5, -2, -10, 5, -4, -2, -1], [14, -20, -20, 0, 0, -3, -3, 0, -8, 3, 8, 3], [6, 0, 16, 4, 6, 6, 0, -6, 6, 4, 0, -6], [0, 2, -2, -3, 3, -14, 14, -28, 3, 12, 3, -12], [-32, -2, 0, -5, -6, -2, 3, -32, 2, 5, -3, -6], [0, 8, -8, 2, -2, 5, -5, 8, -5, -7, -5, 7], [2, -5, -5, 0, 0, 1, 1, 0, 8, 17, -8, 17], [0, 24, -24, 14, -14, 20, -20, 40, -10, -16, -10, 16], [36, 16, 0, 20, 3, 6, -4, 36, -6, -20, 4, 3], [14, 8, 0, 3, 9, 7, -1, 14, -7, -3, 1, 9], [0, 14, -14, 4, -4, -10, 10, 26, 6, -6, 6, 6], [-6, 0, 2, 4, -12, 14, 8, 6, 14, 4, 8, 12], [20, 8, 8, 12, 12, -3, -3, 0, -16, -5, 16, -5], [0, 14, -14, 8, -8, -2, 2, -8, 4, -6, 4, 6], [32, 0, 28, 7, 11, -3, 9, -32, -3, 7, 9, -11], [-24, -28, -28, 3, 3, 1, 1, 0, 11, 2, -11, 2], [-4, -20, 0, -6, 14, 4, -4, -4, -4, 6, 4, 14], [0, 12, -12, 18, -18, 6, -6, 30, 6, 2, 6, -2], [14, 0, -18, -6, 1, 9, 4, -14, 9, -6, 4, -1], [4, -14, -14, -6, -6, -6, -6, 0, 4, -2, -4, -2], [-26, 0, -28, 3, -21, 9, -17, 26, 9, 3, -17, 21], [0, 14, -14, -1, 1, 4, -4, -4, -5, -11, -5, 11], [0, -14, 14, -16, 16, -22, 22, -12, 8, -4, 8, 4], [-16, 0, -36, -7, -11, 9, -9, 16, 9, -7, -9, 11], [14, 0, 14, 4, 9, -11, 6, -14, -11, 4, 6, -9], [0, 0, 0, -4, 4, -14, 14, 2, 14, 0, 14, 0], [-16, -28, 0, -7, 5, -5, 9, -16, 5, 7, -9, 5], [0, -4, 4, 6, -6, 0, 0, 22, 0, -14, 0, 14], [10, 0, -22, 0, -14, 0, -7, -10, 0, 0, -7, 14], [32, 8, 8, 2, 2, 1, 1, 0, -10, -13, 10, -13], [-6, 4, 4, 11, 11, -13, -13, 0, 7, -5, -7, -5], [-8, 0, -16, -2, -15, 2, 2, 8, 2, -2, 2, 15], [-20, -34, 0, -8, 0, -20, 5, -20, 20, 8, -5, 0], [12, 0, 38, 0, 18, -8, 14, -12, -8, 0, 14, -18], [11, 6, 0, 0, 5, 3, -14, 11, -3, 0, 14, 5], [-16, -28, 0, -12, 10, 4, 4, -16, -4, 12, -4, 10], [0, 0, 0, 0, 0, -1, 1, -2, 8, 10, 8, -10], [4, 6, 0, -4, -3, -12, 4, 4, 12, 4, -4, -3], [8, -26, 0, 1, -4, -4, -7, 8, 4, -1, 7, -4], [0, -8, 8, -4, 4, -1, 1, -16, 11, 7, 11, -7], [-30, -12, -12, -18, -18, -2, -2, 0, 0, 12, 0, 12], [0, -14, 14, -3, 3, -11, 11, -6, 1, -9, 1, 9], [22, -20, 0, -11, -5, -1, 7, 22, 1, 11, -7, -5], [14, 0, 26, 4, 14, -6, 12, -14, -6, 4, 12, -14], [34, 24, 24, 6, 6, 10, 10, 0, -15, -2, 15, -2], [14, 0, 20, 1, 7, -21, 13, -14, -21, 1, 13, -7], [18, 22, 0, 16, -8, 8, -2, 18, -8, -16, 2, -8], [0, 22, -22, 11, -11, 9, -9, 10, -5, -13, -5, 13], [2, 0, -10, -14, 2, -8, 10, -2, -8, -14, 10, -2], [8, 24, 0, 9, -5, 7, -17, 8, -7, -9, 17, -5], [-6, 8, 0, -3, -9, 3, 9, -6, -3, 3, -9, -9], [0, 20, -20, 15, -15, 9, -9, 6, -13, -5, -13, 5], [-8, 7, 7, 3, 3, 18, 18, 0, -1, 2, 1, 2], [0, 14, -14, 7, -7, 2, -2, -4, -5, 9, -5, -9], [10, 0, -20, 1, 3, 5, 5, -10, 5, 1, 5, -3], [18, 15, 15, 10, 10, -5, -5, 0, -18, -13, 18, -13], [22, 32, 0, 15, -1, -7, -3, 22, 7, -15, 3, -1], [-2, 22, 22, -7, -7, 2, 2, 0, -9, -9, 9, -9], [4, 0, -4, 1, 10, 9, 3, -4, 9, 1, 3, -10], [0, 8, -8, 1, -1, 6, -6, -6, -5, -2, -5, 2], [-3, 20, 0, 8, -4, -8, -8, -3, 8, -8, 8, -4], [30, 10, 0, 10, -8, 6, -10, 30, -6, -10, 10, -8], [0, -20, 20, 5, -5, -8, 8, -8, -13, -10, -13, 10], [-2, -12, -12, 2, 2, 4, 4, 0, 16, 6, -16, 6], [0, -26, 26, 2, -2, -20, 20, -10, 0, 14, 0, -14], [-8, 24, 0, 14, -11, 8, 2, -8, -8, -14, -2, -11], [28, 0, -6, 2, 14, -2, 6, -28, -2, 2, 6, -14], [0, 10, -10, -1, 1, -2, 2, 20, -5, 5, -5, -5], [-22, -14, 0, -17, 12, -6, 3, -22, 6, 17, -3, 12], [22, 22, 22, 8, 8, 9, 9, 0, -22, -12, 22, -12], [-4, 2, 2, -2, -2, 5, 5, 0, 8, 1, -8, 1], [-16, 0, -2, -4, -4, 19, -4, 16, 19, -4, -4, 4], [4, 2, 2, -2, -2, 11, 11, 0, 12, -2, -12, -2], [0, 0, 44, 3, 20, -5, 9, 0, -5, 3, 9, -20], [14, -14, 0, -6, 0, 8, -6, 14, -8, 6, 6, 0], [-18, -14, 0, -8, 12, 4, 12, -18, -4, 8, -12, 12], [0, 5, -5, -1, 1, 10, -10, -10, 11, 10, 11, -10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1050_j_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1050_2_j_c();".
// 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_j_c(:prec:=12)
    chi := MakeCharacter_1050_j();
    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_j_c();".
// 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_j_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1050_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![4096, 0, 20480, 0, 29440, 0, 12288, 0, 1732, 0, 76, 0, 1]>,<17,R![10863616, -10758144, 5326848, 4593664, 1388928, 198656, 93440, 52032, 16868, 3240, 392, 28, 1]>,<29,R![64, 1600, 976, 32, -60, -4, 1]>],Snew);
    return Vf;
end function;