
// Make newform 1950.2.t.c in Magma, downloaded from the LMFDB on 30 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1950_t();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1950_t_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_1950_2_t_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_1950_2_t_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, 108, 0, 565, 0, 726, 0, 227, 0, 26, 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], [172, 0, -4615, 0, -5428, 0, -1724, 0, -203, 0, -8, 0], [0, 2674, 0, 9125, 0, 7920, 0, 2203, 0, 240, 0, 9], [0, -12038, 0, 45161, 0, 134312, 0, 47815, 0, 5776, 0, 229], [6014, -23228, 63643, -196378, 102796, -273340, 33821, -86810, 3968, -10004, 155, -386], [-5690, 18764, -78961, 196378, -139156, 273340, -46295, 86810, -5408, 10004, -209, 386], [6014, 23228, 63643, 196378, 102796, 273340, 33821, 86810, 3968, 10004, 155, 386], [-5690, -18764, -78961, -196378, -139156, -273340, -46295, -86810, -5408, -10004, -209, -386], [-4942, 32154, -61115, 182805, -89936, 246312, -28717, 77835, -3304, 8952, -127, 345], [6534, -18184, 27375, -184064, 23760, -276608, 6609, -89176, 720, -10336, 27, -400], [-6534, -18184, -27375, -184064, -23760, -276608, -6609, -89176, -720, -10336, -27, -400], [-4942, -32154, -61115, -182805, -89936, -246312, -28717, -77835, -3304, -8952, -127, -345]];
    Rf_basisdens := [1, 558, 496, 4464, 4464, 4464, 4464, 4464, 1488, 1488, 1488, 1488];
    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_1950_t();"
function MakeCharacter_1950_t()
    N := 1950;
    order := 4;
    char_gens := [1301, 1327, 301];
    v := [4, 1, 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_1950_t_Hecke();"
function MakeCharacter_1950_t_Hecke(Kf)
    N := 1950;
    order := 4;
    char_gens := [1301, 1327, 301];
    char_values := [[1, 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]];
    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, 0, -1, 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], [1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1], [-1, 0, -1, 0, -1, 0, 1, 1, -1, 0, 0, 0], [1, 0, -1, 0, -1, -1, 0, 0, 1, -1, 0, 0], [0, 0, 0, 0, 1, 1, 0, 0, 1, -1, 0, 0], [0, 1, 0, -1, 0, 0, 1, -1, 0, 0, 0, 0], [0, 0, -1, 1, 0, -1, 0, 1, 1, 0, 0, -1], [0, -1, 0, 1, 0, 0, -2, 0, 0, 0, 1, 1], [0, -2, 0, 0, -2, 1, -2, 1, 1, 0, 0, 1], [1, -1, 1, 1, 0, 0, -2, 0, 0, 0, -1, -1], [-1, 1, -1, -1, 0, 0, 4, 0, 0, 0, 0, 0], [0, -2, 0, 0, -1, 1, -1, 1, 1, 1, -1, 1], [1, -1, -1, -1, -2, 0, 0, 0, -1, 1, -2, 2], [1, -1, 1, 1, 0, 0, -2, 0, 0, 0, 1, 1], [2, 0, 0, 0, -1, -1, -1, -1, 2, 0, 0, 2], [0, 0, -4, 0, 0, 1, 0, -1, 2, -1, -1, -2], [0, 0, 0, 0, 0, 0, -6, 0, 1, 1, 1, 1], [0, 0, 1, -1, -4, -2, 4, 2, 1, -1, -1, -1], [0, 0, -2, 0, 5, -1, -5, 1, 1, 1, 1, -1], [-2, 0, 0, 0, -2, -2, -2, -2, 1, 1, -1, 1], [3, -1, 3, 1, 0, 0, 2, -2, 1, 1, 2, 2], [0, 0, -7, -1, -2, 0, 2, 0, 1, -1, -1, -1], [0, 0, 1, -1, 0, 1, 0, -1, 1, -2, -2, -1], [-4, 0, -4, 0, 0, 0, -4, -2, 0, 0, -1, -1], [-4, 0, -4, 0, 0, 0, -2, -2, 2, 2, 0, 0], [-3, 1, 3, 1, 3, 1, 0, 0, 0, 0, 0, 0], [3, 0, -3, 0, -3, -3, 0, 0, -1, 1, 0, 0], [-5, -1, 5, -1, -2, 0, 0, 0, 1, -1, 0, 0], [5, -1, 0, 0, 0, 1, 0, 1, -1, 0, 0, -1], [10, -2, 0, 0, 3, -1, 3, -1, 3, 1, -1, 3], [0, 0, 0, 4, -2, 0, 2, 0, 1, -1, -1, -1], [4, 0, -4, 0, 2, 2, 0, 0, -2, 2, 0, 0], [8, -1, -8, -1, 8, -2, 0, 0, 1, -1, -2, 2], [-1, 1, -1, -1, 0, 0, -4, 0, 1, 1, 0, 0], [0, 0, -4, 0, 0, 1, 0, -1, -2, -1, -1, 2], [2, 2, 0, 0, -6, -2, -6, -2, 0, 0, 0, 0], [3, -1, 3, 1, 0, 0, 8, -2, -2, -2, -1, -1], [9, 3, 0, 0, -6, -3, -6, -3, 1, 0, 0, 1], [0, 0, -8, -2, -5, -3, 5, 3, -2, -2, -2, 2], [-2, -2, 0, 0, -6, 2, -6, 2, 0, 0, 0, 0], [0, 0, 5, -1, 6, 0, -6, 0, 1, -1, -1, -1], [0, 0, -6, 0, -6, 0, 6, 0, -2, 0, 0, 2], [2, -2, 0, 0, 5, 1, 5, 1, 0, -2, 2, 0], [0, 0, 0, 0, 3, -3, 3, -3, 2, -2, 2, 2], [-3, -1, 0, 0, -6, 2, -6, 2, -3, 1, -1, -3], [0, 0, 4, 0, -4, 2, 4, -2, 0, 2, 2, 0], [-1, -1, -1, 1, 0, 0, -1, 1, -4, -4, 2, 2], [1, 2, 1, -2, 0, 0, -11, -3, 0, 0, 3, 3], [-2, 0, 2, 0, -4, -4, 0, 0, 2, -2, 2, -2], [-1, -2, 1, -2, -6, -2, 0, 0, 0, 0, -4, 4], [0, 0, -3, 1, -10, 1, 10, -1, 1, 0, 0, -1], [-5, 2, 5, 2, 1, -3, 0, 0, -3, 3, 0, 0], [-2, 1, 2, 1, 15, 1, 0, 0, 0, 0, 2, -2], [0, 0, 9, -1, -8, -1, 8, 1, -1, 2, 2, 1], [0, 3, 0, 3, -8, -2, 0, 0, 1, -1, -2, 2], [1, 1, -1, 1, 8, -2, 0, 0, -1, 1, -2, 2], [-7, -1, 7, -1, 18, 2, 0, 0, 0, 0, 1, -1], [9, -1, 9, 1, 0, 0, -4, -2, 3, 3, 1, 1], [0, 0, -4, 2, -4, 0, 4, 0, 3, 3, 3, -3], [-12, 4, 0, 0, 4, 1, 4, 1, -1, 0, 0, -1], [0, 0, 6, 2, 4, 2, -4, -2, 4, 2, 2, -4], [3, 3, -3, 3, 8, 0, 0, 0, 0, 0, 4, -4], [0, 0, 4, 4, 4, 0, -4, 0, -2, 2, 2, 2], [2, -4, 2, 4, 0, 0, -13, 7, -2, -2, 1, 1], [-4, 2, 4, 2, 6, 2, 0, 0, 0, 0, 2, -2], [4, 4, 4, -4, 0, 0, -6, -2, 0, 0, -2, -2], [-3, 5, -3, -5, 0, 0, 13, 3, -2, -2, -2, -2], [-2, -2, 0, 0, -9, -1, -9, -1, 3, 1, -1, 3], [0, -2, 0, 2, 0, 0, -14, -4, 3, 3, -1, -1], [5, 1, 5, -1, 0, 0, 2, 0, -2, -2, 3, 3], [-3, 1, 3, 1, 0, 2, 0, 0, -3, 3, 2, -2], [4, -4, -4, -4, -9, -1, 0, 0, 3, -3, -2, 2], [0, -4, 0, 0, -2, 2, -2, 2, -3, -3, 3, -3], [-11, 1, 0, 0, 6, -1, 6, -1, 0, 1, -1, 0], [-4, 6, 0, 0, 10, -1, 10, -1, 0, -3, 3, 0], [1, -3, -1, -3, -10, 8, 0, 0, 3, -3, -2, 2], [-3, 6, 3, 6, -2, -2, 0, 0, 2, -2, 4, -4], [0, 0, -7, -3, 4, 1, -4, -1, 0, -3, -3, 0], [3, 1, 3, -1, 0, 0, -1, -3, -2, -2, 0, 0], [14, -2, 14, 2, 0, 0, -4, 0, 0, 0, 4, 4], [2, -4, 2, 4, 0, 0, 6, 2, 2, 2, 0, 0], [0, -6, 0, 0, -6, 0, -6, 0, 3, 1, -1, 3], [2, 0, -2, 0, 10, -2, 0, 0, 6, -6, -2, 2], [5, 1, 5, -1, 0, 0, 2, -6, 1, 1, 0, 0], [0, 0, -5, 1, -12, 0, 12, 0, -5, -1, -1, 5], [-11, 1, -11, -1, 0, 0, -18, 4, -1, -1, -1, -1], [0, 0, 9, 5, -6, 2, 6, -2, -2, 2, 2, 2], [0, 4, 0, 4, 10, 2, 0, 0, -4, 4, 2, -2], [-2, -4, -2, 4, 0, 0, 4, 0, 0, 0, 0, 0], [0, 0, -13, 3, -4, -2, 4, 2, -3, -1, -1, 3], [0, 0, 1, 5, 12, 1, -12, -1, 0, 5, 5, 0], [-4, 0, 4, 0, -13, -1, 0, 0, 1, -1, 0, 0], [8, -3, -8, -3, 7, 1, 0, 0, 4, -4, 0, 0], [-9, 3, 9, 3, -6, 0, 0, 0, 3, -3, -1, 1], [10, 0, 0, 0, 10, -2, 10, -2, 4, 4, -4, 4], [-2, 2, 2, 2, 16, -6, 0, 0, -3, 3, 1, -1], [1, 1, -1, 1, 23, -3, 0, 0, 0, 0, -2, 2], [-8, 0, -8, 0, 0, 0, 8, 2, -3, -3, 1, 1], [0, 0, 0, 0, -6, 0, -6, 0, 0, 2, -2, 0], [-4, -2, -4, 2, 0, 0, 14, 6, -4, -4, 4, 4], [6, 0, 0, 0, -6, 0, -6, 0, -2, 0, 0, -2], [0, 0, 2, 2, 4, 2, -4, -2, -4, 2, 2, 4], [0, 0, -7, -1, 6, 0, -6, 0, 1, -5, -5, -1], [0, 0, 4, 0, -10, 2, 10, -2, 2, 2, 2, -2], [0, 0, 0, 4, -7, 1, 7, -1, 1, -1, -1, -1], [0, 0, 4, -4, 0, -2, 0, 2, 2, -4, -4, -2], [-8, 0, 0, 0, 7, 1, 7, 1, 0, 4, -4, 0], [9, 1, 9, -1, 0, 0, -14, -4, 0, 0, 1, 1], [0, 6, 0, 0, -8, -3, -8, -3, 4, -3, 3, 4], [0, 0, 6, 4, 2, -2, -2, 2, 0, 4, 4, 0], [2, -2, 2, 2, 0, 0, -15, 1, 4, 4, -3, -3], [-6, 5, 6, 5, 10, 0, 0, 0, 3, -3, 4, -4], [0, 0, 4, -2, -26, 0, 26, 0, 0, -2, -2, 0], [6, -2, 0, 0, -12, -5, -12, -5, 3, 2, -2, 3], [-2, -3, 2, -3, -1, 5, 0, 0, 2, -2, 2, -2], [-1, 3, 1, 3, 10, -8, 0, 0, 1, -1, -2, 2], [0, 0, 15, -5, -10, 1, 10, -1, 2, 1, 1, -2], [1, 1, -1, 1, -19, 3, 0, 0, -2, 2, -4, 4], [5, 3, 5, -3, 0, 0, 2, -2, -8, -8, 0, 0], [7, 3, 7, -3, 0, 0, 2, -4, 2, 2, -1, -1], [0, 0, -2, 4, -8, -6, 8, 6, 1, -5, -5, -1], [6, 2, 6, -2, 0, 0, -11, -3, 2, 2, 5, 5], [0, 0, -3, -5, -10, -5, 10, 5, -7, -2, -2, 7], [-7, 1, -7, -1, 0, 0, -3, -5, -4, -4, -2, -2], [8, 4, 0, 0, 10, -6, 10, -6, 4, 4, -4, 4], [-1, -3, 1, -3, 4, -2, 0, 0, -1, 1, -6, 6], [26, -4, 0, 0, 2, 1, 2, 1, -1, 2, -2, -1], [2, 6, 2, -6, 0, 0, 3, -9, -2, -2, 1, 1], [-12, -2, 0, 0, -1, 1, -1, 1, -2, -2, 2, -2], [0, 0, -4, -2, 12, 2, -12, -2, 5, -1, -1, -5], [10, -2, 10, 2, 0, 0, 2, 4, 0, 0, -5, -5], [1, -5, -1, -5, -6, 4, 0, 0, -1, 1, 0, 0], [-9, -2, 9, -2, 2, 6, 0, 0, -4, 4, 2, -2], [-4, 6, 0, 0, 8, -4, 8, -4, 3, 3, -3, 3], [-10, -6, 0, 0, 0, 5, 0, 5, -1, 0, 0, -1], [3, -5, -3, -5, 4, 2, 0, 0, -1, 1, -2, 2], [0, 0, 6, 0, 6, 0, -6, 0, -3, 7, 7, 3], [12, -2, 12, 2, 0, 0, -13, 3, 2, 2, -1, -1], [0, 8, 0, -8, 0, 0, 2, -2, -4, -4, -4, -4], [4, 8, 4, -8, 0, 0, 8, -6, 2, 2, -1, -1], [6, -4, 0, 0, -14, -4, -14, -4, -1, 5, -5, -1], [-6, 4, 0, 0, 0, -4, 0, -4, 2, 2, -2, 2], [-2, 0, -2, 0, 0, 0, 8, 8, 0, 0, -2, -2], [0, 0, -4, -6, -4, 7, 4, -7, 2, -3, -3, -2], [5, 0, -5, 0, 1, 1, 0, 0, 3, -3, -4, 4], [0, 0, 8, 0, 1, -5, -1, 5, 2, 2, 2, -2], [-30, 6, 0, 0, 6, 6, 6, 6, -6, -2, 2, -6], [0, 0, -16, -6, 0, 1, 0, -1, 4, -3, -3, -4], [0, 0, -12, -2, 2, -2, -2, 2, -4, -4, -4, 4], [-22, 2, -22, -2, 0, 0, 0, 2, -1, -1, 1, 1], [-2, -7, -2, 7, 0, 0, 1, 3, 2, 2, 2, 2], [3, 1, -3, 1, -8, -10, 0, 0, 7, -7, -7, 7], [8, 0, 0, 0, 20, 2, 20, 2, -2, 0, 0, -2], [0, 0, -8, 2, -6, 4, 6, -4, 1, -5, -5, -1], [13, 3, -13, 3, -10, 2, 0, 0, 6, -6, -1, 1], [-15, -1, -15, 1, 0, 0, -6, -2, 2, 2, 0, 0], [1, 5, 1, -5, 0, 0, -8, -6, -3, -3, -1, -1], [2, -8, 0, 0, -8, 0, -8, 0, 1, 1, -1, 1], [3, 0, 3, 0, 0, 0, 9, -3, 4, 4, 1, 1], [0, 0, -3, 1, 18, 4, -18, -4, 2, 0, 0, -2], [1, 3, 0, 0, -20, -5, -20, -5, 6, -3, 3, 6], [0, 0, -4, -8, -21, -1, 21, 1, 3, -3, -3, -3], [0, 0, -4, -2, -9, 5, 9, -5, 0, -4, -4, 0], [-10, -2, 0, 0, 5, 1, 5, 1, -6, 0, 0, -6], [1, -1, 1, 1, 0, 0, -8, 0, 9, 9, -2, -2], [11, 0, 11, 0, 0, 0, -24, 0, -2, -2, 0, 0], [17, 3, 17, -3, 0, 0, -20, -2, 4, 4, -3, -3], [-7, -5, 7, -5, -4, 6, 0, 0, 3, -3, -2, 2], [-21, 1, 21, 1, 5, 3, 0, 0, -6, 6, 2, -2], [0, 0, 0, 4, -16, -2, 16, 2, 0, -2, -2, 0], [-11, -7, 0, 0, 10, 6, 10, 6, -3, 1, -1, -3], [26, 4, 0, 0, -5, -3, -5, -3, -1, -1, 1, -1], [0, 0, 22, 0, -4, -4, 4, 4, -3, -7, -7, 3], [-14, 2, 14, 2, -19, 1, 0, 0, -1, 1, 8, -8], [-11, 5, 11, 5, 16, -6, 0, 0, -1, 1, -5, 5], [-8, -2, -8, 2, 0, 0, -10, 4, 6, 6, -1, -1], [0, 0, 0, -6, -3, 3, 3, -3, 0, -4, -4, 0], [13, -1, 0, 0, -14, -6, -14, -6, 9, -3, 3, 9], [0, 0, 27, 7, -4, 1, 4, -1, 2, -1, -1, -2], [14, 2, -14, 2, -14, -4, 0, 0, 3, -3, -2, 2], [0, 0, -4, -4, 5, -3, -5, 3, 1, -1, -1, -1], [0, 0, -28, -4, -4, 0, 4, 0, 7, -5, -5, -7], [13, 1, 0, 0, 18, -1, 18, -1, -6, -5, 5, -6], [-10, -6, 10, -6, 6, 4, 0, 0, 1, -1, 2, -2], [24, 0, 0, 0, -4, 3, -4, 3, -3, 4, -4, -3], [13, -4, 13, 4, 0, 0, 8, 4, 8, 8, 6, 6], [22, -2, -22, -2, -12, -8, 0, 0, 0, 0, -6, 6], [-18, 2, 18, 2, -16, 4, 0, 0, 2, -2, 0, 0], [-8, 8, 0, 0, 8, 0, 8, 0, 0, 0, 0, 0], [-12, 0, 0, 0, 16, 4, 16, 4, -4, 0, 0, -4], [11, 3, -11, 3, -8, -2, 0, 0, -7, 7, 1, -1], [4, -2, -4, -2, 0, -8, 0, 0, 2, -2, 0, 0], [1, 0, 1, 0, 0, 0, -13, -13, 4, 4, -7, -7], [9, 2, 9, -2, 0, 0, -20, 0, 4, 4, -2, -2], [5, 3, 5, -3, 0, 0, 4, 4, -7, -7, -2, -2], [20, -2, 0, 0, -20, 0, -20, 0, -2, 2, -2, -2], [-26, 7, 26, 7, 15, 1, 0, 0, 0, 0, 6, -6], [-5, 3, -5, -3, 0, 0, 2, 8, -1, -1, -5, -5], [-12, 8, 0, 0, -16, 0, -16, 0, 0, -4, 4, 0], [0, 0, 16, 6, 0, 5, 0, -5, 4, 1, 1, -4], [0, 0, 6, 8, 5, 3, -5, -3, -6, 6, 6, 6], [-5, 1, -5, -1, 0, 0, 6, -8, 10, 10, 1, 1], [-9, 1, -9, -1, 0, 0, 28, 2, -4, -4, -1, -1], [10, -1, -10, -1, -16, -14, 0, 0, -1, 1, -4, 4], [8, 0, -8, 0, 10, -2, 0, 0, -2, 2, -4, 4], [-9, -1, 9, -1, 14, 4, 0, 0, 1, -1, 4, -4], [0, 0, -10, 2, -12, 4, 12, -4, 7, -1, -1, -7], [-2, -4, 2, -4, 10, 6, 0, 0, -4, 4, 6, -6], [13, 1, 0, 0, 12, -1, 12, -1, -2, -7, 7, -2], [-16, 2, 16, 2, 22, 8, 0, 0, -3, 3, 4, -4], [-10, 8, 0, 0, 0, -2, 0, -2, -5, -7, 7, -5], [16, 0, -16, 0, -16, -4, 0, 0, 8, -8, 2, -2], [-1, -4, 1, -4, 16, 0, 0, 0, 4, -4, 6, -6], [-16, -2, -16, 2, 0, 0, -2, 0, 0, 0, -7, -7], [16, -18, 0, 0, -20, 4, -20, 4, 5, 1, -1, 5], [0, 0, 12, 2, 7, 5, -7, -5, -4, 8, 8, 4], [0, 0, 20, 0, -8, -2, 8, 2, -8, 6, 6, 8], [0, 6, 0, 0, 26, 8, 26, 8, -4, -6, 6, -4], [4, 3, 4, -3, 0, 0, -26, -8, 4, 4, 1, 1], [7, 3, -7, 3, 26, 8, 0, 0, 5, -5, 10, -10], [1, 11, 0, 0, 6, 0, 6, 0, -5, -7, 7, -5], [6, -4, -6, -4, -4, 4, 0, 0, -6, 6, -8, 8], [0, 2, 0, 0, -17, -7, -17, -7, 4, 4, -4, 4], [15, 4, -15, 4, 1, 5, 0, 0, -3, 3, -4, 4], [0, 0, 32, 4, -6, 8, 6, -8, 2, 4, 4, -2], [11, 5, -11, 5, -12, -10, 0, 0, -5, 5, -6, 6], [11, -7, -11, -7, 18, -4, 0, 0, 5, -5, 0, 0], [-11, -7, -11, 7, 0, 0, -4, 6, 2, 2, -3, -3], [0, 0, -4, 12, 14, 2, -14, -2, -2, 6, 6, 2], [10, -5, 10, 5, 0, 0, -14, 8, -2, -2, -9, -9], [0, 0, 18, 0, -12, 0, 12, 0, 7, 7, 7, -7], [15, -3, -15, -3, 0, 6, 0, 0, 7, -7, 3, -3], [0, 0, 14, 4, 15, -7, -15, 7, 1, 1, 1, -1], [3, -4, 3, 4, 0, 0, -12, -4, 4, 4, 8, 8], [0, 0, -14, -2, -8, 6, 8, -6, -1, 9, 9, 1], [-19, -9, 0, 0, 20, 5, 20, 5, -1, 0, 0, -1], [6, -6, 0, 0, 18, 0, 18, 0, 6, -8, 8, 6], [6, -4, 0, 0, 22, 2, 22, 2, -5, -5, 5, -5], [0, 0, 10, 10, 12, -4, -12, 4, 3, 3, 3, -3], [-2, -4, 2, -4, -10, -12, 0, 0, 3, -3, 0, 0], [-9, 3, 9, 3, 5, -1, 0, 0, -4, 4, -2, 2], [36, -8, 0, 0, -1, -5, -1, -5, 7, 5, -5, 7], [2, -10, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0], [-31, 3, 0, 0, 6, -4, 6, -4, -1, 1, -1, -1], [-2, 2, 2, 2, -26, -6, 0, 0, 0, 0, -5, 5], [14, 0, -14, 0, -27, -15, 0, 0, 3, -3, -6, 6], [4, 5, 4, -5, 0, 0, -17, -3, 0, 0, 2, 2], [12, 10, 0, 0, -2, 1, -2, 1, 4, -1, 1, 4], [-3, -5, -3, 5, 0, 0, -2, 2, 5, 5, 2, 2], [-10, -2, 0, 0, 4, 12, 4, 12, 1, -7, 7, 1], [0, 6, 0, 0, -2, -2, -2, -2, -1, -9, 9, -1], [-13, 3, 13, 3, -26, 4, 0, 0, 1, -1, 8, -8], [25, -1, 25, 1, 0, 0, -8, -6, 0, 0, 3, 3], [-10, 8, 0, 0, -9, 7, -9, 7, 2, 0, 0, 2], [0, 0, 2, -6, -4, 1, 4, -1, -8, -11, -11, 8], [1, 7, -1, 7, 18, -8, 0, 0, -3, 3, -2, 2], [0, 0, 15, 9, 4, 0, -4, 0, -3, -3, -3, 3], [-19, -1, -19, 1, 0, 0, 0, -10, -4, -4, -3, -3], [22, -2, 22, 2, 0, 0, 24, -8, -2, -2, -2, -2], [-7, -5, 7, -5, 1, 11, 0, 0, 0, 0, 0, 0], [6, -8, -6, -8, -2, 2, 0, 0, 10, -10, -1, 1], [-8, -2, 0, 0, 30, 2, 30, 2, -4, -4, 4, -4], [0, 0, -10, -2, -3, -7, 3, 7, 0, 2, 2, 0], [7, -1, -7, -1, -26, 0, 0, 0, -7, 7, -5, 5], [-7, -9, -7, 9, 0, 0, -2, -2, -1, -1, -4, -4], [0, 0, -12, 4, 24, 1, -24, -1, -2, 3, 3, 2], [-5, -1, -5, 1, 0, 0, -26, -12, 10, 10, 5, 5], [0, 0, -20, 2, 2, 0, -2, 0, -2, 8, 8, 2], [0, 0, -14, -6, 4, -1, -4, 1, -10, -3, -3, 10], [0, 0, 25, 3, 10, -4, -10, 4, 11, 1, 1, -11], [0, 10, 0, 0, 8, -2, 8, -2, -5, -7, 7, -5], [-7, 11, -7, -11, 0, 0, 10, 2, -6, -6, -10, -10], [27, 1, 0, 0, 18, -2, 18, -2, 8, 0, 0, 8], [0, 0, 24, 8, -8, 2, 8, -2, -4, 2, 2, 4], [-7, 3, -7, -3, 0, 0, -19, -1, -4, -4, -4, -4], [-9, -8, 9, -8, -24, 4, 0, 0, 2, -2, -2, 2], [0, 0, -7, -3, -8, 7, 8, -7, -3, 4, 4, 3], [10, 1, -10, 1, 21, 7, 0, 0, -2, 2, 4, -4], [-14, -11, 14, -11, -4, 6, 0, 0, 3, -3, 2, -2], [-8, 9, -8, -9, 0, 0, 5, -1, -10, -10, -10, -10], [5, -1, 5, 1, 0, 0, -13, -11, -2, -2, 6, 6], [8, 0, 0, 0, -8, 5, -8, 5, -7, 2, -2, -7], [0, 0, 6, -14, -4, 4, 4, -4, -6, -10, -10, 6], [25, 1, -25, 1, -4, -8, 0, 0, 2, -2, -8, 8], [0, 0, 16, 2, 6, -2, -6, 2, -1, -1, -1, 1], [4, -9, 4, 9, 0, 0, -6, 12, 4, 4, 1, 1], [36, 2, 0, 0, -2, -4, -2, -4, 3, 1, -1, 3], [-5, 15, 0, 0, 4, 1, 4, 1, -4, 1, -1, -4], [-16, 0, -16, 0, 0, 0, -14, 10, -6, -6, -8, -8], [-12, 14, 0, 0, 22, 2, 22, 2, -4, 0, 0, -4], [6, -2, -6, -2, 10, 14, 0, 0, -4, 4, 1, -1], [-26, 4, 26, 4, -18, -2, 0, 0, 0, 0, 3, -3], [-39, 5, 0, 0, 10, 5, 10, 5, -4, -9, 9, -4], [-14, -6, 0, 0, 5, 5, 5, 5, 4, 10, -10, 4], [5, -3, 5, 3, 0, 0, 10, -8, 4, 4, 5, 5], [0, 0, 9, -3, 6, 9, -6, -9, 9, 8, 8, -9], [30, -2, 0, 0, 16, 1, 16, 1, -9, -8, 8, -9], [-3, 5, -3, -5, 0, 0, -24, 4, -2, -2, 6, 6], [-10, 12, 0, 0, -16, 2, -16, 2, 4, -2, 2, 4], [-22, 12, 0, 0, 5, -1, 5, -1, 1, -7, 7, 1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1950_t_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1950_2_t_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_1950_2_t_c(:prec:=12)
    chi := MakeCharacter_1950_t();
    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_1950_2_t_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_1950_2_t_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1950_t();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-16, 16, 100, 48, -18, -4, 1]>],Snew);
    return Vf;
end function;