
// Make newform 1152.2.i.i in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1152_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1152_i_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_1152_2_i_i();".
// 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_1152_2_i_i();".
// 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 := [729, -486, 243, -216, 198, -126, 51, -42, 22, -8, 3, -2, 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], [-486, 243, -216, 198, -126, 51, -42, 22, -8, 3, -2, 1], [162, -135, 54, -81, 108, -51, 24, -13, 8, -3, 2, -1], [-1701, -405, -324, 36, -279, 15, 0, 40, -5, 9, -2, 4], [1215, -297, 189, -234, 243, -69, 33, -38, 13, -3, 1, -2], [4374, -567, 1404, -693, 954, -327, 192, -179, 52, -33, 4, -11], [-6075, -324, -1377, 306, -1035, 240, -135, 196, -29, 36, 1, 16], [891, -216, 234, -108, 165, -48, 26, -30, 7, -4, 0, -2], [9234, -972, 2538, -1125, 1494, -444, 210, -275, 46, -42, -2, -17], [-13122, 1944, -4482, 1845, -2430, 870, -462, 463, -92, 78, -2, 25], [-4050, -54, -918, 513, -756, 210, -138, 161, -34, 24, -4, 11], [918, -135, 234, -108, 165, -48, 26, -30, 7, -4, 0, -2]];
    Rf_basisdens := [1, 243, 162, 486, 162, 486, 486, 54, 486, 486, 162, 27];
    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_1152_i();"
function MakeCharacter_1152_i()
    N := 1152;
    order := 3;
    char_gens := [127, 901, 641];
    v := [3, 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_1152_i_Hecke();"
function MakeCharacter_1152_i_Hecke(Kf)
    N := 1152;
    order := 3;
    char_gens := [127, 901, 641];
    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, -1, 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, 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, 1, -1, -1, 0, 0, 0, 0, 0], [-2, -1, -1, 0, 0, 1, -1, 0, 0, 1, 1, 0], [0, 1, 1, -1, -1, 1, 2, 0, 1, 0, -1, 0], [1, 1, 0, -1, 1, 2, 1, 1, -1, -1, -1, -1], [0, 1, 0, 0, 1, 0, 1, 0, -1, -1, 1, 1], [-2, -2, -1, 0, 1, 0, -2, -1, -1, -1, 1, 0], [0, 2, 0, 0, 0, -2, 0, -1, 1, 0, 1, 0], [1, 2, 2, 0, -1, 0, 1, 0, 0, -1, -1, 1], [0, 0, 0, -2, 0, 0, 2, 1, -1, -2, -1, 0], [0, -1, 1, 1, 0, 0, 0, -1, 1, 0, 0, 0], [-1, -1, 0, 1, -2, 0, 0, -1, 1, 1, 1, 1], [1, 2, 2, 0, -2, 0, 3, -1, 0, 0, 0, 1], [2, -1, 1, 0, -2, -1, 1, 0, 0, 1, 1, 0], [4, 3, 1, 1, 0, 0, 2, 1, 1, 0, 0, 0], [-2, -3, 0, 0, 0, -2, -3, -1, 0, 0, 0, 2], [1, -2, -2, 0, 1, -2, -1, 0, 0, 1, 1, -1], [0, 3, 0, 1, -2, -1, 1, -2, 2, 1, 2, 0], [1, -1, -3, 0, 1, 0, -3, 0, -1, -1, 1, -1], [2, -3, -2, 0, 1, 0, -1, -2, -1, -1, 1, 1], [-3, 0, 0, -1, -1, 2, -2, 1, 1, 1, 0, 1], [-6, 0, -1, 1, -1, 3, -2, -1, -1, 2, 3, 2], [-6, -3, -1, -1, 0, 0, -2, -1, -1, 0, 0, 0], [-4, -4, -1, 2, 0, 4, -2, 0, -2, 1, 3, -1], [-4, -2, -1, 0, -1, 3, -1, 0, 0, 2, 2, 0], [0, 0, 0, 1, 0, -7, 0, 0, 1, 1, 1, 0], [-2, 2, -1, -2, -1, 0, 4, 1, -1, 1, -1, 2], [2, 1, 3, 1, -2, 0, 0, 1, 3, 2, -2, -2], [-3, -5, 0, -2, -1, 1, -1, 0, -1, -2, -1, 3], [8, 4, 6, 0, -2, 0, 4, 2, 2, 2, -2, 0], [-1, -1, 0, 1, 2, -3, -2, 0, 2, 1, 2, 1], [4, 2, 1, 0, 2, -2, 2, 0, 0, -3, -3, -1], [0, 1, 0, 0, 4, -8, -5, -1, 0, 0, 0, 0], [1, 1, 0, -3, 1, 0, 1, 1, -3, -3, -3, -1], [-7, 0, 2, 1, -1, 8, -2, 1, -1, -1, 0, 1], [0, 4, 0, 0, -3, 3, 3, -2, 2, 0, 2, 0], [10, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 0, 3, 0, -5, 2, 2, 1, 3, 1, -2], [-4, 4, 1, 0, 1, 3, -1, 0, 0, -2, -2, 2], [10, 4, 4, 4, 0, 0, 4, 0, 4, 0, 0, 0], [-8, -5, -1, -1, 2, 0, -2, -3, -3, -2, 2, 2], [12, 1, 5, -2, -2, -5, 5, 2, 2, -3, -5, -2], [3, 1, 0, 1, 2, -4, 2, 3, -1, 1, -1, -3], [0, -4, -2, 0, 0, 0, -2, -2, 0, 0, 0, 0], [7, 1, -3, 0, 1, 0, -5, 2, -1, -1, 1, -3], [1, -1, 0, 0, -2, -2, 2, 1, -2, 0, -2, -1], [-4, -3, -3, 2, 6, 3, -7, 2, -2, -3, -1, -2], [-2, 5, 1, -1, 3, 3, 2, 0, 1, -4, -5, 0], [-2, 2, 0, -2, -1, 13, 1, -2, 2, -2, 2, 2], [8, 2, 1, -3, -1, 0, -1, 3, -2, 1, -1, -1], [0, 2, 0, 2, 0, 0, -4, -3, 1, 2, 1, 0], [1, 2, 0, -4, 2, 1, 2, 2, 4, -2, -6, -2], [-10, 0, 5, 2, -1, 0, 2, -1, 3, 1, -1, 0], [-4, -8, 0, 0, 0, -3, -8, -2, -2, 0, -2, 4], [-7, -2, 0, -1, -1, 6, -6, 3, 1, 1, 0, 1], [10, 9, 7, 1, -2, 0, 10, 3, 3, 2, -2, 2], [6, 4, -2, -2, 2, 0, 2, 2, -4, -2, 2, 2], [6, 4, -3, -2, 3, -9, 3, -2, 2, 0, -2, 0], [5, -3, 0, 1, -2, -6, -4, 1, -1, 2, 3, 1], [1, 1, 0, -3, 0, -9, 0, 0, -4, -3, -4, -1], [0, 4, 0, 0, -5, -5, 5, -2, 2, 0, 2, 0], [10, -2, 5, 2, -3, 0, 0, -1, 5, 3, -3, -2], [2, 4, 0, -4, 2, -4, -2, -1, -5, -4, -5, -2], [-3, 6, 0, 2, 4, 1, 0, -2, -2, -4, -2, 2], [-1, 4, -6, 0, -1, -12, -5, -4, 0, 7, 7, 7], [-7, -1, -2, 2, 0, 2, -7, 0, -2, 2, 4, 3], [4, 4, 0, 2, 0, -1, 8, 4, 2, 2, 2, -4], [0, -1, 0, 5, -6, 9, 1, -2, 2, 5, 2, 0], [10, 2, 5, -4, -9, -7, 7, 0, 4, 4, 0, 2], [10, 6, 7, 0, -4, -6, 4, 0, 0, -3, -3, 3], [2, 6, 6, 0, 0, 0, 6, 2, 0, 0, 0, 2], [-7, 6, 2, -3, -1, 8, 6, -1, 3, -1, -4, 1], [1, 1, 0, 1, 1, 16, 1, 1, 1, 1, 1, -1], [-1, -1, -4, 4, 0, 0, 3, -3, 4, 0, 0, 1], [-2, -4, 0, 5, -4, 9, 2, 0, 5, 5, 5, 2], [-4, -2, -1, 0, 1, 5, 1, 0, 0, 0, 0, -2], [0, 7, 5, 1, 0, 0, -2, 5, 1, 0, 0, -4], [-1, -3, 0, 3, 2, -6, -6, -1, 1, 3, 1, 1], [-4, -4, 0, 2, -6, -1, 2, -2, 4, 2, 4, 4], [3, 1, 0, 6, -4, 10, 8, 3, 4, 6, 4, -3], [1, 0, -6, 2, 3, -10, -7, -2, -2, 3, 5, 3], [-2, 4, 4, 2, 4, 0, 2, 0, -2, -4, 4, 2], [4, 3, 2, -6, 1, 0, -1, 4, -7, -1, 1, 1], [-5, 8, -4, -1, 1, -6, -4, -3, 1, 3, 2, 7], [15, 2, 2, 0, 0, -12, 5, -1, 0, -2, -2, -1], [-2, -3, -8, -2, 3, 0, -7, 0, -5, -3, 3, -1], [-16, -4, -5, 0, 2, 12, -4, 0, 0, 3, 3, -1], [-1, 0, 0, 2, 5, 2, -5, 2, -2, -5, -3, -1], [0, 4, 0, 0, 6, -8, -4, -1, 3, 0, 3, 0], [14, 2, 3, 2, 1, 0, 4, -1, 1, -1, 1, 2], [-11, -4, -12, 5, 5, -6, -16, -3, -5, 7, 12, 5], [14, -6, -2, 0, 0, 0, -6, -2, 0, 0, 0, -2], [2, 7, 0, -6, -2, 4, 9, 1, -2, -6, -2, -2], [-2, 1, 0, -4, -2, -2, 3, -1, 0, -4, 0, 2], [7, -13, 1, -2, -5, 0, -1, -6, 3, 5, -5, 1], [1, -3, 0, 1, -5, 4, 3, 1, -3, 1, -3, -1], [-4, -2, -3, -1, -1, 0, -5, 1, 0, 1, -1, -3], [9, 5, -1, 0, -1, 0, 5, 2, 1, 1, -1, 1], [0, 8, -2, 4, 0, 0, 6, 2, 4, 0, 0, 0], [-2, 3, 3, -5, -1, 7, 6, 2, 5, -2, -7, -2], [4, -1, 7, 3, -2, 0, 0, -1, 5, 2, -2, -2], [2, 1, 0, -5, 2, -15, 5, 4, -4, -5, -4, -2], [7, -4, 2, 0, -2, -1, 6, 0, 0, 0, 0, -4], [0, -3, 0, 3, 2, 1, 3, 4, 4, 3, 4, 0], [12, 5, 0, 6, 1, 0, 3, 0, 5, -1, 1, -1], [3, 6, -4, 0, 2, -10, 3, -5, 0, 2, 2, 3], [4, -3, -1, 1, -2, 0, -2, -1, 3, 2, -2, -2], [-4, 2, 0, 5, -4, -15, -6, -8, 7, 5, 7, 4], [-6, 2, 3, -2, -6, 6, 2, 0, 2, 3, 1, 3], [-4, 2, 0, 1, -6, -25, -4, -8, 3, 1, 3, 4], [-8, 3, 5, 1, 4, 0, -2, 1, -3, -4, 4, 0], [-1, 1, 0, -7, -4, -4, 4, -1, -5, -7, -5, 1], [-14, 3, -1, -1, 1, 13, 4, -2, 1, 0, -1, 0], [6, 6, 4, 0, 2, 0, 10, 0, -2, -2, 2, 6], [-17, -8, -6, 2, 4, 15, -4, 0, -2, 2, 4, -4], [0, -3, 0, -5, 0, -3, 5, 4, -4, -5, -4, 0], [18, 14, 2, 0, -4, 0, 14, 6, 4, 4, -4, 2], [4, 4, 0, 4, -1, 3, 1, 0, 0, 4, 0, -4], [-9, -3, -8, 0, 2, 0, -5, -2, 0, 6, 6, 1], [-2, -6, 0, 2, -3, 1, -1, 0, 0, 2, 0, 2], [7, 11, 12, -3, -10, -2, 6, 1, 3, -2, -5, 7], [-6, 8, 5, -2, 1, 13, 13, -2, 2, -6, -8, -2], [-6, 4, 9, 4, -1, 0, -4, 3, 5, 1, -1, -6], [-8, -4, 3, 1, 1, 15, 0, 3, -1, -4, -3, -4], [-2, -3, 5, 7, -4, 0, 0, -3, 11, 4, -4, -4], [2, -4, -1, 0, -1, -5, -11, 4, 0, 2, 2, 2], [-20, -12, -4, 0, 0, 0, 0, -8, 0, 0, 0, 4], [-19, 2, 0, 3, 3, 18, -2, -1, -3, -3, 0, 1], [-1, -1, 0, 5, 5, 0, -3, 1, 7, 5, 7, 1], [-11, 3, -10, 0, 4, 0, -1, 1, -4, -4, 4, 1], [-6, -14, 0, -5, 4, -13, -8, 2, -5, -5, -5, 6], [-2, -4, 0, -7, 0, 5, -2, 0, -7, -7, -7, 2], [-2, 4, -4, 0, 4, 0, -8, 4, -4, -4, 4, -4], [2, 0, 0, 7, -5, -4, 1, -1, 2, 7, 2, -2], [-3, -1, 0, 4, 7, -17, -13, -4, 5, 4, 5, 3], [0, -1, -5, -1, 2, 0, -6, 1, -3, -2, 2, -2], [-1, -5, 0, 9, 6, -23, -10, 0, 6, 9, 6, 1], [-3, 1, 0, -5, -5, -18, -1, -5, -3, -5, -3, 3], [6, 1, 2, 6, 5, 0, -3, -2, 1, -5, 5, -1], [-22, -10, -7, -4, -1, 0, -6, -3, -3, 1, -1, 0], [9, -4, -6, 6, 5, -16, -9, -2, -6, 1, 7, 1], [-4, -4, 0, -4, 0, 14, -2, -1, -1, -4, -1, 4], [2, 0, -4, -4, -4, 0, 8, 0, 0, 4, -4, 4], [-6, 1, 5, -1, 0, 0, 6, -1, -1, 0, 0, 4], [-6, 1, -3, 4, 10, 3, -11, 2, -4, -7, -3, 0], [1, 4, -2, -2, -3, -8, -1, -2, 2, 5, 3, 5], [-21, 5, 0, -5, -4, 18, 4, -1, 5, 4, -1, 3], [-4, -3, 0, 3, -8, -1, 1, -4, 4, 3, 4, 4], [10, 18, 16, 8, -2, 0, 18, 4, 10, 2, -2, 2], [1, 5, 0, -5, -5, 2, 7, -1, -3, -5, -3, -1], [16, 3, 9, -1, -6, 0, 4, 3, 5, 6, -6, -2], [-2, -4, 5, 0, -3, 0, 4, -3, 3, 3, -3, 2], [2, -2, 0, 4, -6, -12, 0, -1, -3, 4, -3, -2], [15, -8, 4, 0, 4, -4, -7, 9, 0, -8, -8, -7], [1, 10, 14, -7, -13, 12, 22, -1, 7, -1, -8, 1], [-6, -8, -4, 6, 0, 0, 2, -8, 6, 0, 0, 2], [15, -13, 0, 1, 2, -6, -2, 5, -1, -2, -1, -9], [10, 3, -7, 1, 0, 0, -4, 3, 1, 0, 0, -4], [2, 10, 0, -4, 5, -5, 3, 0, 2, -4, 2, -2], [-16, -9, -1, -1, -3, 19, 0, 2, 1, 4, 3, -4], [-4, 11, 16, 2, -5, 0, 7, 6, 7, 5, -5, -3], [4, 10, 0, -2, -2, -2, 10, 1, 1, -2, 1, -4], [-9, -5, 1, -4, -3, 0, 3, -2, -1, 3, -3, 3], [4, 10, 0, 6, 2, -9, -4, -4, 4, 6, 4, -4], [7, 12, 12, 3, -11, -2, 16, -7, -3, -1, 2, 7], [6, -2, 8, 0, -6, 0, 2, 0, 6, 6, -6, -2], [8, -2, 3, -8, -1, 3, 11, 4, 8, -2, -10, -8], [8, 5, 7, -3, 2, 0, 14, -1, -5, -2, 2, 10], [-3, -3, 0, 3, -5, -2, -1, -3, 3, 3, 3, 3], [-11, -4, -2, 0, 8, 18, 5, 1, 0, -6, -6, -9], [-3, 1, 0, -9, -5, 6, 7, -1, -3, -9, -3, 3], [-2, -10, 0, 8, 2, -20, -4, 3, 5, 8, 5, 2], [14, -4, 13, -2, -2, 8, 8, 8, 2, -11, -13, -9], [4, 4, 0, 1, 6, -5, 2, 4, 1, 1, 1, -4], [-5, -3, 0, 2, -1, -15, -5, -4, 5, 2, 5, 5], [21, 10, 14, 0, -10, -12, 11, -1, 0, -4, -4, 5], [28, -1, -1, -1, -2, 0, 4, -1, 1, 2, -2, 2], [-4, -6, 2, -2, -4, 0, -12, 2, 2, 4, -4, -8], [12, 16, 3, -2, -3, -17, 9, -6, 2, 0, -2, 8], [-2, 9, 3, -10, -2, 5, 9, 2, 10, -1, -11, 0], [1, -10, -10, 8, 16, -6, -19, 3, -8, -6, 2, -5], [6, 10, 0, 10, 3, 9, 5, 2, 10, 10, 10, -6], [4, -3, 6, 6, 3, 0, 3, -6, 3, -3, 3, 3], [2, -6, 0, 3, 10, -3, -14, 2, -5, 3, -5, -2], [15, 10, 14, -6, -13, -6, 11, 2, 6, -1, -7, 5], [-6, -12, -6, -6, -6, 0, -6, -2, 0, 6, -6, -2], [-14, 4, 5, -3, -7, 17, 8, -1, 3, 2, -1, 2], [-4, -8, 0, 0, 2, -7, -2, 2, 2, 0, 2, 4], [4, 2, 2, -10, -4, 0, 0, 6, -6, 4, -4, 0], [4, 3, -6, -8, -3, 0, 5, 4, -5, 3, -3, 3], [22, 9, 7, -1, -5, -21, 2, 0, 1, -2, -3, 6], [-18, 8, -1, 6, 5, 0, 2, 1, 1, -5, 5, 0], [30, -2, 1, 4, 7, -23, 3, 0, -4, -8, -4, -6], [-6, -14, -5, 2, 5, 0, -8, -9, -3, -5, 5, 2], [0, -10, 0, 2, 8, -5, -2, 8, 0, 2, 0, 0], [-1, -1, 0, -1, 0, 22, 2, 1, 1, -1, 1, 1], [-4, 0, -1, 6, 1, -1, -7, -2, -6, 0, 6, 4], [1, -2, 2, 4, 4, 5, 2, 0, -4, -6, -2, -4], [-24, 0, 6, -4, -6, 0, 4, 2, 2, 6, -6, 0], [2, 6, 0, 2, -1, -31, 5, 0, 4, 2, 4, -2], [26, -1, 1, 0, -2, -29, -11, 4, 0, 1, 1, 4], [6, 14, 0, 4, 1, 7, 11, 2, 8, 4, 8, -6], [-11, 10, 2, 0, 0, 13, 18, -8, 0, -2, -2, 0], [18, 14, 1, 4, 3, 0, -6, 9, 1, -3, 3, -8], [14, 5, -3, -1, 4, 0, 4, 1, -5, -4, 4, 4], [6, -2, -4, 0, 10, 0, -14, 0, -10, -10, 10, -2], [-8, -11, 0, 4, -14, 8, 1, -5, 4, 4, 4, 8], [-23, 4, -4, 2, 2, 11, -14, 0, -2, 2, 4, 8], [-2, -9, 0, -6, -4, 26, 7, 5, -6, -6, -6, 2], [-4, -11, 0, -8, -3, 0, -11, 0, -5, 3, -3, -3], [-5, -9, 0, -7, -7, 28, 1, -1, -7, -7, -7, 5], [30, 13, 13, -2, -8, -25, 3, 2, 2, -5, -7, 8], [19, 15, 8, -10, -16, -20, 17, -4, 10, 8, -2, 9], [0, 10, 0, -7, -4, -3, 14, 0, 3, -7, 3, 0], [13, 0, 2, 2, -8, -17, -2, -2, -2, 6, 8, 6], [-12, -20, 0, 2, -4, 0, -18, -7, -1, 2, -1, 12], [-14, -12, -5, 4, 6, 16, -6, 2, -4, -1, 3, -7], [26, -10, 2, 0, -4, 0, -10, -2, 4, 4, -4, -6], [3, 3, 0, 11, 3, 6, -9, -3, 5, 11, 5, -3], [7, 11, 0, -9, 1, -10, 13, 5, -7, -9, -7, -7], [2, 0, 0, 9, 6, 3, 0, 4, 9, 9, 9, -2], [3, 1, 0, 3, -6, -30, 10, 3, 1, 3, 1, -3], [-22, -4, 4, -6, 0, 0, -2, 0, -6, 0, 0, 2], [-8, -9, 0, 6, 6, -16, -21, -7, 6, 6, 6, 8], [16, 2, 7, 4, 1, -11, -7, 4, -4, -8, -4, 2], [8, 11, 21, -7, -18, 11, 24, 1, 7, -3, -10, 2], [-6, 12, -2, 6, 2, 0, 10, 2, 4, -2, 2, 2], [-13, -8, 0, 9, 3, 14, -10, 1, -9, -3, 6, -1], [7, 7, 10, -6, -10, -2, 3, 4, 6, 0, -6, 5], [-6, -6, 0, 10, -17, -7, 1, -8, 8, 10, 8, 6], [2, -12, -6, 4, 8, 0, -12, -8, -4, -8, 8, 0], [-2, 4, 0, -12, 0, 2, 12, 3, -1, -12, -1, 2], [0, 2, 0, 4, 4, 5, -2, 0, 6, 4, 6, 0], [-19, -13, -12, 6, 6, 8, -17, 0, -6, 6, 12, -1], [18, -15, -1, 2, 2, -7, 5, 2, -2, -1, 1, -12], [-4, 6, 0, 7, -5, -8, -11, -13, 8, 7, 8, 4], [-6, 9, 5, 7, 0, 0, 8, 1, 7, 0, 0, 0], [2, 9, 0, -4, 8, 6, 5, 3, 4, -4, 4, -2], [4, 4, 6, 12, -2, 0, 4, -2, 14, 2, -2, -4], [13, -2, -6, -4, 4, -18, -7, 3, 4, 2, -2, -1], [-22, -8, -8, -12, -4, 0, 0, 0, -8, 4, -4, 4], [2, 4, 0, 5, -2, -5, 4, 0, 5, 5, 5, -2], [-20, -21, -17, -5, 6, 0, -16, -9, -11, -6, 6, 2], [7, -1, 0, 1, 8, 20, 2, 9, -5, 1, -5, -7], [-8, 15, 1, -2, 6, 11, 21, -8, 2, -7, -9, -2], [-20, 8, 0, 0, 4, 0, 0, 4, -4, -4, 4, 0], [-4, -9, 0, 5, -4, -27, -5, -2, 2, 5, 2, 4], [3, 5, 0, 12, 0, -4, -4, -3, 8, 12, 8, -3], [26, -2, 6, -4, 0, 0, -6, 2, -4, 0, 0, -2], [1, -3, 0, 11, 5, -6, -15, -3, 3, 11, 3, -1], [-6, 1, 0, 8, -1, 0, 7, -4, 9, 1, -1, 1], [4, 0, 0, 8, 0, 10, 10, 7, 7, 8, 7, -4], [2, -3, -9, 7, 12, -11, -14, -1, -7, -3, 4, 0], [2, 6, -5, -6, 3, -7, 5, -2, 6, 2, -4, 0], [-17, -7, -4, -2, 6, 0, 3, -7, -8, -6, 6, 9], [30, 1, -1, -3, 2, 0, -12, 5, -5, -2, 2, -6], [-9, 3, 0, 2, 0, 12, 17, -8, -2, 0, 2, -3], [10, 6, 8, -8, -6, 0, 20, 2, -2, 6, -6, 10], [-4, -7, -2, 4, 7, 0, -19, -2, -3, -7, 7, -7], [35, 0, -2, 3, 11, -32, -2, 1, -3, -9, -6, -5], [-49, 4, -2, 0, 0, 44, 1, -3, 0, 2, 2, 3], [-2, -6, 0, -4, 9, -13, -9, 2, -4, -4, -4, 2], [8, -5, 3, -1, -6, 0, 6, -3, 5, 6, -6, 2], [-8, -10, 0, 9, -1, -6, -19, -9, 6, 9, 6, 8], [-10, -3, -5, -1, 5, 9, 0, 0, 1, 0, -1, -4], [32, -1, 1, 1, 0, 0, -12, 3, 1, 0, 0, -8], [-3, -2, 6, 10, 6, 9, -12, 4, -10, -12, -2, 0], [-2, -3, 0, -1, 4, 5, -5, 0, 0, -1, 0, 2], [5, 5, 0, -11, 9, 8, 9, 9, -7, -11, -7, -5], [-3, 1, 0, 1, 3, -40, -13, -7, 1, 1, 1, 3], [2, 10, 10, -2, 0, 0, -8, 10, -2, 0, 0, -8], [13, 4, 0, -2, -6, -17, 6, -4, 2, 6, 4, 4], [4, 4, 0, 3, -8, 23, 12, 2, 1, 3, 1, -4], [3, 15, 0, -7, 4, 0, -2, -5, -3, -7, -3, -3], [20, 12, 0, -12, -4, 0, 16, 8, -8, 4, -4, 8], [-7, -7, 0, 1, -12, 7, -4, -8, 0, 1, 0, 7], [-9, 8, 4, -6, 5, 18, 11, 2, 6, -9, -15, -5], [-8, 3, 4, -4, -5, 0, 9, 2, 1, 5, -5, 3], [16, 17, 3, -1, 7, -15, 10, -4, 1, -10, -11, 2], [0, 4, 0, 7, -12, 23, 4, -6, 5, 7, 5, 0], [2, 6, 0, 1, 2, -1, -6, -4, -1, 1, -1, -2], [-3, -20, -10, 5, 3, 0, -10, 1, -5, 7, 12, -7], [-8, 10, 9, 2, -9, 11, 13, -6, -2, 0, 2, 6], [-2, 9, 0, -12, -6, 8, 15, -1, 0, -12, 0, 2], [-31, -5, -9, -4, 3, 0, -3, -2, -7, -3, 3, 3], [-4, 11, 1, 11, 6, 0, -4, 3, 5, -6, 6, -6], [-6, -6, 0, 4, 1, 15, -5, -2, 8, 4, 8, 6], [-20, -9, -3, -2, 2, 23, -3, 4, 2, 1, -1, -6], [0, 18, -9, -3, 1, 0, -1, 13, -4, -1, 1, -5], [-19, 10, -6, 8, 1, 2, -1, -12, -8, 5, 13, 11], [11, -5, -2, -10, 2, 0, -11, 3, -12, -2, 2, -1], [-4, -2, 0, 0, 8, -15, -14, -4, 2, 0, 2, 4], [-6, -9, 0, 1, -4, 33, -3, -2, 2, 1, 2, 6], [-12, 16, 9, -8, -9, 17, 21, -4, 8, 0, -8, 4], [6, -6, 0, -8, 7, 27, 9, 14, -12, -8, -12, -6], [-11, 16, 0, -11, -11, 2, 16, -7, 11, 11, 0, 9], [-1, -14, -18, 2, 9, -16, -23, 2, -2, 9, 11, -1], [-26, -2, 0, -6, 8, 0, 0, -2, -14, -8, 8, 8], [-2, 8, -3, 5, 13, 1, 4, -5, -5, -10, -5, -2], [0, 8, 0, -4, 2, 13, 2, -2, 2, -4, 2, 0], [2, -3, 1, 3, 8, 0, 4, -7, -5, -8, 8, 8], [-28, 2, 2, 6, 0, 0, 4, -2, 6, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1152_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1152_2_i_i();".
// 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_1152_2_i_i(:prec:=12)
    chi := MakeCharacter_1152_i();
    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_1152_2_i_i();".
// 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_1152_2_i_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1152_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2304, -4224, 9424, -1720, 4665, 1674, 2928, 948, 465, 60, 24, 2, 1]>,<7,R![394384, 324048, 294516, 117452, 67353, 21192, 10164, 2400, 861, 152, 48, 6, 1]>],Snew);
    return Vf;
end function;