
// Make newform 1152.2.i.g 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_g();".
// 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_g();".
// 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 := [243, -162, 27, 0, 27, -36, 9, 0, 1, -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, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-567, 27, 0, 0, -63, 45, 12, 4, 1, 1], [81, -270, 162, 81, 9, -18, 15, -4, -1, -1], [81, 297, 0, 0, 9, 9, -12, -4, -1, -1], [162, -27, 0, -27, 36, -9, 0, -1, 2, -1], [-567, -297, -648, 162, -63, 45, -24, 28, 7, 7], [1053, -189, 0, 0, 117, -99, -12, -10, 11, -7], [324, -108, -81, 81, 36, -45, 6, 11, -4, -4], [2349, -459, -324, 162, 369, -171, -60, 22, 19, -17]];
    Rf_basisdens := [1, 1, 486, 243, 162, 81, 486, 486, 243, 486];
    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], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 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, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [-1, -1, 1, 1, 1, 1, 0, 0, 1, 0], [0, -1, 0, 0, 1, 1, 1, 0, 0, -1], [0, 0, -1, 0, 1, -1, 0, 0, 0, 0], [-1, 1, 0, 1, 0, 0, 0, -1, 0, 1], [2, 0, 0, 0, -1, 1, 0, 2, 0, -1], [0, 0, 1, 0, 0, -1, -1, 0, 1, 0], [1, -1, -1, 2, 2, 2, 2, 1, 2, -2], [0, -2, -1, 0, 2, 1, 1, 2, 3, 0], [3, -3, 0, -1, 1, -1, 0, 1, 0, -2], [0, 4, -2, 0, -1, -3, 0, -4, -1, 0], [-2, -1, 2, 2, 1, 1, 2, 0, 2, -2], [-1, 1, 1, -1, 1, 1, 2, 2, -1, -2], [-1, 3, 0, -1, -1, 3, 0, 1, 0, 0], [0, -3, -2, 0, 2, 3, 2, 3, 4, 0], [-3, -3, 3, 2, 6, 6, 2, 3, 2, -2], [0, 2, -4, 0, -2, -1, -1, -2, 0, 0], [0, 3, 0, 2, -2, 4, 0, 3, 0, -1], [-3, -1, 0, -1, 0, -2, 0, -1, 0, 1], [-1, -1, 1, -3, 3, 3, 3, 2, -3, -3], [2, -2, -2, 0, 1, 1, -1, -1, 0, 1], [3, 5, 0, 1, -3, 5, 0, 3, 0, 0], [4, -1, -4, -1, 4, 4, 0, 3, -1, 0], [-2, 6, 2, 1, -2, -2, 0, 4, 1, 0], [0, -3, -3, 0, 2, -1, -2, 3, -3, 0], [6, -4, 0, -2, -1, -3, 0, 2, 0, -1], [3, 1, 0, -1, 3, -1, 0, -5, 0, 2], [0, -3, 5, 0, -1, 3, -1, 3, 4, 0], [4, -4, 0, 0, 4, -4, 0, -4, 0, 0], [0, -4, 6, 0, -1, 2, -3, 4, 0, 0], [0, 7, 0, -1, -2, -2, 0, 5, -1, 0], [0, 3, -8, 0, -2, 3, 4, -3, 0, 0], [0, 6, 5, 0, -3, -3, 0, -6, -2, 0], [-3, 3, 3, 1, -1, -1, -3, 2, 1, 3], [0, -2, -6, 0, 6, -4, 0, 2, 3, 0], [2, 4, 0, 6, -2, 2, 0, 0, 0, 2], [0, 7, 1, 0, -4, -3, 0, -7, -3, 0], [-4, 0, 4, -1, -4, -4, -6, -4, -1, 6], [-2, 0, 0, 0, -2, -2, 0, 0, 0, 2], [3, 1, 0, 3, -1, -3, 0, -3, 0, 4], [-5, 3, 5, 1, -3, -3, -2, 0, 1, 2], [0, -4, 8, 0, 5, -1, 0, 4, 9, 0], [0, 6, 0, 2, 0, 6, 0, 0, 0, 0], [8, -3, 0, -2, 2, -4, 0, -3, 0, 1], [0, 1, -2, 0, -5, 2, -2, -1, 0, 0], [-9, -3, 9, 1, 1, 1, -4, -2, 1, 4], [10, 1, -10, -4, -3, -3, -1, -2, -4, 1], [0, -2, -8, 0, 0, 4, 2, 2, 1, 0], [-4, 4, 0, -2, -1, 5, 0, 2, 0, -1], [0, 2, -7, 0, 0, -1, 1, -2, -3, 0], [5, -2, -5, 4, 8, 8, 2, 6, 4, -2], [-4, 2, 0, 4, 3, -3, 0, -8, 0, 5], [0, 0, -9, 0, 4, -2, 2, 0, -2, 0], [3, 7, -3, -3, -9, -9, -3, -2, -3, 3], [3, 3, 0, -1, -5, -3, 0, -1, 0, 6], [10, 4, 0, 4, -2, 2, 0, 0, 0, 2], [-6, 0, 6, -7, -6, -6, -2, -6, -7, 2], [-6, 2, 6, 5, 2, 2, 1, 4, 5, -1], [0, -2, 0, 0, -3, 0, -5, 2, -4, 0], [0, 6, -8, 0, -2, -4, 0, -6, 1, 0], [10, 0, 0, 2, 1, -1, 0, -2, 0, 1], [0, -4, 15, 0, 4, -1, -1, 4, -3, 0], [11, 0, -11, -2, -4, -4, 2, -4, -2, -2], [-3, -7, 3, 2, 8, 8, 4, 1, 2, -4], [-2, 2, 2, 4, -3, -3, -2, -1, 4, 2], [0, 2, 7, 0, 4, -2, 4, -2, -6, 0], [0, 0, -10, 0, -4, 5, 1, 0, 0, 0], [0, 6, 0, -1, 2, 2, 6, 8, -1, -6], [2, -5, -2, 1, 2, 2, 4, -3, 1, -4], [-8, 2, 0, -6, -10, 2, 0, 10, 0, 0], [-15, 1, 15, 1, -1, -1, -3, 0, 1, 3], [0, -2, -3, 0, -7, 5, -4, 2, 2, 0], [12, -1, 0, 4, 1, -1, 0, -1, 0, 0], [0, 1, -11, 0, -4, -3, -6, -1, -7, 0], [-6, -6, 6, 1, 6, 6, 6, 0, 1, -6], [11, 3, 0, 7, 1, 1, 0, -3, 0, 2], [0, -2, -2, 0, 1, 5, 4, 2, -5, 0], [0, 2, 7, 0, -4, -4, -6, -2, -2, 0], [0, 1, 12, 0, 5, 0, 6, -1, 4, 0], [-5, 1, 5, 4, -2, -2, 2, -1, 4, -2], [6, -4, 0, 4, 8, -4, 0, -8, 0, 0], [-7, 1, 0, -1, 2, 4, 0, 1, 0, -3], [-11, 7, 11, 5, -1, -1, 5, 6, 5, -5], [-4, 5, 4, 10, 1, 1, -2, 6, 10, 2], [-3, -1, 0, -1, -2, -4, 0, -1, 0, 3], [10, 1, -10, -11, -8, -8, 0, -7, -11, 0], [5, 1, -5, -10, -6, -6, -2, -5, -10, 2], [0, 6, -13, 0, -4, -9, -7, -6, 1, 0], [-14, 0, 0, -12, -3, 3, 0, 6, 0, -3], [-15, -3, 15, 3, -1, -1, 3, -4, 3, -3], [0, -10, 0, 2, 2, -10, 0, -2, 0, 0], [0, -1, -10, 0, -4, 1, -4, 1, 0, 0], [0, -5, -16, 0, 12, -3, 4, 5, 2, 0], [2, -9, 0, -12, 2, -8, 0, -1, 0, -1], [0, -12, 3, 0, 7, 7, 2, 12, 12, 0], [4, 2, 0, 2, 1, -3, 0, -6, 0, 5], [2, -1, 0, 10, 6, 2, 0, -3, 0, -3], [0, -2, 0, -2, 2, 4, 0, 4, 0, -6], [-14, -5, 14, 10, 9, 9, 5, 4, 10, -5], [7, -5, 0, 7, 9, -7, 0, -11, 0, 2], [0, 6, 6, 0, 0, -3, 3, -6, -6, 0], [3, 4, -3, -10, -12, -12, -4, -8, -10, 4], [0, -2, -6, 0, 2, -1, -1, 2, 6, 0], [-13, 1, 0, 9, 2, -2, 0, -5, 0, 3], [-24, 1, 24, 4, 1, 1, -2, 2, 4, 2], [1, -1, 0, -13, -9, -11, 0, -1, 0, 10], [0, -9, -5, 0, 14, 3, 8, 9, 3, 0], [0, -3, 0, 15, 2, 2, 0, -1, 15, 0], [0, -7, 3, 0, -4, 9, -2, 7, -3, 0], [7, 3, 0, 3, -7, 5, 0, 9, 0, -2], [0, -4, 6, 0, 1, -1, -4, 4, -1, 0], [-10, -13, 10, -2, 13, 13, 3, 0, -2, -3], [12, 6, 0, 6, -2, 14, 0, 10, 0, -8], [-3, -10, 3, 4, 4, 4, -4, -6, 4, 4], [0, -4, -4, 0, 4, 5, 5, 4, -4, 0], [-16, 2, 0, 2, -2, 2, 0, 2, 0, 0], [0, -10, -6, 0, -2, 6, -6, 10, 3, 0], [10, 2, -10, 0, 9, 9, 4, 11, 0, -4], [0, 12, -12, 0, -2, -8, 2, -12, -7, 0], [2, -2, -2, -7, 0, 0, 1, -2, -7, -1], [-10, -4, 10, 1, 6, 6, 0, 2, 1, 0], [0, 6, 0, 8, -3, 7, 0, 4, 0, -1], [0, -12, 0, 12, 17, 17, 3, 5, 12, -3], [3, -5, 0, -5, -5, 1, 0, 11, 0, -6], [-2, 4, 2, -7, 0, 0, 6, 4, -7, -6], [20, -12, 0, 0, 6, -10, 0, -4, 0, -2], [-9, 13, 9, -11, -9, -9, -1, 4, -11, 1], [0, 6, -17, 0, -3, -3, 0, -6, -4, 0], [10, -5, 0, -6, 3, -7, 0, -5, 0, 2], [0, 3, 19, 0, -10, 7, 0, -3, -5, 0], [0, 1, 5, 0, -8, 5, -2, -1, 1, 0], [18, 0, 0, -8, -8, 4, 0, 12, 0, -4], [0, -5, 14, 0, 4, 10, 9, 5, -1, 0], [0, 11, 3, 0, -9, -7, -5, -11, -6, 0], [23, -3, 0, -1, -1, -3, 0, 1, 0, 0], [0, 4, 2, 0, -5, -2, -3, -4, 0, 0], [0, 0, 19, 0, 1, 1, 2, 0, 0, 0], [1, -13, 0, 7, 6, -4, 0, 3, 0, -9], [10, -4, 0, 0, 1, -9, 0, -6, 0, 5], [23, -3, -23, 0, 2, 2, 0, -1, 0, 0], [0, 18, -13, 0, -12, -5, 1, -18, -5, 0], [18, 4, 0, 4, 2, 2, 0, -4, 0, 2], [7, 5, 0, -1, -11, 5, 0, 11, 0, 0], [15, 5, -15, -9, -15, -15, 0, -10, -9, 0], [3, 15, -3, -8, -14, -14, -6, 1, -8, 6], [12, -4, -12, 3, 8, 8, 3, 4, 3, -3], [0, -2, -18, 0, 10, -1, 7, 2, 0, 0], [12, -2, 0, 6, -2, 2, 0, 6, 0, -4], [0, -2, -5, 0, -5, 3, -4, 2, -4, 0], [-17, -5, 0, 5, 5, 5, 0, 5, 0, -10], [0, -6, 0, 2, 15, -7, 0, -16, 0, 1], [0, -8, 13, 0, 14, -15, -9, 8, -1, 0], [-8, 15, 8, -12, -11, -11, -4, 4, -12, 4], [-19, 3, 19, -11, -5, -5, -5, -2, -11, 5], [-14, 20, 0, 4, -10, 14, 0, 4, 0, 6], [-6, -2, 6, 9, 4, 4, -3, 2, 9, 3], [-3, -11, 0, -1, 11, 1, 0, 1, 0, -12], [0, 8, -14, 0, -4, 0, 4, -8, -9, 0], [28, 5, -28, 2, -11, -11, -3, -6, 2, 3], [1, -7, 0, -5, 2, -12, 0, -7, 0, 5], [0, -12, -3, 0, 2, 5, -5, 12, -5, 0], [6, 13, 0, -6, -6, 2, 0, -5, 0, 11], [0, 6, -5, 0, 10, -8, 8, -6, 6, 0], [17, -3, -17, -3, -1, -1, -9, -4, -3, 9], [-16, 2, 0, -10, -4, -8, 0, -6, 0, 10], [-6, -4, 6, 9, 2, 2, -6, -2, 9, 6], [-3, 3, 0, 7, -1, 9, 0, 7, 0, -6], [0, -2, 25, 0, -1, 1, -2, 2, -6, 0], [-14, 5, 14, -10, 1, 1, 8, 6, -10, -8], [0, 4, 7, 0, -7, 1, -2, -4, -8, 0], [0, 6, 5, 0, -16, -1, -11, -6, -5, 0], [2, 7, -2, -3, -4, -4, 4, 3, -3, -4], [0, 15, 1, 0, 4, -7, 12, -15, -5, 0], [0, -13, -1, 0, -7, 9, -11, 13, 6, 0], [10, -5, -10, -10, 1, 1, 2, -4, -10, -2], [-9, -5, 0, 7, 13, -11, 0, -19, 0, 6], [6, 2, 0, 2, -4, -8, 0, -6, 0, 10], [8, 4, -8, 3, -6, -6, 6, -2, 3, -6], [5, -15, -5, 7, 5, 5, 2, -10, 7, -2], [14, -5, -14, -6, 1, 1, -4, -4, -6, 4], [0, 6, 4, 0, -4, -8, -6, -6, 5, 0], [-13, 1, 0, -11, -2, -6, 0, -5, 0, 7], [0, -9, -5, 0, 6, -1, -4, 9, -3, 0], [31, -9, -31, 0, -4, -4, 4, -13, 0, -4], [10, -4, 0, 0, -10, -10, 0, 4, 0, 6], [18, 10, -18, -14, -19, -19, -9, -9, -14, 9], [0, 0, 1, 0, 4, -14, -10, 0, -4, 0], [22, 6, 0, 2, -8, 0, 0, 2, 0, 6], [-11, -3, 0, 3, 4, -6, 0, -7, 0, 3], [-2, 1, 2, -2, 9, 9, -3, 10, -2, 3], [0, 10, 0, 12, -7, 11, 0, 8, 0, -1], [-24, 2, 24, 3, -2, -2, -10, 0, 3, 10], [-10, 4, 0, 8, -7, 3, 0, 6, 0, 1], [0, -10, 13, 0, 6, 4, 0, 10, 16, 0], [0, -10, -12, 0, -1, 5, -6, 10, 1, 0], [24, 14, -24, -5, -18, -18, -8, -4, -5, 8], [-9, 6, 9, 2, 6, 6, 8, 12, 2, -8], [-24, 8, 0, 12, -8, 8, 0, 8, 0, 0], [0, 6, 6, 0, 4, -6, 4, -6, 3, 0], [3, 5, -3, -9, -3, -3, 2, 2, -9, -2], [0, -4, -12, 0, 8, 2, 6, 4, 1, 0], [-5, 6, 5, 6, -6, -6, -4, 0, 6, 4], [10, -4, 0, -20, -3, -13, 0, -6, 0, 9], [-21, 23, 0, -5, -13, 13, 0, 3, 0, 10], [-4, -2, 0, -10, -6, -6, 0, 2, 0, 4], [0, 5, 22, 0, -4, -3, -2, -5, -6, 0], [3, -6, -3, -6, -8, -8, -8, -14, -6, 8], [0, 3, 14, 0, -8, 9, 4, -3, -2, 0], [-1, -3, 0, -11, -8, -16, 0, -5, 0, 13], [0, -4, 13, 0, -1, 9, 4, 4, 6, 0], [-23, 5, 23, 9, 1, 1, -10, 6, 9, 10], [-8, 0, 8, 2, -15, -15, -6, -15, 2, 6], [0, 13, -5, 0, -16, 5, 2, -13, 3, 0], [1, 2, -1, -4, 6, 6, 6, 8, -4, -6], [0, -6, -1, 0, 2, 13, 9, 6, 7, 0], [-6, 17, 6, -15, -18, -18, 2, -1, -15, -2], [-4, -2, 0, 6, 24, 0, 0, -22, 0, -2], [0, 8, 19, 0, -19, 3, -8, -8, -8, 0], [0, -6, 21, 0, 3, 1, -2, 6, -2, 0], [0, 17, 9, 0, 4, -19, 2, -17, -5, 0], [0, 14, 6, 0, -7, -15, -8, -14, -13, 0], [2, -8, 0, 0, 4, 12, 0, 16, 0, -20], [0, -5, 2, 0, 10, -5, 0, 5, -12, 0], [-6, -2, 6, 13, 22, 22, 6, 20, 13, -6], [-31, -1, 31, 0, 4, 4, -3, 3, 0, 3], [-22, 0, 0, -16, -4, -4, 0, 0, 0, 4], [11, -21, -11, 3, 15, 15, 3, -6, 3, -3], [-24, -4, 24, 8, 1, 1, 2, -3, 8, -2], [0, -14, 10, 0, 4, 10, 0, 14, 7, 0], [-26, -12, 0, -2, 6, -10, 0, -4, 0, -2], [0, -8, 31, 0, 8, -9, -9, 8, 3, 0], [0, 4, -1, 0, 10, -10, 4, -4, -14, 0], [2, 6, -2, 6, -15, -15, -6, -9, 6, 6], [-3, 5, 3, -1, 5, 5, 6, 10, -1, -6], [0, 5, 0, 0, -10, 20, 15, -5, 1, 0], [-33, -1, 0, 11, 1, 7, 0, 7, 0, -8], [0, -11, 22, 0, 6, 7, 2, 11, 0, 0], [0, 8, 0, -8, -12, 16, 0, 20, 0, -8], [6, -5, -6, -14, -15, -15, -4, -20, -14, 4], [-30, 12, 0, 8, -2, 6, 0, -4, 0, 6], [0, -3, 17, 0, 8, 1, 6, 3, -9, 0], [-9, -9, 0, -5, -5, 3, 0, 17, 0, -12], [0, 8, -14, 0, -13, 7, 2, -8, -5, 0], [-1, -11, 1, 9, 21, 21, 22, 10, 9, -22], [20, -8, 0, -4, 0, -16, 0, -8, 0, 8], [0, -2, -20, 0, 10, -9, -1, 2, -6, 0], [0, -17, 40, 0, 9, 12, 4, 17, 8, 0], [12, -2, 0, 22, 12, 12, 0, 2, 0, -14], [0, -6, -17, 0, 7, 1, 2, 6, -12, 0], [17, 7, 0, -13, 0, 2, 0, -5, 0, 5], [0, 4, 39, 0, -6, 1, -1, -4, -1, 0], [-7, -9, 7, 0, 20, 20, 3, 11, 0, -3], [-32, 2, 32, 7, 4, 4, 10, 6, 7, -10], [-2, -7, 0, 2, 21, -1, 0, -15, 0, -6], [-9, 1, 0, -9, -9, 3, 0, 11, 0, -2], [24, 8, -24, 6, 1, 1, 6, 9, 6, -6], [40, 8, 0, 2, -2, 4, 0, -2, 0, 4], [-5, -7, 0, -7, 4, -20, 0, -17, 0, 13], [-3, -17, 3, 11, 19, 19, 9, 2, 11, -9], [-16, -9, 16, 10, 5, 5, 6, -4, 10, -6], [0, 4, 4, 0, 20, -18, 6, -4, 13, 0], [-17, -3, 0, 11, 17, -5, 0, -19, 0, 2], [0, 15, 10, 0, -2, -18, -5, -15, -5, 0], [2, -11, -2, -2, 13, 13, 19, 2, -2, -19], [-9, -3, 0, -5, -1, 13, 0, 17, 0, -16], [33, 12, -33, 8, -10, -10, -8, 2, 8, 8], [0, -4, 4, 0, -18, 7, -15, 4, 0, 0], [0, 4, -51, 0, 7, -5, 6, -4, -2, 0], [0, -12, 3, 0, 7, 7, 2, 12, 24, 0], [0, 18, 0, -2, -16, 16, 0, 14, 0, 2], [-1, 18, 1, -6, -4, -4, 0, 14, -6, 0], [0, -7, -3, 0, 8, -3, -2, 7, 21, 0], [0, -12, 0, 0, 3, -11, -20, 12, -7, 0], [-16, 4, 0, -12, -4, 4, 0, 4, 0, 0], [0, -2, -28, 0, 1, 0, -1, 2, -12, 0], [1, 3, -1, 6, -14, -14, -8, -11, 6, 8], [25, 5, 0, -9, -4, -6, 0, -7, 0, 11], [12, -5, -12, 22, 21, 21, 11, 16, 22, -11], [0, 13, 9, 0, -12, -11, -10, -13, -3, 0], [0, -13, 31, 0, 4, 17, 8, 13, 1, 0], [5, 1, -5, 21, 11, 11, -1, 12, 21, 1], [22, 0, -22, -3, 4, 4, -6, 4, -3, 6], [0, 29, -26, 0, -18, -13, -2, -29, -22, 0], [-16, 3, 0, -18, -4, 6, 0, 7, 0, -3], [11, -9, 0, 1, 5, -15, 0, -11, 0, 6], [0, -2, 40, 0, 14, -8, 4, 2, 13, 0], [23, 5, -23, 5, 9, 9, 10, 14, 5, -10], [-28, 12, 0, 2, -1, 13, 0, 2, 0, -1], [5, -9, -5, 10, 14, 14, 8, 5, 10, -8], [4, -17, 0, 0, -7, -13, 0, 11, 0, -4], [0, 12, 7, 0, 6, 2, 20, -12, -6, 0], [0, 2, 24, 0, 8, -7, 3, -2, -18, 0], [16, -16, -16, 15, 28, 28, 12, 12, 15, -12], [0, 22, -14, 0, -10, -4, 8, -22, -9, 0], [29, 3, -29, -13, -9, -9, -11, -6, -13, 11], [23, -3, -23, 4, 4, 4, -4, 1, 4, 4], [-24, 10, 0, -4, 0, 8, 0, -2, 0, 2], [22, 6, -22, -6, -3, -3, 3, 3, -6, -3], [0, 4, -7, 0, 0, 6, 10, -4, 4, 0], [-1, -5, 0, 19, 23, 5, 0, -13, 0, -10], [-2, 2, 0, 14, 8, 4, 0, -6, 0, -2]];
    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_g();".
// 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_g(:prec:=10)
    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_g();".
// 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_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1152_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![36, 162, 717, 198, 328, -18, 117, -4, 12, 0, 1]>,<7,R![4, -2, 33, 48, 240, 138, 129, 0, 24, 4, 1]>],Snew);
    return Vf;
end function;