
// Make newform 1456.2.cc.c in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1456_cc();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1456_cc_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_1456_2_cc_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_1456_2_cc_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 := [64, 0, -80, -16, 60, 4, -30, 2, 15, -2, -5, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -80, -16, 60, 4, -30, 2, 15, -2, -5, 0, 1], [96, -48, 16, -4, -28, -6, 30, -13, -6, 7, 4, -3], [-64, -64, 24, 84, 0, -42, 4, 21, -8, -11, 2, 3], [-64, -56, 68, 60, -38, -26, 23, 7, -13, -3, 3, 1], [64, -32, 48, -20, -68, 10, 42, -23, -10, 13, 4, -5], [-32, 16, 120, -68, -88, 50, 44, -39, -8, 17, 2, -5], [-32, 24, 52, -28, -34, 20, 13, -13, -3, 5, 1, -1], [-80, -16, 108, 8, -70, 12, 35, -13, -13, 5, 3, -1], [-64, 96, 96, -92, -68, 66, 18, -37, -2, 15, 0, -3], [-160, 16, 192, -20, -116, 34, 50, -27, -18, 9, 4, -1]];
    Rf_basisdens := [1, 1, 32, 32, 32, 16, 32, 32, 8, 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_1456_cc();"
function MakeCharacter_1456_cc()
    N := 1456;
    order := 6;
    char_gens := [911, 1093, 1249, 561];
    v := [6, 6, 6, 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_1456_cc_Hecke();"
function MakeCharacter_1456_cc_Hecke(Kf)
    N := 1456;
    order := 6;
    char_gens := [911, 1093, 1249, 561];
    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], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 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, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, -1, 0, -1, 0, -1, 0, 0, 0, -1], [0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0], [0, 1, -1, 2, 0, 1, 1, 0, -1, 0, 0, -1], [1, 0, -1, 1, 2, 0, 2, 1, -2, -1, -1, 2], [-1, 1, 0, 0, 0, -1, 1, 0, 0, 1, 0, -1], [0, 1, 0, 2, 0, 2, -1, 0, 2, 0, 0, -2], [0, -2, -2, -2, 0, -4, 2, 0, -4, 2, 0, 2], [0, -1, 0, 2, -1, 0, 1, -2, 0, 0, 1, 2], [-1, -2, -2, 0, 0, -1, 2, -3, -1, 2, 2, 0], [-4, 0, -3, 2, 3, 1, 0, 0, -1, 2, 3, -1], [4, 0, 1, 2, -5, 1, 0, 0, -1, -2, -1, -1], [1, 4, 3, 2, 1, 2, 1, -1, -2, -1, -3, 2], [3, -2, -2, 2, 0, -1, 2, -3, -3, -6, 2, 2], [-3, -2, 2, -2, 0, 0, -2, 0, -2, 0, 2, 2], [-1, 2, -2, -3, 1, -3, 0, 1, -3, -1, -2, 3], [2, 1, 0, -3, -1, 2, 1, 1, 3, -2, 0, 2], [2, 2, -1, -4, 5, -1, 2, 0, 1, -1, -1, 3], [2, 0, -2, -2, 0, 0, 2, 0, -2, 2, -2, 0], [2, 1, 1, 1, 0, 1, 1, 5, 0, -4, 1, 1], [4, 0, 0, 2, 0, 0, -4, 0, 2, 0, 4, -2], [1, -2, -2, -2, 0, 1, 0, 5, 3, -2, 0, -2], [0, 0, 3, 2, 6, 2, 0, 0, -2, 0, -3, 0], [1, -1, -4, -3, 5, -1, 5, 5, -3, 1, -4, 1], [0, -6, -8, 0, -3, 1, 6, -6, 1, 2, -2, -1], [3, -2, 2, -6, 2, -3, 0, 1, -3, 0, 0, 6], [0, 1, 3, 2, -4, -3, -1, -8, -3, -6, 2, 5], [1, 3, 3, 1, 0, 1, 3, -3, 0, -2, 3, 1], [1, -4, -2, 0, -2, -2, -2, 2, 0, -1, 2, -2], [0, -5, -4, -2, -1, 0, 5, -2, 0, 1, 1, -2], [-1, -1, 1, 6, -2, 5, 3, -1, 1, 0, -3, -6], [4, 6, -3, 0, -5, 4, -3, -5, 0, 4, -3, -4], [1, -3, -4, 5, -5, 3, 1, 5, -5, -1, 4, 3], [0, 2, -3, -2, 1, 0, 1, 1, -2, 0, -3, 0], [-3, 11, 11, -1, 0, 1, -3, -5, 2, 6, -3, -1], [-1, -4, 4, -3, -12, -4, -2, -6, 1, 0, 2, 3], [0, 2, 4, 8, -4, 8, -6, -4, 8, 0, 4, -8], [6, 2, 4, -3, -3, 0, 2, 0, 0, -3, -6, 3], [5, -2, -2, -1, 3, -1, 0, -3, 1, -5, 2, -1], [0, 2, 5, 2, -1, -3, -2, -2, -3, 7, 3, 5], [-6, -1, 1, -7, -2, -5, 5, -1, -2, 0, -5, 7], [-4, -7, -5, -3, 2, -2, -2, -2, 3, 4, 5, -2], [12, -4, 1, 6, 3, 1, -4, 0, -1, -6, 3, -5], [-4, -3, -5, -2, 2, -5, -3, 0, 5, 2, 8, -3], [-3, -12, -6, -5, -5, -5, -6, 5, 5, 3, 6, -5], [0, 0, -5, 2, -1, 3, 0, -2, 3, -3, -5, -1], [2, 2, 5, -3, 3, -4, 2, 0, 4, -1, -7, -1], [3, 0, -2, 3, 11, 1, 2, 11, 3, 3, -2, -1], [-10, 6, 6, 2, 0, -2, -2, -2, -4, 20, -2, 2], [6, 0, 0, 11, 0, 11, -5, 0, 0, 0, 5, -11], [-5, 3, 3, -5, 0, -5, 3, 3, 0, 10, 3, -5], [-1, 2, -4, -4, 10, -9, 2, 10, -4, -1, -4, 9], [-2, -2, 0, 0, -8, -2, -2, 8, 0, 2, 0, -2], [0, 8, 6, 3, 2, 5, -8, 4, 5, 9, -2, -2], [0, -2, -5, 4, -7, 5, 2, -14, 5, -3, -3, -1], [-11, -7, -6, 5, -3, -7, -1, 3, -5, 11, 6, -7], [14, -4, -2, 1, 5, 2, -4, 0, -2, -7, 6, 1], [7, 8, 6, 6, 0, 6, 2, 0, -6, -7, -6, 6], [-8, -1, -1, -5, 0, -3, -1, 5, 2, 16, -1, -5], [0, -6, 0, 2, -2, -2, 6, -4, -2, 10, 6, 4], [0, -1, 10, 7, -5, 0, -9, -5, 7, 0, 10, 0], [-5, -5, -5, -2, 0, -3, 3, 3, -1, 10, 3, -2], [5, 3, -3, 10, 2, 11, 3, 1, -1, 0, -3, -10], [10, 8, -8, 4, -8, 6, 2, -4, -2, 0, -2, -4], [3, -2, -2, 2, 0, 4, 0, -14, 2, -6, 0, 2], [-2, 3, -3, -7, 6, -2, 0, 6, -7, -2, -3, 2], [-8, -3, 3, -4, -2, -10, -2, -1, 6, 0, 2, 4], [10, 2, -2, 10, 0, -2, 4, 0, -10, -10, 2, -2], [-2, -4, 6, -5, -7, 0, -4, 0, 0, 1, -2, 5], [0, 0, 12, -6, -7, -3, 0, 0, 3, 0, -12, 3], [-3, -3, -3, 0, 0, -2, -4, 5, -2, 6, -4, 0], [0, 4, 6, 6, -4, -6, -4, -8, -6, 2, 2, 12], [3, -1, -1, 1, -10, 6, 0, 10, -1, -3, 1, 6], [-12, 1, 3, 0, 8, -5, 1, 0, 5, 6, -4, -5], [1, 5, 0, 1, 9, 1, -5, 9, 1, 1, 0, -1], [0, 1, -1, 9, 18, 5, -5, 9, 4, 0, 5, -9], [-4, -10, 10, -6, -8, -14, 0, -8, -6, -4, 10, 14], [8, -1, 0, 0, -1, 4, -1, 0, -4, -4, 1, 4], [8, -2, 8, 7, 11, 12, -6, 11, 7, 8, 8, -12], [0, -8, -8, -3, -5, 2, 8, -10, 2, 17, 0, -5], [2, 9, 9, 3, 0, 1, -3, 13, -2, -4, -3, 3], [22, 5, 2, 6, 5, 10, 5, 0, -10, -11, -7, 4], [9, -13, -6, -2, -4, -9, -7, 4, 2, -9, 6, -9], [0, 4, 15, 9, -3, 12, -4, -6, 12, -1, 11, -3], [16, 8, -8, 8, -8, 8, 12, -4, 0, 0, -12, -8], [6, 0, -6, 0, -18, 4, 6, -18, 0, 6, -6, -4], [8, 16, -1, 6, 1, 3, 16, 0, -3, -4, -15, -3], [6, 6, -6, 7, 1, 12, 0, 1, 7, 6, -6, -12], [1, -1, -1, 5, 0, 5, 11, 1, 0, -2, 11, 5], [11, -4, 4, -4, -2, 1, -4, -1, -5, 0, 4, 4], [-12, 6, -1, 6, 10, -4, 6, 0, 4, 6, -5, -10], [-9, 6, -5, 0, -3, 0, -1, -3, 0, -9, -5, 0], [0, 1, 4, 4, 7, -8, -1, 14, -8, -9, 3, 12], [1, 5, 5, 10, 0, 4, 4, -9, -6, -2, 4, 10], [-2, -5, 4, -4, -8, -2, -9, 8, 4, 2, -4, -2], [-2, 4, -5, -3, -4, 1, 4, 0, -1, 1, 1, 4], [6, -5, 5, -5, 2, -3, -3, 1, -2, 0, 3, 5], [0, 12, 14, -2, 4, -2, -12, 8, -2, -4, 2, 0], [0, -7, -7, 4, 0, -2, 2, 15, -6, 0, 2, 4], [-11, 1, -1, 1, -18, 5, -3, -9, -4, 0, 3, -1], [6, 9, -1, -12, 12, -1, 9, 0, 1, -3, -8, 11], [-10, -6, 2, -4, -6, 6, -8, 6, 4, 10, -2, 6], [0, 2, 2, -8, 2, -4, -2, 4, -4, -8, 0, -4], [12, 2, -2, 1, 0, 1, 3, 0, 0, 0, -3, -1], [1, 3, 3, -1, 0, -2, 7, -12, -1, -2, 7, -1], [2, 0, -11, 11, 21, 2, 0, 0, -2, -1, 11, -9], [3, 6, 6, 11, 0, -6, -2, 8, -17, -6, -2, 11], [9, -11, 11, 2, -10, -2, -14, -5, 4, 0, 14, -2], [0, 6, 6, 8, -5, 7, -6, -10, 7, -12, 0, 1], [5, 4, 6, -5, -9, -7, -2, 9, 5, -5, -6, -7], [-10, 3, -1, 0, 6, 3, 3, 0, -3, 5, -2, 3], [-5, -2, 4, -4, 16, -6, -2, 16, -4, -5, 4, 6], [-8, -8, -8, -10, 0, -6, 10, 16, 4, 16, 10, -10], [2, -3, -5, -5, 2, -14, 8, 2, -5, 2, -5, 14], [-1, -11, -13, -1, -4, -4, 2, 4, 1, 1, 13, -4], [-11, -2, 0, -3, 3, -3, 2, 3, -3, -11, 0, 3], [0, 8, 6, 4, -4, 8, -8, -8, 8, -14, -2, -4], [0, 1, -6, -4, -9, -8, -1, -18, -8, 5, -7, 4], [-8, -10, 4, -8, -12, -2, -14, 12, 8, 8, -4, -2], [-16, -10, 1, -8, -7, -13, -10, 0, 13, 8, 9, -5], [0, -4, -4, -8, 4, -16, 4, 8, -16, -1, 0, 8], [2, 7, -7, 6, 4, 6, -5, 2, 0, 0, 5, -6], [-1, 7, -6, 3, 7, 12, -1, 7, 3, -1, -6, -12], [-4, 16, 1, -12, 0, 2, 16, 0, -2, 2, -17, 14], [10, 8, -8, -6, 32, 2, 18, 16, -8, 0, -18, 6], [8, -8, 3, 0, 1, 8, 5, 1, 0, 8, 3, -8], [13, 4, -2, 17, -3, 9, 6, 3, -17, -13, 2, 9], [-3, 4, -4, 4, 26, 9, 6, 13, -5, 0, -6, -4], [3, 20, 20, 5, 0, 12, -4, 2, 7, -6, -4, 5], [12, 4, 2, 6, 10, 0, 4, 0, 0, -6, -6, -6], [-2, -7, 4, 4, 9, 0, -7, 0, 0, 1, 3, -4], [0, -4, -7, -18, -5, -17, 4, -10, -17, 3, -3, -1], [0, 16, 10, 6, -2, 16, -16, -4, 16, -20, -6, -10], [-5, -7, 6, -3, -21, -7, 1, -21, -3, -5, 6, 7], [-10, 2, -4, 1, 13, -8, 2, 0, 8, 5, 2, -9], [7, -2, -8, -5, 5, -9, 10, 5, -5, 7, -8, 9], [-2, -4, 4, 0, 0, 4, 0, 0, 0, -2, 4, -4], [30, -10, -10, 14, 0, 0, 0, 0, -14, -30, 10, 0], [0, -6, 2, -2, -8, -4, 6, -16, -4, -5, 8, 2], [8, 13, 13, 4, 0, 6, -9, -12, 2, -16, -9, 4], [10, 2, -11, 6, 19, -1, 2, 0, 1, -5, 9, -7], [-5, 1, 6, -1, 3, 6, -5, -3, 1, 5, -6, 6], [3, 7, 7, -6, 0, -4, 6, 7, 2, -6, 6, -6], [0, -4, 0, -2, -5, -1, 4, -10, -1, 0, 4, -1], [0, -5, 10, 12, 8, 4, -5, 8, 12, 0, 10, -4], [3, 5, 5, 1, 0, 16, -5, 0, 15, -6, -5, 1], [-3, 2, -2, -11, 0, -8, 14, 0, -3, 0, -14, 11], [6, 10, -10, 10, 24, 0, 2, 12, 10, 0, -2, -10], [4, -8, -8, -6, 0, -4, -8, 28, 2, -8, -8, -6], [-9, 6, 8, 0, 0, -2, -14, 0, 0, -9, 8, 2], [0, 6, 7, -1, -6, 15, -6, -12, 15, -1, 1, -16], [-7, 1, -1, -5, -18, -7, 1, -9, 2, 0, -1, 5], [0, 14, 18, 16, -2, 20, -14, -4, 20, -10, 4, -4], [0, -10, -12, 6, -4, 0, 10, -8, 0, -4, -2, 6], [12, -8, 8, -7, 0, -13, 3, 0, 6, 0, -3, 7], [-20, 16, 6, -4, 2, 2, 10, -2, 4, 20, -6, 2], [-3, -3, 2, 2, 28, -7, 1, 28, 2, -3, 2, 7], [-9, 5, -5, -11, -8, -12, 5, -4, 1, 0, -5, 11], [-7, -4, -4, -6, 0, -19, -6, -7, -13, 14, -6, -6], [-20, 1, -3, -10, -8, 1, 1, 0, -1, 10, 2, 11], [-11, 5, 10, -17, -1, 0, -5, 1, 17, 11, -10, 0], [1, 9, 9, -9, 0, 3, -7, -5, 12, -2, -7, -9], [17, 4, 12, -11, -5, 7, -8, 5, 11, -17, -12, 7], [-2, -6, 0, 6, -18, 6, -6, 0, -6, 1, 6, 0], [9, 1, 1, 4, 0, -7, -5, 9, -11, -18, -5, 4], [0, 1, -11, 0, 2, -7, -1, 4, -7, -12, -12, 7], [13, -27, -16, 2, -10, -1, -11, 10, -2, -13, 16, -1], [1, 5, 5, 12, 0, 20, 6, 1, 8, -2, 6, 12], [21, -3, 3, 5, 8, 8, 9, 4, -3, 0, -9, -5], [-7, -1, -1, -4, 0, -4, -6, -9, 0, 14, -6, -4], [1, -4, 2, 2, 22, 7, 2, 22, 2, 1, 2, -7], [-5, -2, -3, -12, -11, -14, 1, 11, 12, 5, 3, -14], [3, 11, -13, 3, -2, 14, 2, -2, 3, 3, -13, -14], [2, -3, 3, -2, 4, 6, -11, 2, -8, 0, 11, 2], [1, 9, 12, -8, -4, 11, -3, 4, 8, -1, -12, 11], [8, -16, 4, -4, 6, 6, -16, 0, -6, -4, 12, 10], [1, 6, 6, 17, 0, -9, -5, 16, -26, -2, -5, 17], [0, 2, -5, -5, 1, 24, -2, 2, 24, 23, -7, -29], [0, 8, -3, 1, 9, 8, -8, 18, 8, 5, -11, -7], [5, 1, 1, -3, 0, 5, -1, 19, 8, -10, -1, -3], [0, -13, 13, 2, -28, -6, -5, -14, 8, 0, 5, -2], [-33, -8, 8, -8, 4, -16, -6, 2, 8, 0, 6, 8], [10, 3, 3, 0, 0, 10, -3, -4, 10, -20, -3, 0], [12, -5, -5, -24, 6, -13, -5, 0, 13, -6, 10, 11], [5, -9, 0, -8, 0, 3, -9, 0, 8, -5, 0, 3], [-4, 6, -6, 0, 20, 4, 4, 10, -4, 0, -4, 0], [-3, 11, 11, -4, 0, 7, -1, 5, 11, 6, -1, -4], [4, -16, 13, -14, -7, 3, -16, 0, -3, -2, 3, 17], [12, 11, 8, 4, -6, 4, -19, -6, 4, 12, 8, -4], [9, 21, 12, 6, -10, 1, 9, 10, -6, -9, -12, 1], [3, 12, -9, 14, 19, 14, -3, 19, 14, 3, -9, -14], [-2, -2, 2, 9, -12, 3, -21, -6, 6, 0, 21, -9], [16, -18, 3, 20, -30, 2, -18, 0, -2, -8, 15, -18], [-6, 19, 9, 3, -6, 4, 10, 6, -3, 6, -9, 4], [0, 2, 0, 3, 13, 16, -2, 26, 16, 3, -2, -13], [-1, -12, -12, 11, 0, -3, 9, 6, -14, 2, 9, 11], [-35, -9, -1, -21, 4, -10, -8, -4, 21, 35, 1, -10], [4, -3, -3, 11, 0, 7, -1, 5, -4, -8, -1, 11], [3, 6, -6, -2, 2, 3, -10, 1, -5, 0, 10, 2], [15, 1, -13, 5, -12, 6, 12, -12, 5, 15, -13, -6], [4, -1, -6, -14, -4, -4, 5, 4, 14, -4, 6, -4], [26, 12, 5, 11, 12, 15, 12, 0, -15, -13, -17, 4], [-14, -14, 14, -1, -10, -15, -34, -5, 14, 0, 34, 1], [26, -4, -8, 29, 5, 22, -4, 0, -22, -13, 12, -7], [0, 14, 1, -1, 0, 17, -14, 0, 17, 15, -13, -18], [-4, -8, -8, -2, 0, 10, -6, 8, 12, 8, -6, -2], [-3, -21, -16, 15, -7, 6, -5, 7, -15, 3, 16, 6], [2, -13, 17, -8, -10, -1, -13, 0, 1, -1, -4, 7], [-8, -18, -4, 8, -4, -6, -14, 4, -8, 8, 4, -6], [0, 5, 6, -10, 1, -12, -5, 2, -12, 14, 1, 2], [21, 11, -11, -4, 2, 7, 27, 1, -11, 0, -27, 4], [0, 10, 7, 16, 7, 19, -10, 14, 19, -23, -3, -3], [-9, 4, 6, 3, 3, 11, -10, 3, 3, -9, 6, -11], [19, -1, -6, -5, -15, 1, 7, -15, -5, 19, -6, -1], [-1, -9, -9, -5, 0, 2, 11, -16, 7, 2, 11, -5], [-15, -7, 7, 9, -18, 1, -7, -9, 8, 0, 7, -9], [-31, 12, 2, 12, 10, 3, 10, -10, -12, 31, -2, 3], [-26, 13, 0, 12, 1, 4, 13, 0, -4, 13, -13, -8], [-6, 2, 2, -13, 0, -5, 5, 14, 8, 12, 5, -13], [0, -17, -18, -6, -5, 4, 17, -10, 4, 18, -1, -10], [-15, -6, -6, 4, 0, 3, -16, 23, -1, 30, -16, 4], [-4, 13, 13, 11, 0, -3, -5, -5, -14, 8, -5, 11], [11, 16, -23, 10, 25, 10, 7, 25, 10, 11, -23, -10], [0, 8, 13, 10, -12, -14, -8, -24, -14, 2, 5, 24], [0, -7, 7, 8, -18, -4, -16, -9, 12, 0, 16, -8], [0, 4, 13, 9, -8, 21, -4, -16, 21, 17, 9, -12], [-15, -6, 13, -4, -11, 2, -19, 11, 4, 15, -13, 2], [11, 3, -6, -1, 3, 13, 9, -3, 1, -11, 6, 13], [1, -31, -31, 7, 0, -1, 15, -7, -8, -2, 15, 7], [0, -6, 3, 8, -11, 13, 6, -22, 13, 22, 9, -5], [0, -4, -12, -9, 7, 0, 4, 14, 0, 39, -8, -9], [6, -3, 6, 16, 15, 18, -3, 0, -18, -3, -3, 2], [-10, -3, 3, -19, -22, -7, 7, -11, -12, 0, -7, 19], [-6, 10, -10, 2, 0, -8, 22, 0, 10, 0, -22, -2], [29, 2, 2, 6, 0, -1, 8, -1, -7, -58, 8, 6], [5, 9, 10, 15, -7, 21, -19, -7, 15, 5, 10, -21], [-22, 0, 2, -12, -18, 2, 0, 0, -2, 11, -2, 14], [4, -10, -4, -2, 6, -8, -6, -6, 2, -4, 4, -8], [0, -6, 14, -3, -7, -14, 6, -14, -14, 5, 20, 11], [-16, 11, -1, 8, 18, 3, 11, 0, -3, 8, -10, -5], [0, 20, 10, 7, 13, 14, -20, 26, 14, 23, -10, -7], [-1, 23, 18, 1, -7, 21, 5, 7, -1, 1, -18, 21], [-14, 2, 0, 16, 24, 2, 2, 0, -2, 7, -2, -14], [12, -9, 18, -7, 13, 0, -9, 13, -7, 12, 18, 0], [-14, -18, 18, 10, -4, 0, -24, -2, 10, 0, 24, -10], [8, 2, -6, -4, 18, 12, 4, 18, -4, 8, -6, -12], [-36, -6, 9, -6, 12, -6, -6, 0, 6, 18, -3, 0], [-8, -10, -16, -12, 0, -18, 26, 0, -12, -8, -16, 18], [0, -7, -6, -24, -7, -14, 7, -14, -14, 3, 1, -10], [22, 13, -2, -18, 5, -6, 13, 0, 6, -11, -11, 12], [-42, 10, -7, 3, -13, -2, 10, 0, 2, 21, -3, -5], [9, -9, -9, 7, 0, 13, 7, 21, 6, -18, 7, 7], [0, -4, -15, -3, 16, -7, 4, 32, -7, -13, -11, 4], [23, 4, -4, -8, 2, -9, 0, 1, 1, 0, 0, 8], [5, 1, 18, -1, -9, 4, -19, -9, -1, 5, 18, -4], [-2, -7, -5, 7, -28, -4, -2, 28, -7, 2, 5, -4], [6, -12, 6, -11, 29, 2, -12, 0, -2, -3, 6, 13], [11, 10, -10, 4, 10, 9, 4, 5, -5, 0, -4, -4], [-7, 4, -4, -26, 24, -41, 0, 24, -26, -7, -4, 41], [-3, 17, -17, 27, 18, 13, -5, 9, 14, 0, 5, -27], [0, -6, 5, -14, -15, -21, 6, -30, -21, 5, 11, 7], [9, 1, 1, -3, 0, 24, -13, 20, 27, -18, -13, -3], [0, -8, 5, 11, -2, -11, 8, -4, -11, -15, 13, 22], [-28, -5, 10, -1, -21, 0, -5, -21, -1, -28, 10, 0], [-22, -14, -20, 4, -12, -10, 6, 12, -4, 22, 20, -10], [-9, 15, -6, 5, 21, 18, -9, 21, 5, -9, -6, -18], [-1, 26, 26, 3, 0, 8, 6, -10, 5, 2, 6, 3], [3, 15, 18, 19, 23, 8, -33, 23, 19, 3, 18, -8], [-1, -12, -6, 6, 4, -10, -6, -4, -6, 1, 6, -10], [-7, 0, 0, -3, 12, -6, -28, 6, 3, 0, 28, 3], [5, -15, -15, -4, 0, -4, 2, 25, 0, -10, 2, -4], [-4, 8, 15, 14, 5, 15, 8, 0, -15, 2, -23, 1], [-13, -1, 12, -13, 13, -12, -13, -13, 13, 13, -12, -12], [0, 30, 12, 6, 6, 22, 18, -6, -6, 0, -12, 22], [-16, -2, -9, -8, 20, 0, -2, 0, 0, 8, 11, 8], [5, -1, 8, -15, -37, -20, -7, -37, -15, 5, 8, 20], [-12, -7, -7, -21, 0, -3, -9, 19, 18, 24, -9, -21], [-10, -11, 2, -9, -9, -32, 9, -9, -9, -10, 2, 32], [-7, 2, -6, -9, -9, 1, 8, 9, 9, 7, 6, 1], [0, 6, 5, -5, -9, 22, -6, -18, 22, 17, -1, -27], [22, 9, 2, 4, -1, -2, 9, 0, 2, -11, -11, -6], [-14, 11, -11, -18, 24, -6, 1, 12, -12, 0, -1, 18], [20, -4, 11, -2, 13, -5, -4, 0, 5, -10, -7, -3], [-13, -12, -12, 0, 0, -11, 0, 3, -11, 26, 0, 0], [-22, 6, -6, -28, 24, -28, 14, 12, 0, 0, -14, 28], [5, 18, -18, -1, 12, -1, 7, 6, 0, 0, -7, 1], [13, -14, -14, -5, 0, -3, 13, 4, 2, -26, 13, -5], [-20, 6, -10, 6, -24, 6, 4, -24, 6, -20, -10, -6], [14, 32, 20, 17, 3, 20, 12, -3, -17, -14, -20, 20], [0, -28, -37, 0, -3, -5, 28, -6, -5, -21, -9, 5], [15, 1, -8, -5, 13, -11, 9, -13, 5, -15, 8, -11], [8, -6, 1, 2, -37, 3, -6, 0, -3, -4, 5, 1], [-23, -11, 0, -21, -13, -35, 11, -13, -21, -23, 0, 35], [-38, -8, -8, -12, 12, -2, 0, -12, 12, 38, 8, -2], [-19, -2, 2, -7, 48, -12, 6, 24, 5, 0, -6, 7], [36, -3, 3, -4, -48, -2, -1, -24, -2, 0, 1, 4], [0, 12, 12, 1, -14, -1, -12, -28, -1, -23, 0, 2], [0, -22, -5, 2, 4, -16, 22, 8, -16, -2, 17, 18], [4, 24, 17, -8, 6, 4, 24, 0, -4, -2, -41, 12], [-16, -13, -16, 9, -13, -14, 3, 13, -9, 16, 16, -14], [8, -1, -1, 18, 0, -2, -2, 9, -20, -16, -2, 18], [0, 0, -4, 8, 18, 22, 0, 36, 22, -18, -4, -14]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1456_cc_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1456_2_cc_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_1456_2_cc_c(:prec:=12)
    chi := MakeCharacter_1456_cc();
    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_1456_2_cc_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_1456_2_cc_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1456_cc();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![16, 80, 300, 516, 709, 390, 287, 42, 96, 4, 11, 0, 1]>],Snew);
    return Vf;
end function;