
// Make newform 1728.2.r.f in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1728_r();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1728_r_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_1728_2_r_f();".
// 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_1728_2_r_f();".
// 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 := [36, 144, 288, 384, 304, 96, 0, -24, -16, 0, 0, 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], [-96192, -26082, -44964, -51252, -27262, 5062, 3300, 1747, 1148, -650, 300, -275], [275832, 708624, 1654212, 921960, 209952, -31862, -131336, -22648, 8976, -10784, 7708, 839], [15984, 49284, -132080, 98118, 57708, 11380, 14442, -13990, -2220, 2887, -2392, 962], [-5172, -12468, -22776, -27758, -10408, -1364, 1416, 1866, 32, -53, 90, -118], [133704, 364968, 348980, 562962, 219348, 17238, -7530, -44568, -2808, 4065, -3602, 2793], [268416, 645714, 1177668, 1432764, 530594, 65542, -72588, -93803, -2632, 3454, -4740, 6049], [105120, 282372, 400856, 420176, 151968, 12648, -24587, -26378, -684, 840, -1054, 1688], [-1037028, -1622652, -2984520, -3664246, -1340808, -234336, 188112, 267718, -11340, -681, 10866, -15676], [-1369008, -3656232, -5685648, -5346606, -1841148, -103124, 348166, 317744, 1848, -3611, 7150, -19630], [4860, 29640, 53952, 65498, 26820, 2664, -3408, -4304, -192, 159, -222, 284], [-462528, -1237836, -1805344, -1820914, -637476, -40656, 108978, 110866, 1476, -2205, 3388, -6916]];
    Rf_basisdens := [1, 51972, 103944, 51972, 1704, 34648, 51972, 12993, 103944, 103944, 1464, 34648];
    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_1728_r();"
function MakeCharacter_1728_r()
    N := 1728;
    order := 6;
    char_gens := [703, 325, 1217];
    v := [6, 3, 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_1728_r_Hecke();"
function MakeCharacter_1728_r_Hecke(Kf)
    N := 1728;
    order := 6;
    char_gens := [703, 325, 1217];
    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, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [0, 0, -1, 1, 0, -1, 0, -2, 0, -2, 0, -2], [-1, 1, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0], [0, 0, 1, -1, 0, 1, 0, 2, 0, 0, 0, 0], [2, 0, 0, 0, 2, 0, 0, 0, 1, 0, -1, 0], [0, 0, 0, 0, 3, 0, 1, 0, 0, 0, 0, 0], [0, 0, 1, -2, 0, 2, 0, 2, 0, 2, 0, 1], [0, 0, -1, 1, 0, 2, 0, 1, 0, -2, 0, 1], [-2, -1, 0, 0, -4, 0, -2, 0, 0, 0, 0, 0], [0, 0, 1, 3, 0, -1, 0, 0, 0, 1, 0, -1], [1, 1, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0], [0, 0, 1, 2, 0, -1, 0, 0, 0, 2, 0, -2], [0, 0, -1, -4, 0, -1, 0, 2, 0, -2, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0], [0, 0, 0, -1, 0, 1, 0, -1, 0, -1, 0, 1], [0, 0, 1, -3, 0, 2, 0, 3, 0, 2, 0, 1], [4, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0], [-1, 3, 0, 0, -1, 0, 3, 0, 0, 0, 0, 0], [-5, -1, 0, 0, 2, 0, 1, 0, -3, 0, 0, 0], [0, 0, 1, 2, 0, 1, 0, -4, 0, -1, 0, -1], [0, 0, -2, 0, 0, 0, 0, -6, 0, -2, 0, -2], [6, -1, 0, 0, 8, 0, -2, 0, 2, 0, -2, 0], [2, 0, 0, 0, 6, 0, -3, 0, 2, 0, -1, 0], [0, 0, 2, 4, 0, -2, 0, -2, 0, 0, 0, -2], [0, 0, 1, -3, 0, -1, 0, 0, 0, -1, 0, 1], [-13, 2, 0, 0, -12, 0, 2, 0, -1, 0, 2, 0], [0, 0, 1, -2, 0, -1, 0, 0, 0, -1, 0, 1], [6, 2, 0, 0, 3, 0, 1, 0, 0, 0, 0, 0], [0, 0, -2, 2, 0, 1, 0, 2, 0, -1, 0, -1], [8, 2, 0, 0, 4, 0, 1, 0, 0, 0, 1, 0], [0, 0, -4, 0, 0, -2, 0, 2, 0, -2, 0, -4], [0, 0, 3, 2, 0, 3, 0, -4, 0, 3, 0, 3], [-7, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0], [-2, 0, 0, 0, -2, 0, 0, 0, -1, 0, 1, 0], [0, 0, 1, 2, 0, -2, 0, 2, 0, 2, 0, -1], [4, 0, 0, 0, -1, 0, -3, 0, 4, 0, -2, 0], [0, -1, 0, 0, -6, 0, -2, 0, 3, 0, -3, 0], [0, 0, -1, -8, 0, 3, 0, 4, 0, 2, 0, 1], [0, 0, -1, -1, 0, 1, 0, 0, 0, -2, 0, 2], [0, 0, 3, -2, 0, -2, 0, -2, 0, 2, 0, 1], [0, 0, 0, 3, 0, 1, 0, -3, 0, 1, 0, 1], [2, -4, 0, 0, 1, 0, -2, 0, 0, 0, -2, 0], [-6, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 5, 0, -1, 0, -10, 0, -2, 0, -2], [9, 0, 0, 0, 7, 0, 0, 0, 2, 0, -4, 0], [0, 0, 0, 0, 0, -2, 0, -6, 0, 2, 0, 0], [-2, 0, 0, 0, 12, 0, -1, 0, -2, 0, 1, 0], [0, 0, -4, 3, 0, -1, 0, 3, 0, 1, 0, -5], [-6, 2, 0, 0, -6, 0, 4, 0, -3, 0, 3, 0], [0, 0, -4, 2, 0, 2, 0, -1, 0, -2, 0, 4], [1, -1, 0, 0, -2, 0, 1, 0, -1, 0, 0, 0], [-9, 0, 0, 0, -6, 0, 0, 0, -3, 0, 6, 0], [0, 0, -3, -2, 0, 3, 0, 0, 0, 3, 0, -3], [16, 2, 0, 0, 8, 0, 1, 0, 0, 0, 3, 0], [0, 0, 4, 0, 0, 2, 0, -2, 0, 2, 0, 4], [0, 0, -3, 2, 0, -3, 0, -4, 0, 0, 0, 0], [-15, 0, 0, 0, -13, 0, 0, 0, -2, 0, 4, 0], [-3, 0, 0, 0, 6, 0, 0, 0, 3, 0, 0, 0], [0, 0, -2, 5, 0, -7, 0, -5, 0, -7, 0, -5], [-2, 0, 0, 0, -2, 0, 3, 0, -2, 0, 1, 0], [0, 0, 1, 3, 0, -1, 0, 0, 0, 4, 0, -4], [-3, -3, 0, 0, 2, 0, 3, 0, -1, 0, 0, 0], [-17, -2, 0, 0, -18, 0, -2, 0, 1, 0, -2, 0], [0, 0, 4, 2, 0, -2, 0, -1, 0, 2, 0, -4], [0, 0, -4, 2, 0, -1, 0, 2, 0, 1, 0, -5], [6, -2, 0, 0, 3, 0, -1, 0, 0, 0, 2, 0], [0, 0, -7, 0, 0, -5, 0, -2, 0, -2, 0, -7], [0, 0, -1, 2, 0, -1, 0, -4, 0, 1, 0, 1], [-5, 1, 0, 0, 8, 0, -1, 0, 3, 0, 0, 0], [6, 1, 0, 0, 16, 0, 2, 0, -2, 0, 2, 0], [-2, 0, 0, 0, -18, 0, -2, 0, -2, 0, 1, 0], [-4, 0, 0, 0, -7, 0, 0, 0, -4, 0, 2, 0], [0, 0, -1, 0, 0, 1, 0, 0, 0, 4, 0, -4], [-3, -3, 0, 0, -1, 0, 3, 0, -4, 0, 0, 0], [0, 0, -2, 2, 0, 4, 0, -1, 0, 2, 0, 2], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0], [0, 0, -2, -6, 0, 1, 0, -6, 0, -1, 0, -1], [0, 0, 1, -9, 0, 6, 0, 9, 0, 6, 0, 5], [-4, 2, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0], [-13, -3, 0, 0, -11, 0, -3, 0, -2, 0, 4, 0], [3, -3, 0, 0, -3, 0, 3, 0, 0, 0, 0, 0], [0, 0, -1, -4, 0, -1, 0, 8, 0, 0, 0, 0], [0, 0, -4, 0, 0, 0, 0, 3, 0, -4, 0, -4], [0, 0, 6, 3, 0, -3, 0, 3, 0, 3, 0, 3], [0, 0, 4, 14, 0, -6, 0, -7, 0, -2, 0, -4], [0, 0, -7, 3, 0, 7, 0, 0, 0, -7, 0, 7], [0, 0, -5, 5, 0, 5, 0, 0, 0, -1, 0, 1], [0, 0, 1, -8, 0, 1, 0, 4, 0, 2, 0, -1], [-2, 2, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0], [-6, -5, 0, 0, 0, 0, 0, 0, -3, 0, -3, 0], [0, 0, -3, 0, 0, 3, 0, -8, 0, -6, 0, -3], [8, -3, 0, 0, 14, 0, -6, 0, 1, 0, -1, 0], [0, 0, -4, -4, 0, 1, 0, 4, 0, 1, 0, 5], [6, -3, 0, 0, 0, 0, -6, 0, 6, 0, -6, 0], [0, 0, -1, -12, 0, 1, 0, 0, 0, -4, 0, 4], [-9, -2, 0, 0, -9, 0, -2, 0, 0, 0, 0, 0], [0, 0, 1, 4, 0, -1, 0, 0, 0, 2, 0, -2], [12, 3, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0], [0, 0, 1, -6, 0, 4, 0, 6, 0, 4, 0, 3], [4, -3, 0, 0, 0, 0, 0, 0, -2, 0, -2, 0], [0, 0, -5, -2, 0, -5, 0, 4, 0, -7, 0, -7], [5, 0, 0, 0, 4, 0, 0, 0, 1, 0, -2, 0], [0, 0, -1, 3, 0, -1, 0, -6, 0, -7, 0, -7], [10, 1, 0, 0, 20, 0, 2, 0, 0, 0, 0, 0], [4, 0, 0, 0, -15, 0, -1, 0, 4, 0, -2, 0], [0, 0, 2, 2, 0, -5, 0, -2, 0, -5, 0, -7], [0, 0, -2, -6, 0, 0, 0, 3, 0, -2, 0, 2], [-5, 4, 0, 0, -3, 0, 4, 0, -2, 0, 4, 0], [0, 0, 5, 4, 0, -5, 0, 0, 0, 4, 0, -4], [0, 0, -2, 2, 0, -8, 0, -1, 0, -10, 0, 2], [-6, 8, 0, 0, -3, 0, 4, 0, 0, 0, 0, 0], [0, 0, 0, 15, 0, -3, 0, -15, 0, -3, 0, -3], [16, -6, 0, 0, 8, 0, -3, 0, 0, 0, -3, 0], [-9, 3, 0, 0, -7, 0, 3, 0, -2, 0, 4, 0], [11, 1, 0, 0, -14, 0, -1, 0, -3, 0, 0, 0], [0, 0, -8, 0, 0, -2, 0, -6, 0, -6, 0, -8], [0, 0, 3, -4, 0, 0, 0, 4, 0, 0, 0, -3], [18, -4, 0, 0, 30, 0, -8, 0, 3, 0, -3, 0], [19, 0, 0, 0, -15, 0, 0, 0, 4, 0, 0, 0], [0, 0, 2, 12, 0, -10, 0, -6, 0, -8, 0, -2], [0, 0, 10, 3, 0, -5, 0, 3, 0, 5, 0, 5], [-2, -6, 0, 0, -1, 0, -3, 0, 0, 0, -10, 0], [0, 0, 1, 0, 0, -1, 0, -14, 0, 2, 0, 1], [0, 0, 3, 2, 0, 3, 0, -4, 0, -3, 0, -3], [0, 0, 1, 6, 0, 1, 0, -12, 0, 3, 0, 3], [8, 4, 0, 0, 16, 0, 8, 0, 0, 0, 0, 0], [2, 0, 0, 0, 0, 0, -6, 0, 2, 0, -1, 0], [8, 0, 0, 0, -1, 0, 0, 0, 8, 0, -4, 0], [-2, -2, 0, 0, 2, 0, -4, 0, -3, 0, 3, 0], [0, 0, 3, -11, 0, -3, 0, 0, 0, 3, 0, -3], [2, 2, 0, 0, 1, 0, 1, 0, 0, 0, -6, 0], [-20, 4, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0], [0, 0, 4, 0, 0, 8, 0, 1, 0, -4, 0, 4], [11, 1, 0, 0, -11, 0, -1, 0, 0, 0, 0, 0], [0, 0, 5, -5, 0, 5, 0, 10, 0, 2, 0, 2], [0, 0, 2, 0, 0, -2, 0, -3, 0, 4, 0, 2], [10, -3, 0, 0, 0, 0, -6, 0, 10, 0, -10, 0], [0, 0, -9, -5, 0, 6, 0, -5, 0, -6, 0, -3], [-19, -1, 0, 0, 19, 0, 1, 0, 0, 0, 0, 0], [17, 3, 0, 0, 15, 0, 3, 0, 2, 0, -4, 0], [0, 0, -1, 5, 0, 1, 0, 0, 0, 10, 0, -10], [0, 0, 6, 14, 0, -12, 0, -7, 0, -6, 0, -6], [-14, 14, 0, 0, -7, 0, 7, 0, 0, 0, 4, 0], [-32, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0], [0, 0, 1, -6, 0, 1, 0, 12, 0, -4, 0, -4], [0, 0, 3, -14, 0, 0, 0, 14, 0, 0, 0, -3], [0, 0, -8, -8, 0, 1, 0, -8, 0, -1, 0, -7], [0, 0, 6, 6, 0, 0, 0, -3, 0, 6, 0, -6], [1, -3, 0, 0, -3, 0, -3, 0, 4, 0, -8, 0], [-24, 3, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0], [-34, 0, 0, 0, -17, 0, 0, 0, 0, 0, 6, 0], [0, 0, 0, 9, 0, 1, 0, -9, 0, 1, 0, 1], [12, 2, 0, 0, 0, 0, 0, 0, -3, 0, -3, 0], [0, 0, 3, 4, 0, 3, 0, -8, 0, 0, 0, 0], [0, 0, -5, 0, 0, 5, 0, 0, 0, -10, 0, -5], [6, 0, 0, 0, -12, 0, 1, 0, 6, 0, -3, 0], [0, 0, 4, 6, 0, 1, 0, 6, 0, -1, 0, 5], [0, 0, -2, 2, 0, -8, 0, -1, 0, -10, 0, 2], [-3, 5, 0, 0, 6, 0, -5, 0, 3, 0, 0, 0], [16, -9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0], [0, 0, 3, 0, 0, -2, 0, 0, 0, -2, 0, -5], [42, -2, 0, 0, 21, 0, -1, 0, 0, 0, 0, 0], [0, 0, 1, -8, 0, 1, 0, 16, 0, 2, 0, 2], [27, 3, 0, 0, 23, 0, 3, 0, 4, 0, -8, 0], [0, 0, 11, 7, 0, 11, 0, -14, 0, 5, 0, 5], [-2, 0, 0, 0, 0, 0, 3, 0, -2, 0, 1, 0], [0, 0, -1, -4, 0, 4, 0, 4, 0, 4, 0, 5], [-2, 1, 0, 0, 8, 0, 2, 0, -6, 0, 6, 0], [0, 0, 2, -16, 0, -2, 0, 8, 0, 0, 0, -2], [1, 5, 0, 0, 4, 0, -5, 0, 5, 0, 0, 0], [0, 0, -3, -6, 0, -4, 0, -6, 0, 4, 0, -7], [0, 0, 3, 6, 0, 2, 0, -6, 0, 2, 0, -1], [6, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0], [-12, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 5, 3, 0, 5, 0, -6, 0, 7, 0, 7], [-3, 0, 0, 0, -6, 0, 0, 0, -9, 0, 0, 0], [-8, 4, 0, 0, -16, 0, 8, 0, 0, 0, 0, 0], [0, 0, -5, 10, 0, -16, 0, -10, 0, -16, 0, -11], [-6, 0, 0, 0, 16, 0, 3, 0, -6, 0, 3, 0], [0, 0, -8, 16, 0, 4, 0, -8, 0, -4, 0, 8], [8, -6, 0, 0, 0, 0, 0, 0, -3, 0, -3, 0], [0, 0, -4, 6, 0, -13, 0, -6, 0, -13, 0, -9], [0, 0, -16, 0, 0, -2, 0, -20, 0, -14, 0, -16], [11, -5, 0, 0, -14, 0, 5, 0, -3, 0, 0, 0], [0, 0, 6, 0, 0, -2, 0, 9, 0, 8, 0, 6], [6, 0, 0, 0, -24, 0, -7, 0, 6, 0, -3, 0], [0, 0, 0, 0, 13, 0, -9, 0, 0, 0, 0, 0], [-7, -3, 0, 0, 12, 0, 3, 0, 5, 0, 0, 0], [-23, 5, 0, 0, -18, 0, 5, 0, -5, 0, 10, 0], [0, 0, -4, 8, 0, -4, 0, -4, 0, -8, 0, 4], [-4, -6, 0, 0, -2, 0, -3, 0, 0, 0, -3, 0], [0, 0, -5, 7, 0, 0, 0, 7, 0, 0, 0, -5], [48, 2, 0, 0, 24, 0, 1, 0, 0, 0, 1, 0], [-5, 9, 0, 0, 2, 0, 9, 0, -7, 0, 14, 0], [0, 0, -6, 0, 0, -2, 0, -9, 0, -4, 0, -6], [-2, 4, 0, 0, 2, 0, 8, 0, -3, 0, 3, 0], [0, 0, -4, -14, 0, 6, 0, 7, 0, 2, 0, 4], [0, 0, 3, -18, 0, -3, 0, 0, 0, 1, 0, -1], [15, -4, 0, 0, 15, 0, -4, 0, 0, 0, 0, 0], [0, 0, 5, 14, 0, -5, 0, 0, 0, -5, 0, 5], [-8, -3, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0], [-32, 0, 0, 0, -16, 0, 0, 0, 0, 0, 9, 0], [0, 0, -1, -11, 0, -4, 0, -11, 0, 4, 0, -5], [0, 0, 10, 3, 0, 11, 0, -3, 0, 11, 0, 1], [0, 0, -5, -12, 0, -5, 0, 24, 0, -1, 0, -1], [-13, 9, 0, 0, -14, 0, 9, 0, 1, 0, -2, 0], [0, 0, -3, -11, 0, -3, 0, 22, 0, -4, 0, -4], [11, -4, 0, 0, 12, 0, -4, 0, -1, 0, 2, 0], [0, 0, 10, 6, 0, -8, 0, -3, 0, 2, 0, -10], [-4, 0, 0, 0, -2, 0, 0, 0, 0, 0, 3, 0], [-48, -4, 0, 0, -24, 0, -2, 0, 0, 0, 7, 0], [0, 0, -1, 9, 0, -1, 0, -18, 0, 1, 0, 1], [-2, 0, 0, 0, 18, 0, 8, 0, -2, 0, 1, 0], [0, 0, 2, 14, 0, 12, 0, -7, 0, 14, 0, -2], [0, 0, 1, 3, 0, -1, 0, 0, 0, 1, 0, -1], [-7, 1, 0, 0, 12, 0, -1, 0, 5, 0, 0, 0], [-1, 6, 0, 0, -3, 0, 6, 0, 2, 0, -4, 0], [0, 0, -2, -6, 0, 4, 0, 3, 0, 2, 0, 2], [0, 0, 12, 6, 0, -5, 0, 6, 0, 5, 0, 7], [0, 0, -10, 0, 0, 1, 0, 0, 0, 1, 0, 11], [-8, -2, 0, 0, -4, 0, -1, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 6, 0, 8, 0, -6, 0, 0], [0, 0, -11, -5, 0, -11, 0, 10, 0, -2, 0, -2], [0, 0, 0, 0, 21, 0, -14, 0, 0, 0, 0, 0], [0, 0, 4, -8, 0, 5, 0, 8, 0, 5, 0, 1], [0, 0, -1, 8, 0, -4, 0, 8, 0, 4, 0, -5], [-2, 0, 0, 0, -8, 0, 0, 0, -2, 0, 1, 0], [14, 0, 0, 0, 10, 0, 0, 0, 9, 0, -9, 0], [0, 0, 9, 3, 0, -9, 0, 0, 0, 7, 0, -7], [0, 0, 6, 0, 0, 6, 0, 0, 0, 12, 0, -6], [0, 0, -1, -6, 0, 6, 0, 6, 0, 6, 0, 7], [0, 0, -12, 0, 0, 0, 0, -1, 0, -12, 0, -12], [0, 0, -5, -7, 0, -5, 0, 14, 0, -1, 0, -1], [-22, 5, 0, 0, -44, 0, 10, 0, 0, 0, 0, 0], [-6, 0, 0, 0, -12, 0, 4, 0, -6, 0, 3, 0], [0, 0, -7, -8, 0, 2, 0, -8, 0, -2, 0, -5], [0, 0, 2, 8, 0, 10, 0, -4, 0, 12, 0, -2], [0, 0, 1, -3, 0, -1, 0, 0, 0, -8, 0, 8], [0, 0, -3, -2, 0, 3, 0, 0, 0, -12, 0, 12], [0, 0, 2, 6, 0, 8, 0, -3, 0, 10, 0, -2], [-36, -4, 0, 0, -18, 0, -2, 0, 0, 0, -3, 0], [8, -8, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0], [0, 0, 3, 11, 0, 3, 0, -22, 0, 0, 0, 0], [-7, -12, 0, 0, -13, 0, -12, 0, 6, 0, -12, 0], [-15, -2, 0, 0, 15, 0, 2, 0, 0, 0, 0, 0], [0, 0, -5, 0, 0, -3, 0, 18, 0, -2, 0, -5], [-14, 3, 0, 0, -4, 0, 6, 0, -12, 0, 12, 0], [0, 0, 6, 23, 0, 9, 0, -23, 0, 9, 0, 3], [28, -3, 0, 0, 38, 0, -6, 0, 9, 0, -9, 0], [48, -3, 0, 0, 0, 0, 0, 0, -2, 0, -2, 0], [0, 0, 2, 5, 0, 3, 0, 5, 0, -3, 0, 5], [0, 0, 9, 9, 0, 2, 0, -9, 0, 2, 0, -7], [32, -10, 0, 0, 16, 0, -5, 0, 0, 0, -3, 0], [10, 2, 0, 0, 22, 0, 4, 0, -1, 0, 1, 0], [8, 0, 0, 0, 39, 0, -3, 0, 8, 0, -4, 0], [0, 0, -8, -14, 0, -7, 0, 14, 0, -7, 0, 1], [22, 2, 0, 0, 26, 0, 4, 0, 9, 0, -9, 0], [-27, -1, 0, 0, -30, 0, -1, 0, 3, 0, -6, 0], [0, 0, -11, 8, 0, 11, 0, 0, 0, -4, 0, 4], [34, 6, 0, 0, 17, 0, 3, 0, 0, 0, 6, 0], [2, -6, 0, 0, 1, 0, -3, 0, 0, 0, -4, 0], [0, 0, 4, 0, 0, 8, 0, -11, 0, -4, 0, 4], [25, 0, 0, 0, 23, 0, 0, 0, 2, 0, -4, 0], [0, 0, 9, -14, 0, 9, 0, 28, 0, 15, 0, 15], [-8, 0, 0, 0, -19, 0, 3, 0, -8, 0, 4, 0], [0, 0, 0, 8, 0, 0, 0, -4, 0, 0, 0, 0], [0, 0, -1, -12, 0, 1, 0, 0, 0, 7, 0, -7], [13, -1, 0, 0, -23, 0, 1, 0, -10, 0, 0, 0], [12, -6, 0, 0, 0, 0, 0, 0, -5, 0, -5, 0], [0, 0, -6, 21, 0, -13, 0, -21, 0, -13, 0, -7], [0, 0, -1, 6, 0, -1, 0, -12, 0, 1, 0, 1], [0, 0, -9, 7, 0, -9, 0, -14, 0, -7, 0, -7], [0, 0, 0, 6, 0, -3, 0, 6, 0, 3, 0, -3], [-13, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 8, 0, -1, 0, 0, 0, 5, 0, -5], [0, 0, 7, 0, 0, -5, 0, 0, 0, 2, 0, -7], [40, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0], [4, 10, 0, 0, 2, 0, 5, 0, 0, 0, 9, 0], [0, 0, -9, 4, 0, 6, 0, 4, 0, -6, 0, -3], [34, 4, 0, 0, 0, 0, 0, 0, 4, 0, 4, 0], [7, -5, 0, 0, -13, 0, 5, 0, -6, 0, 0, 0], [0, 0, -1, 0, 0, -3, 0, -6, 0, 2, 0, -1], [4, 0, 0, 0, 39, 0, 4, 0, 4, 0, -2, 0], [0, 0, -7, -15, 0, 7, 0, 0, 0, -6, 0, 6], [-33, 2, 0, 0, 33, 0, -2, 0, 0, 0, 0, 0], [12, -2, 0, 0, 6, 0, -1, 0, 0, 0, -3, 0], [0, 0, -8, -13, 0, 5, 0, -13, 0, -5, 0, -3], [-39, 0, 0, 0, 42, 0, 0, 0, 3, 0, 0, 0], [0, 0, 16, 0, 0, 2, 0, 6, 0, 14, 0, 16], [0, 0, 1, 2, 0, 14, 0, -2, 0, 14, 0, 13], [6, 0, 0, 0, 22, 0, 9, 0, 6, 0, -3, 0], [22, 4, 0, 0, 26, 0, 8, 0, 9, 0, -9, 0], [0, 0, 4, 28, 0, -12, 0, -14, 0, -8, 0, -4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1728_r_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1728_2_r_f();".
// 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_1728_2_r_f(:prec:=12)
    chi := MakeCharacter_1728_r();
    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_1728_2_r_f();".
// 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_1728_2_r_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1728_r();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![3, -3, 1]>],Snew);
    return Vf;
end function;