
// Make newform 720.2.t.d in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_720_t();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_720_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_720_2_t_d();".
// 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_720_2_t_d();".
// 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 := [1024, 0, -512, -512, 448, 512, 96, -288, -168, 80, 136, 40, -42, -36, 6, 16, 7, -4, -2, 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, 0, 0, 0, 0, 0, 0, 0], [0, 512, 512, -448, -512, -96, 288, 168, -80, -136, -40, 42, 36, -6, -16, -7, 4, 2, 0, -1], [512, 512, -448, -512, -96, 288, 168, -80, -136, -40, 42, 36, -6, -16, -7, 4, 2, 0, -1, 0], [5632, -2816, -8448, -7040, 2976, 5536, 1024, -3808, -3332, -152, 1426, 768, -364, -616, -199, 94, 104, -8, -33, -18], [-5632, -11264, -8448, 1024, 5440, 992, -4448, -4816, -1184, 1356, 1096, -266, -836, -464, 12, 131, 28, -44, -36, -11], [4096, 7936, 6016, -448, -3744, -800, 3024, 3352, 940, -876, -746, 156, 568, 330, 7, -84, -20, 30, 25, 8], [4096, 8448, 6016, -448, -3744, -800, 3024, 3352, 940, -876, -746, 156, 568, 330, 7, -84, -20, 30, 25, 8], [-22528, -22528, -3584, 15808, 7616, -7840, -12800, -4216, 3592, 3688, -92, -2134, -1324, 90, 438, 133, -132, -94, -22, 11], [-17920, -5120, 11776, 17728, -736, -11680, -7264, 3800, 6636, 2008, -2086, -2082, -64, 946, 497, -67, -204, -30, 39, 31], [28672, 24064, -128, -21760, -7936, 11424, 15248, 3232, -5696, -4436, 652, 2786, 1376, -364, -594, -123, 192, 104, 10, -21], [-26624, -30208, -6912, 18560, 12000, -7840, -15968, -6848, 3412, 4804, 550, -2354, -1808, -120, 483, 205, -116, -116, -35, 7], [-10240, 5632, 17536, 14208, -5440, -11168, -2480, 7392, 6816, 588, -2764, -1598, 632, 1204, 426, -163, -200, 8, 62, 35], [-5632, -12416, -9088, 960, 6048, 1520, -4512, -5136, -1388, 1404, 1198, -212, -844, -486, 3, 130, 32, -42, -35, -10], [16128, 17024, 4480, -10624, -6368, 4752, 9408, 3888, -2080, -2736, -220, 1446, 1056, 54, -286, -119, 80, 70, 22, -5], [-34304, -39168, -14464, 20416, 16032, -7200, -20112, -10808, 2644, 5956, 1338, -2672, -2496, -458, 497, 314, -116, -150, -65, 2], [24064, 14208, -7424, -20480, -2912, 12560, 11296, -992, -6676, -3156, 1666, 2552, 644, -762, -571, -4, 216, 66, -21, -28], [-10752, -14336, -6656, 5440, 6176, -1408, -6720, -4600, 148, 2008, 790, -746, -952, -310, 119, 129, -12, -54, -31, -5], [35840, 26368, -4736, -28096, -7584, 15808, 17680, 1688, -8068, -5032, 1462, 3482, 1348, -722, -753, -85, 264, 110, -7, -31], [-28416, -9216, 16896, 26880, -1056, -18080, -11552, 5456, 10072, 3060, -3224, -3262, -168, 1424, 760, -103, -320, -52, 56, 47], [-96256, -76800, 4480, 73536, 24544, -39040, -49392, -8680, 19708, 14228, -2706, -9168, -4176, 1454, 1963, 338, -652, -318, -11, 74]];
    Rf_basisdens := [1, 512, 256, 512, 512, 256, 256, 512, 512, 512, 512, 512, 256, 256, 512, 256, 128, 256, 256, 512];
    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_720_t();"
function MakeCharacter_720_t()
    N := 720;
    order := 4;
    char_gens := [271, 181, 641, 577];
    v := [4, 1, 4, 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_720_t_Hecke();"
function MakeCharacter_720_t_Hecke(Kf)
    N := 720;
    order := 4;
    char_gens := [271, 181, 641, 577];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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], [-1, 0, 1, -1, 1, 0, -1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1], [0, -1, -1, 1, -1, -1, 1, 0, 0, 0, 0, 0, 1, -1, -1, 0, 0, -1, 0, 0], [0, -1, 1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, -1, 1, 1, -1, 0], [1, 2, 1, 1, 0, 0, -1, -1, 0, 1, 1, 0, -1, 1, 1, 1, -1, 0, 2, 1], [0, -1, 0, 0, -1, 1, -2, 0, -1, 0, 1, -1, -2, 1, 1, 0, -1, 2, 1, 1], [-1, -1, 2, -2, 1, 0, 0, 1, 0, 1, -1, 2, 2, 0, -2, 1, 1, 0, 0, 1], [-1, -1, 0, 0, -1, 1, 0, 0, 1, 1, -1, -1, 2, 0, -1, 1, 1, 0, 0, -1], [-1, 0, 1, 0, 0, 1, -2, -1, 1, 1, 1, 0, -1, 1, 1, 3, 0, 0, 2, 1], [1, -1, -2, 1, 0, -1, 3, 0, 0, 0, 0, -1, 1, -1, 0, 1, 1, -1, -1, -1], [0, 0, 0, 2, 2, -1, 1, 0, 2, 2, 0, -2, 0, 0, 0, 0, -1, -2, 0, 0], [-1, 2, 1, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 1, 1, 2, -1, 0, 0, 1], [1, 0, -1, 0, -1, 0, 1, -1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, -2, 1], [1, 2, 1, 0, 1, 2, -2, 0, 0, 2, 0, -1, 0, 1, 1, -2, -1, 0, 0, 1], [0, -1, 1, -3, 1, 1, -3, 0, 0, 2, 0, 2, 1, 1, 1, 0, 0, 1, 0, 2], [1, 1, -1, 2, 0, 0, 3, -2, -1, 0, -1, 0, 2, -1, 0, -1, 1, -2, 0, -2], [-2, -2, 0, -2, 2, -3, 1, 4, 2, 0, -2, 0, 2, 0, -2, 0, -1, -2, 0, 0], [1, -1, -2, 0, -1, -2, 3, 0, -1, -1, -1, -1, 2, -2, -1, 3, -2, -2, 0, -1], [1, 2, 1, 0, 1, 0, 0, -2, 0, -2, 0, -3, 0, -1, -1, 0, -1, -2, 0, -1], [1, -1, 0, 0, -2, -1, 1, 3, -4, -2, -2, 0, -3, 0, -1, 2, 1, 0, 0, 1], [1, 0, 1, -2, 1, -1, -3, 2, 2, -2, 0, -3, 0, 1, -1, 0, 0, 2, 0, 1], [0, -4, 0, -2, 0, -3, 3, 0, -2, -4, -2, -2, 2, 0, -2, -2, 1, -2, 0, 0], [3, -2, 1, 0, 1, 2, -2, 0, 2, 0, 0, -1, 0, 1, 1, -2, -1, 0, 2, -1], [-2, 0, 0, 2, 4, 2, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0], [-1, 2, -1, 1, -1, 0, -1, 3, 2, 0, 0, 2, 2, -1, 0, 0, 0, 0, 0, 1], [-1, -2, -3, 2, -1, -1, 3, 0, -2, -4, -2, 1, 0, -1, -1, 0, 2, 0, -2, -1], [1, 3, -1, 4, 0, 2, -1, -6, 3, 2, 3, -2, -4, -1, 4, -1, -1, 2, 0, 0], [3, 2, -1, -1, 2, 0, 1, -3, 4, 1, 1, 0, 3, -1, 1, 1, 1, 0, -2, -1], [0, -2, -2, -3, -2, 1, 2, -1, 2, 0, 0, -1, 2, -2, -1, 0, 2, 0, -4, 0], [1, 4, -1, 1, 0, 1, 0, 0, -3, -1, -1, -1, 1, 1, 0, 1, 2, -1, 2, -1], [1, -1, 0, 1, 0, -1, -1, -3, -1, 0, 2, -6, -5, 2, 3, 0, -3, 0, 0, -1], [-2, -1, 0, -4, -3, 0, -1, 0, -3, -4, -3, 1, -2, 1, -1, -2, 0, 0, -1, 1], [5, -3, -2, -4, 1, 1, 0, 0, -1, 3, -1, 3, 0, -2, -3, -3, 3, 2, 0, -1], [1, 3, -2, 0, 4, 0, 4, 1, 0, 0, 0, 2, 3, -4, -1, 0, 0, -4, 0, -1], [2, 1, 1, -1, 3, -1, -3, 0, 0, -2, 0, -4, 1, 4, -1, -3, 0, 1, 3, 0], [3, 2, 1, 0, 1, 1, -3, -2, 0, -2, 4, 3, -6, 1, 3, 0, -2, 2, 0, 3], [0, -2, 0, -2, 0, 1, 0, -1, -3, -6, 0, 1, -2, 2, 1, -2, 2, 2, 0, 0], [2, -3, 1, -2, 0, -3, 4, 2, 3, -3, -1, 4, 2, -3, -2, -1, 0, -2, -4, 0], [-5, 2, -1, 1, 2, 1, -4, 0, 3, 5, 1, -3, -3, 1, 2, 3, -2, 1, 2, 1], [-1, 3, 1, 4, -4, -2, 1, 0, 1, -2, 1, -2, -2, -1, -2, -1, -3, -2, 2, -2], [-5, 1, 2, 2, 1, 0, -3, 0, -1, 1, 1, 1, -2, 2, 1, 1, -2, 0, 4, 1], [4, -2, 0, -2, -4, -2, 2, 2, 2, 0, 0, 4, 2, 0, -2, 0, 0, -4, 0, -2], [-3, 3, 0, 4, 5, 3, 0, 0, 1, -3, 1, -1, 0, 0, 1, -1, 1, 2, 0, -1], [1, -6, -3, 4, 0, -2, 5, -1, 1, -7, -1, 0, -1, 1, 1, -1, 1, 0, 0, -1], [-4, 0, -1, 0, 3, -4, 2, 2, 0, 3, -2, -3, 2, -2, -1, -1, -1, -2, -1, -1], [-2, -2, 0, -5, 4, -1, -4, -1, 4, 0, 0, 5, -2, 0, 1, 0, -4, 0, 0, 2], [1, -4, 1, -2, 1, -3, 1, 2, -2, -2, 0, 1, 0, -3, -1, 0, -2, 0, -4, 1], [3, -3, -1, -2, 2, 2, -1, 0, 1, 2, 1, 2, 0, 1, 4, -1, 3, 2, -2, -2], [-1, 3, 2, 3, 2, 1, -1, 3, 7, 8, 2, 6, 3, -4, -1, 0, 1, 0, 0, 1], [-1, 7, 4, 2, 1, 0, -3, -2, 3, 3, 3, -1, -2, 4, 3, 3, -2, 0, 8, 1], [-8, 1, 1, 6, 2, 1, -2, 0, -5, 1, 1, -2, 0, 1, 0, 3, 2, 2, 2, 0], [2, -1, 3, -3, -1, -1, -1, 0, 0, 2, 0, 0, -3, 1, -1, 2, 0, 1, 2, 0], [-5, 0, 1, -1, -2, -2, 1, 1, 2, -1, -1, 4, -1, 1, 1, -1, -3, -4, 2, -1], [0, 4, -2, 2, 8, 3, 2, 1, 1, 0, 2, 3, -2, 0, 3, 0, 0, -2, 0, 0], [7, -1, 0, 0, -5, -1, 4, 0, -3, -7, -1, -1, 2, -4, -1, -3, 3, 2, -4, -1], [-5, -1, -2, 6, -8, -1, 7, 1, -8, -2, -2, 0, 1, -2, -1, -2, 1, -4, -4, -1], [-5, 3, 4, -1, 4, 3, -3, 0, 2, -2, 2, -3, -1, 3, 4, 1, -1, 1, 1, 1], [4, 2, -6, 0, 0, -4, 8, -2, -4, -4, -2, -6, 4, -6, -2, -2, -2, -8, 0, -4], [2, -2, 0, -2, 2, 1, 1, 0, 2, 4, 2, -4, 2, 0, 2, 0, 1, -2, 0, 0], [-4, -1, -3, -4, -2, -3, 2, 0, 1, -5, 1, 4, 2, -1, 2, 3, 0, 0, -2, -2], [8, 2, 0, 0, -2, 0, -2, 0, -2, -8, 2, -2, -4, -2, 2, 0, 0, 4, -2, 2], [1, 1, 0, 0, -3, 2, -1, -2, -1, 3, -1, 1, 0, 0, -1, -5, 4, 2, 0, -1], [0, -4, -4, 2, -2, -6, 4, 2, 0, 4, -2, -6, 2, 0, 0, -6, 0, -4, 0, 0], [-9, 1, 0, -4, -3, -3, 0, 8, -1, 5, -3, -3, 2, 4, -3, -5, 3, 2, 4, 1], [-2, -3, -2, 6, -1, 0, -1, 0, 1, -4, 1, 5, 0, -1, 1, -2, 2, 2, -1, -3], [-5, 0, -1, -6, -1, 0, 4, 2, 4, -4, -4, -3, -2, -1, 1, 2, 3, 0, -2, -1], [4, 0, -8, -6, -2, -4, 6, 0, 2, 2, 2, 4, 6, -6, -2, 0, 0, -2, -2, -4], [1, 3, 5, -6, 0, 2, -3, 2, -3, -2, 1, 8, 2, 3, -4, 3, 5, 0, -2, 2], [-5, -4, 1, 1, 0, 4, -5, 1, -4, 1, 1, -4, -3, 1, -1, -3, -1, 4, 2, 3], [-2, 0, 6, -4, 0, 2, -6, 0, 0, 2, 2, 2, -2, 2, 0, -2, 2, 4, 0, 2], [2, 4, 2, 5, -2, 1, -2, -3, -2, 4, 4, -1, 0, 2, 1, -4, -4, 0, 4, 2], [-3, 3, -4, -7, 2, 1, 1, 0, 0, -6, 0, 5, 1, -3, -2, -3, 1, 3, -1, -3], [0, 4, 7, -2, 1, 5, -7, 0, 2, 1, 2, -3, -6, 4, 3, -5, -2, 2, 3, -1], [2, -1, 1, 2, -8, 3, 3, 1, -2, -1, -1, -5, -2, 1, 1, 3, 3, -2, 2, 0], [-8, -8, -4, 2, 0, 0, 6, 0, -4, -16, -4, 0, -4, 0, 0, 0, 8, 2, -4, -4], [-8, -1, 3, -3, -1, 6, -12, 0, -6, 8, 6, 2, -9, 4, 3, 7, -3, 7, 1, 6], [4, 0, 2, 2, -4, 1, -1, -2, 0, 0, 0, 2, -4, 2, 4, 0, -5, -4, 4, 2], [-2, -10, 0, -8, 4, -8, 10, 6, -4, 8, -4, 4, 10, -4, -6, 4, 0, -6, 0, 2], [5, -2, 3, -1, 4, 5, -14, -4, -3, -3, 7, -3, -9, 9, 4, 1, -6, 7, 6, 5], [7, -1, 1, -2, -2, -2, -3, 0, -1, 12, -1, 2, 4, -1, -4, 1, -3, -2, 2, 6], [6, -4, -2, 0, 0, -2, 2, 4, -4, -2, -2, 2, 2, -2, -4, -2, 2, 0, -4, -2], [-7, -4, -3, 6, -3, -4, 4, 2, -12, -4, -4, 7, -2, -3, -1, 6, 1, -4, -6, 1], [-1, -4, 1, -8, 4, -4, 3, 1, -7, 5, -1, -4, 1, 1, -1, -1, -1, -4, 0, 1], [-1, 2, -7, 0, -3, -3, 5, 0, 0, -6, 0, 5, 2, -3, 1, 0, 0, 0, -4, -5], [-8, 2, 0, -6, 2, 0, -2, -4, 8, 2, 2, 2, 0, 0, 2, -2, -4, 0, 0, 2], [1, -6, -3, 6, -5, -4, 2, 0, 4, -16, 2, -1, -2, -1, 3, -2, -1, 2, 0, -1], [5, -3, 2, -4, -5, 3, -6, 4, -1, -7, 1, -1, -2, 2, -1, 3, 1, 6, 0, 3], [8, -4, 0, 13, 8, 7, -4, -3, -6, 0, 0, -7, 4, 0, 3, -4, 2, 4, 0, -4], [6, -4, 2, -2, -4, -2, 6, 0, 2, -6, -2, 2, 6, 0, -4, -8, 4, 0, -2, -2], [3, 1, -2, -2, -1, 2, 3, -4, 3, -1, -1, -3, 0, -2, 3, 3, 2, 0, -4, 1], [5, 6, -3, 11, 1, 4, 1, -7, 6, -4, 4, -2, -6, -3, 4, 0, -2, 0, 0, -5], [-10, 1, 1, 7, 5, 3, 1, 0, 0, -10, 0, -4, -1, 1, 1, 0, 2, 1, 0, 0], [4, 2, -1, 2, 11, 2, -8, -2, 12, -3, 2, -1, -4, 8, 3, 5, -7, -4, 9, 1], [2, 0, 0, 0, 2, 5, 0, -7, -5, -8, -2, -3, -6, 6, 1, -4, 4, 2, 0, -2], [-6, 2, 6, -8, -4, 0, -2, 4, 0, 4, 2, 8, 0, 2, -4, -2, 6, 4, -4, 4], [-4, 6, 6, -4, 2, 0, -10, 4, 2, 10, 4, -4, -2, 2, 2, 4, -6, 0, 0, 4], [-9, -6, 3, 6, 3, -1, 1, 0, 0, -8, 0, -1, -2, 1, -1, 6, 4, 2, 2, 1], [7, -1, 1, 0, 6, 0, -3, -2, 3, -6, 3, 4, -4, 1, 2, 3, -5, -2, 0, 2], [3, -4, 3, 2, 1, 4, -8, -10, -6, 2, 6, -7, -4, 3, 3, -4, -3, 8, 0, 1], [0, 3, -3, 2, -2, 3, -4, 2, 9, -1, 1, -2, -6, 5, 4, 1, -2, 2, 8, 2], [-2, -4, -2, 4, -2, 0, 2, -4, 8, 4, 0, -6, 0, -4, 2, -2, -2, 2, -2, -2], [0, -2, 2, 10, -2, -3, 3, 2, 14, -8, -2, 4, 8, -6, -6, 6, -1, -2, 0, 0], [0, -3, 10, -6, -1, 0, -9, 0, 1, 6, 1, 1, -4, 5, 1, 4, -2, 2, 5, 5], [7, 6, 1, 4, -3, -2, -2, 0, -10, 4, 4, -1, -4, 1, -3, 6, 1, 4, 2, 7], [5, -8, -3, -4, 3, 1, -1, 0, -2, 8, -2, 3, 6, -1, -1, 0, 4, 0, -2, 5], [7, -6, -5, -3, 0, -2, 5, 3, -4, -5, -5, 2, 1, -5, -3, 7, 7, 0, -10, -1], [4, 0, -8, 0, -2, 0, 4, -4, -8, -6, 2, -4, -6, 0, 6, -2, -2, 0, 0, -4], [2, 6, -4, 4, -6, 2, 0, 0, 2, 10, -2, 2, 4, -4, -2, -2, 2, 0, 0, -2], [0, -12, -8, -9, -4, 1, 10, 5, 0, -8, -8, -3, 4, -8, -5, 0, 10, 0, -16, -4], [5, 7, -6, 7, 4, -2, 4, 0, 0, 6, 0, 5, 7, -3, 0, 5, -2, 1, -3, 1], [-1, 1, -4, 3, -2, -3, 3, 5, 3, 14, -4, 0, 7, -2, -3, 2, -1, -4, 0, 1], [8, 3, -4, 16, 5, 6, 1, 0, 1, 6, 1, -3, 2, -1, 3, -8, 2, 0, -3, -3], [3, 4, -3, -2, 2, 8, -1, -7, -9, -13, 1, 0, -9, 5, 5, -7, 5, 4, 0, -3], [-6, 2, -2, -4, -6, -10, 10, 10, -4, -4, -4, 6, 2, -2, -8, 0, 4, -4, -4, -4], [7, 0, -5, 4, 1, -7, 15, 2, -10, -8, -6, 3, 6, -5, -1, -6, 0, -10, 0, -5], [2, -2, -6, 4, -2, -3, 10, 1, 1, -14, -8, 1, 10, -4, -7, -6, 6, -6, 0, -2], [9, 0, -5, 4, 7, -8, 14, 2, 12, -10, -8, 1, 10, -7, -3, 0, -1, -14, -2, -7], [5, 10, -5, 9, 2, 7, -2, -8, 3, -1, 1, -3, -3, -1, 6, 9, 0, -3, 4, -3], [-7, 1, -1, -10, -6, -6, -3, 0, 1, -4, 1, 10, 4, -1, 0, 5, -5, 0, 0, 2], [11, -4, -1, -10, 3, 0, -4, -2, 8, 0, 0, 5, -2, -1, 1, 6, -1, 0, -2, 3], [1, -4, 3, -10, -5, -2, -2, 0, -4, -2, -4, 3, -6, 1, -5, 0, 1, 2, 2, 1], [-8, -4, 4, 6, 2, -6, -2, 0, 6, 2, 6, 4, 6, -2, -2, 12, -2, 0, 6, 4], [10, -3, 4, -4, -3, -1, 2, 0, 13, -10, -1, 1, 6, -3, -5, -2, 5, 2, -7, -1], [0, -6, -4, 4, 2, -2, 6, 0, 4, 0, -2, 2, 0, -8, 2, 6, -6, -6, -4, -2], [9, -7, -9, -18, -6, 0, 9, 0, -7, 0, -7, -6, 0, -3, -4, -9, 5, -6, -6, -2], [-8, 4, 8, 8, 2, 4, -10, 2, -6, 2, 2, -6, -10, 8, 4, 2, -4, 4, 16, 4], [1, -4, -1, 5, 1, -8, 9, 3, 8, 2, -2, 0, 10, -9, -6, -2, 0, -8, 0, -1], [-2, -1, 5, -5, 9, 1, -15, 0, 14, 2, 6, -2, -7, 8, 1, 3, -6, 5, 3, 6], [6, -8, -5, 6, -3, 2, 2, -2, -6, -5, 0, -3, -4, -4, 5, 3, -7, 0, 1, -1], [3, 9, -4, 12, 5, 0, 2, -3, 8, 7, 1, -10, -2, -2, 2, 7, -9, -8, 0, -3], [-3, -6, 1, 0, 10, 7, -10, -1, 3, 3, 3, -6, 3, 1, -1, -3, 2, 8, 2, -1], [-9, 3, 8, -15, -2, 3, -5, 0, -4, -12, -4, -3, -11, 9, 6, 1, -1, 3, -1, 1], [0, -8, 4, 4, -2, -4, 4, 0, 8, -8, -2, 4, 6, -4, -6, -2, 4, 0, 0, 0], [8, -5, -1, 14, 0, 5, 4, 2, -13, -7, -7, -8, -2, -1, 4, -9, 0, 0, -2, -4], [-7, 5, -2, -16, -7, -7, 4, 0, -1, -5, -1, -1, 4, -2, -3, 5, -7, -6, 0, 3], [-5, 0, 5, 0, 7, 1, -1, -10, 14, 10, 2, 1, 6, -1, -3, 0, 2, -2, -6, -3], [5, -1, 0, 2, 9, 1, -8, -4, 3, -3, 5, -1, -8, 2, 5, 5, -9, 0, 2, 3], [2, 0, -2, -6, -2, 2, 0, -2, -8, 10, 4, 4, -4, -2, 4, 0, 0, 4, 0, -2], [12, 2, -1, 10, 3, -1, 1, 0, 0, 13, 0, 3, 2, -2, -3, 3, 0, 2, 1, 1], [9, 3, -2, 0, -5, 1, 0, 0, -3, 3, -3, 1, -4, -2, -5, -9, -1, 0, 0, -3], [3, 2, -9, -3, -9, -2, 7, 1, -8, 4, -8, 0, 10, -5, -6, -4, 4, -4, 0, -3], [-8, -2, 2, -6, 2, 6, 0, 0, -4, -12, -4, -8, -6, 2, -2, -8, 4, 0, 0, 0], [3, 3, 1, -4, -2, 0, -1, 6, -21, -6, -1, 4, 0, -1, -2, 5, 3, 0, -2, 2], [-4, -8, 8, 8, -6, -2, -10, 0, 12, 6, 2, -4, -6, 8, 2, 6, -4, 8, 0, 4], [-9, 5, 2, 1, 4, -4, 6, 0, 4, -2, 4, -9, 5, -1, 0, 11, -4, -5, 3, 3], [-7, 3, 12, -17, 2, 3, -13, 5, -7, -6, 0, 2, 1, 6, -3, 10, -3, 4, 0, 7], [-2, 6, -3, -10, -3, -1, -3, 0, 4, 1, 4, 5, 6, -4, -1, -5, -6, -2, 1, -1], [7, -2, 3, -6, 5, 0, -7, 1, 5, -2, -2, 4, -8, 3, 4, 2, -6, 0, 6, 3], [-6, 3, -1, 3, 3, 3, -5, -4, -2, 8, 2, -4, -3, 6, 3, -1, -2, 1, 7, 0], [16, -4, -6, -4, 0, 7, 3, -8, 4, 0, 0, -8, 4, -6, 2, 0, 5, 4, -12, 2], [11, -2, 1, 4, 1, 1, 1, -10, 12, -6, 4, -1, -2, 1, 3, -8, -2, 2, 0, -1], [-4, -8, 6, -2, -4, -3, -4, 5, 5, 16, 2, 7, 2, 0, -1, 8, -2, 6, 0, 4], [-7, 2, 13, 4, 9, 2, -14, 0, 4, 6, 4, 7, 0, 3, -3, 8, -3, 8, 10, 9], [12, 2, 2, 8, 8, 6, -10, -10, 0, 6, 6, -2, 0, 2, 6, -2, -6, 4, 4, 4], [1, -1, 2, 2, -4, -2, -4, -1, 4, 0, 8, 0, -5, -4, 5, 0, -4, 4, 0, -1], [-4, -2, -8, -12, -4, -2, 16, 0, 10, -8, -8, -2, 6, -8, 0, -4, 6, -4, -16, -6], [2, 10, -4, 0, -2, 0, 2, 0, -2, 0, -6, -6, 8, -4, -6, 2, -2, -8, 0, -2], [-4, -4, 8, -12, 0, -4, -4, 8, 6, 0, 0, 4, 4, 4, -8, -4, 4, 4, -4, 4], [-7, 8, 5, -2, 5, 3, -7, -6, 6, 10, 8, 9, -8, 3, 3, 2, 0, 2, -2, 5], [7, -7, -16, 11, -4, -9, 21, 1, 3, -12, -2, 4, 9, -14, -1, -4, 1, -8, 0, -7], [-1, -2, -1, 1, -1, 0, 5, 5, 4, -4, -4, 12, 10, -5, -6, -8, 10, 0, 0, 1], [4, -7, 7, -9, -5, 1, -7, 0, 0, 2, 0, 2, -9, 5, 3, -2, 2, 3, 2, -2], [9, 8, 3, -7, 0, -6, -1, 7, 2, 3, 3, 0, -1, 3, -7, -5, -1, 4, 6, 1], [-7, 5, 6, -12, 3, 2, -12, 9, -4, -5, 5, 4, -2, 4, 4, 7, -5, 4, 0, 7], [13, -5, -4, 6, -4, 3, 5, -1, -10, -4, -4, -4, -1, -4, 1, 0, 3, 0, -8, 1], [7, -1, 6, 5, -2, -4, -4, 0, 6, 14, 6, 1, 1, -1, -2, 5, -4, -1, 7, 1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_720_t_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_720_2_t_d();".
// 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_720_2_t_d(:prec:=20)
    chi := MakeCharacter_720_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(997) - 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_720_2_t_d();".
// 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_720_2_t_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_720_t();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![262144, 0, 76283904, 0, 204587008, 0, 145248256, 0, 49590528, 0, 9697664, 0, 1161104, 0, 86368, 0, 3880, 0, 96, 0, 1]>],Snew);
    return Vf;
end function;