
// Make newform 1600.2.j.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1600_j();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1600_j_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_1600_2_j_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_1600_2_j_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 := [512, 0, 256, -256, -160, -224, -80, 24, 74, 70, 37, 6, -10, -14, -5, -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], [-28416, -21696, -31296, -9920, 848, 13500, 15406, 10894, 4379, -583, -2676, -2550, -1487, -375, 6, 248, 89, 75], [38912, 4736, 8896, -36256, -40656, -42352, -24152, -3462, 10426, 14845, 11894, 6294, 1614, -997, -1252, -1110, -380, -191], [-90496, -75776, -103296, -35040, 4368, 45600, 51206, 35784, 13989, -2433, -9096, -8400, -4757, -1145, 106, 798, 299, 225], [196480, 115392, 165632, -1648, -64880, -126996, -108530, -56742, -5453, 24377, 29756, 20942, 9185, 249, -1810, -2732, -951, -613], [267008, 210048, 295808, 95120, -12944, -130120, -145098, -101032, -39447, 6959, 25688, 23760, 13511, 3215, -198, -2314, -817, -655], [46208, 32320, 45504, 8592, -8624, -26316, -25600, -15672, -4196, 3435, 5762, 4568, 2280, 321, -232, -524, -186, -129], [513280, 383744, 537024, 137024, -63920, -275192, -284460, -185094, -60796, 26869, 57922, 48934, 25980, 4903, -1600, -5234, -1882, -1371], [-528640, -381952, -540992, -125312, 77840, 288616, 291940, 185642, 56748, -32027, -61646, -50762, -26220, -4409, 2000, 5582, 1966, 1413], [-277120, -183872, -260352, -32912, 66320, 166636, 154530, 89642, 19033, -25777, -37596, -28402, -13525, -1369, 1850, 3432, 1211, 813], [-303232, -220288, -313728, -75136, 40416, 164208, 167822, 108104, 34317, -17157, -34920, -29216, -15309, -2757, 1042, 3182, 1131, 829], [-588032, -405376, -575296, -103264, 113296, 337152, 326212, 198282, 51964, -45185, -74354, -58634, -29124, -3963, 3072, 6790, 2410, 1671], [-306816, -209920, -299520, -54624, 56368, 172232, 167186, 102398, 27445, -22322, -37538, -29814, -14917, -2108, 1466, 3432, 1201, 856], [-670720, -466944, -661184, -127424, 119760, 376472, 368520, 226774, 62446, -47679, -81842, -65414, -32910, -4833, 3140, 7454, 2632, 1861], [-372864, -258560, -364800, -67376, 69792, 211368, 205144, 125222, 33350, -27653, -46182, -36566, -18238, -2547, 1844, 4198, 1484, 1039], [-807680, -542976, -767296, -110976, 179280, 476008, 448020, 264246, 61024, -70181, -106298, -81686, -39440, -4527, 4920, 9706, 3438, 2339], [-496256, -355008, -505408, -112112, 74528, 272636, 274106, 173656, 52407, -30930, -58602, -48112, -24827, -4144, 1966, 5350, 1895, 1368], [1253888, 893824, 1258944, 268896, -203344, -692848, -689368, -431518, -125726, 82065, 149166, 121006, 61766, 9767, -5268, -13550, -4780, -3419]];
    Rf_basisdens := [1, 640, 1280, 640, 640, 640, 128, 1280, 1280, 640, 640, 1280, 640, 1280, 640, 1280, 640, 1280];
    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_1600_j();"
function MakeCharacter_1600_j()
    N := 1600;
    order := 4;
    char_gens := [1151, 901, 577];
    v := [2, 1, 3];
    // 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_1600_j_Hecke();"
function MakeCharacter_1600_j_Hecke(Kf)
    N := 1600;
    order := 4;
    char_gens := [1151, 901, 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, 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, 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, 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, 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, -1, 0, 1, 0, 0], [-1, 0, 0, -1, 1, 0, 1, -1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, -1, 0, -1, -1, 0, 0, 0, 0, 0, -1, 0, 0], [-1, -1, 0, -1, 0, 0, 1, -1, 1, -1, 0, 0, 1, 0, -1, 0, 0, 0], [0, 0, 0, 0, 1, -1, 0, 0, 0, 0, -1, 0, 1, 0, 1, 0, 0, 1], [-1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 1, 1, 0, 0, 0, -1, 1], [1, 0, -1, 0, 1, 0, 1, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, -1, 0, -1, 0, 1, 1, 1, 0, 0, -1, 0], [-2, -1, -1, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 0, -1, 1, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1], [0, 0, 1, 0, 0, 0, -2, 1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 1], [2, -1, -2, 1, 1, 2, 2, 0, 0, 0, 2, 0, 2, 0, -1, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, -2, 0, -1, 0, 0, 0, -2, 0, 0, 0, 0], [3, -1, -1, 0, -1, 0, -1, 2, -1, -1, 0, -1, -1, 1, 0, -1, 1, -1], [-2, 0, 1, -2, 0, -2, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0], [1, 0, 0, 0, 0, 0, -1, 2, -1, -1, 0, 0, 0, 2, 0, -1, 0, 0], [-2, 4, 1, 0, 0, -2, 0, -1, 0, 1, 0, 0, 0, 0, 0, -1, -1, -1], [-1, 0, 1, 0, -1, 0, 3, 0, 1, 1, 2, 0, 1, 0, -1, 0, -1, 0], [1, -2, 0, 0, -1, -2, -1, 1, -1, 0, 0, -3, -1, 3, 0, -1, 0, -1], [-1, 3, 0, 0, 0, -1, 1, 0, -1, 1, 1, 0, 0, -2, 2, 1, -1, 0], [-1, 1, 2, -2, 1, -1, -1, 0, 0, 0, -1, 3, -1, 0, 0, 0, 2, 1], [0, -2, 0, -2, 1, 1, 0, 0, 0, 0, 1, -2, -1, 0, 1, 0, 0, -1], [-1, 0, -2, 0, -1, 0, 1, -3, 1, 1, 2, -1, 3, -1, -1, 1, -1, -1], [0, 3, 0, 1, 0, 1, 0, 0, 0, 0, 1, 2, 1, 0, -4, 0, 0, -2], [1, 0, 0, -1, 2, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 2], [2, 2, 0, 1, 0, -1, 2, 0, 0, 0, -1, 1, 1, 0, -1, 0, -3, 0], [0, 0, 0, -1, 1, 2, 0, 0, 0, 0, 2, 2, 0, 0, -2, 0, -3, 0], [1, 2, 0, -1, 0, -2, 1, 0, 0, 0, -2, 0, -1, 0, 0, 0, 2, 0], [0, 1, -4, 2, -2, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, -4, -1], [0, 2, 0, 1, 0, 2, 0, 2, 2, 1, -2, 0, -1, 0, 4, 0, 2, 0], [5, -4, 0, 0, -1, 0, -1, 1, -1, 1, 0, -1, -1, 1, 0, 0, -1, 0], [1, 0, 0, 0, 1, 0, 1, 1, -1, 0, 1, 1, 3, 1, 0, -1, -1, 1], [1, 0, 0, 0, 1, 0, -9, 1, -1, -3, -4, 1, -1, 1, 1, -1, 1, 1], [-1, 2, 0, 2, 0, 1, 1, 3, -3, 3, -1, 0, -2, 0, 3, -1, 1, 0], [-2, 3, 1, 2, -2, 1, -2, 1, -2, 0, 0, -1, -2, 1, 0, 0, 0, 0], [0, -3, 0, 0, 0, 2, 0, 4, -2, -2, -2, 0, 0, 5, -1, -1, 2, 0], [1, -2, 0, -1, 3, -2, 1, 0, 0, 0, -2, -3, 2, 0, 4, 0, 1, 1], [-2, 0, 0, 0, -2, 0, 6, -2, 2, 0, -2, -1, 0, -1, 2, 0, 2, 0], [0, -3, -1, -1, 3, 1, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 1], [2, -4, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 1, 0, 1], [-1, 0, 1, 0, -1, 0, -1, 0, 1, -2, -4, 0, -3, 0, 0, -1, 2, 1], [-6, 2, 2, -1, 1, -2, -6, 0, 0, 0, -2, 0, 0, 0, 0, 0, 1, 2], [-2, 0, 0, -3, 0, -3, 2, -4, 2, 0, 3, 0, 3, -1, -3, 2, -3, 0], [2, -3, 2, -4, 2, 0, 2, -4, 2, -2, 0, 1, 2, -1, 0, 3, 2, 3], [-4, -5, 0, -1, 0, 1, 4, -1, 1, -1, -1, 0, 1, -3, -4, 3, 1, 0], [-3, 2, 0, -1, 0, -2, 3, -4, 3, 5, 2, 0, 1, 0, 0, 3, -2, 0], [-8, 0, 1, -2, 2, -4, 2, -3, 2, 3, 0, 2, 2, -2, 0, 3, -3, 3], [-1, 2, -1, -1, 3, 2, 3, -2, 3, 0, 0, 2, 3, -2, 0, 1, 0, 1], [-6, -2, 0, 0, 0, 0, 6, -2, 4, 4, 0, 0, 0, 1, -2, 3, 0, 0], [-3, -1, 0, -2, 0, 3, 3, -4, 2, -4, -3, 0, 2, 2, 2, 0, 3, 0], [2, -2, 2, 0, -2, 2, 2, 0, 0, 0, 2, 3, -2, 0, 0, 0, 0, 4], [0, 5, 2, 0, 0, -1, 0, 0, 0, 0, -1, 2, 0, 0, -4, 0, -2, 4], [1, 0, -1, 0, 1, 0, -3, 0, -1, -2, 2, 2, 3, 2, -6, -2, -4, 2], [-5, 1, -1, -4, 3, -1, 3, -2, 3, 2, 0, 0, 3, 0, 0, 3, -2, 3], [-3, 0, -1, 0, -3, 0, 5, -4, 3, -3, -2, 0, 2, 0, -4, -1, -1, 1], [6, 5, 0, 2, -2, -2, -2, 2, -2, 2, 0, 1, -2, -1, 0, -1, -2, -1], [-1, 0, 2, 0, -1, 0, 7, 1, 1, 0, -1, 3, 1, 3, -4, -1, 1, 1], [-12, -1, 1, -2, 2, 0, 2, -3, 2, 2, 0, -2, 2, 2, 0, 2, -2, 2], [3, 4, 2, -2, 1, 2, 1, -3, 1, -1, 0, -1, 1, 1, 0, 0, 1, 0], [5, -4, 0, 2, 0, 0, -5, 2, -2, -4, 0, 0, -2, -2, -4, 2, 0, 0], [-2, 0, 2, 0, -2, 0, -8, 0, 2, 1, 0, 1, -7, 1, 4, 1, 1, -1], [-5, 2, 0, -3, 0, -2, 5, -5, 5, 3, 2, 0, 3, -6, 0, 2, -2, 0], [-1, -3, 0, 1, 0, 1, 1, 4, -2, 2, -1, 0, -1, 2, -2, 0, 1, 0], [-3, 0, 1, 0, -3, 0, 13, -2, 3, -1, -2, 2, -1, 2, 1, 0, 1, 0], [1, -5, -2, 3, -1, 1, 1, 0, 0, 0, 1, -1, 2, 0, 4, 0, 0, 1], [-3, -2, 4, -2, -2, -2, -3, 0, 0, 0, -2, 2, -4, 0, 4, 0, -2, 2], [-2, 0, 0, 0, -2, 0, 10, -2, 2, 0, 0, -1, 0, -1, 0, 3, 0, -3], [-4, -4, 2, 1, 0, 1, -4, 0, 0, 0, 1, -1, 1, 0, 3, 0, 3, 0], [-1, 0, -3, 0, -1, 0, 3, -4, 1, -1, -1, -2, -1, -2, 4, 1, 0, -1], [2, -1, 2, 0, -6, 0, 2, 0, 0, 0, 0, -5, -6, 0, 1, 0, 4, -3], [4, 4, 0, 1, 0, -1, -4, -2, 2, 4, 1, 0, -1, 1, 3, 2, -1, 0], [-2, -3, 0, -2, 0, -1, 2, -6, 4, 1, 1, 0, 2, 2, -4, 0, -1, 0], [-1, 0, 0, 0, -1, 0, 1, -1, 1, 3, 2, 2, 0, 2, -4, -2, 1, 2], [0, 6, 3, 0, -2, -2, -2, -1, -2, 1, 0, -1, -2, 1, 0, 2, -1, 2], [-4, 0, 1, 1, -2, 0, -2, 1, -2, -1, 0, 5, -2, -5, 0, 2, 1, 2], [-4, 3, 0, 0, 0, 0, 4, 0, 4, 4, 0, 0, 0, 1, 3, -1, 0, 0], [2, -1, -4, -1, 0, 1, 2, 0, 0, 0, 1, -6, -1, 0, 0, 0, -4, -2], [4, 0, 3, 0, 4, 0, -10, 7, -4, 1, -2, -1, 0, -1, 10, -1, 3, 1], [4, 4, -1, 3, -1, 0, 4, 0, 0, 0, 0, -2, 2, 0, -4, 0, -1, -5], [2, 0, 0, 0, 2, 0, -10, 2, -2, 0, 4, 3, 0, 3, 0, 1, -4, -1], [1, -1, 2, -4, -1, -2, -1, -1, -1, -1, 0, -2, -1, 2, 0, 2, 1, 2], [-4, 0, 4, 0, -4, 0, 2, 0, 4, 1, 4, 2, -1, 2, 0, 0, -3, 0], [-1, 8, 0, 2, 0, -4, -1, 0, 0, 0, -4, 0, 2, 0, -4, 0, -2, 0], [4, -1, 0, 0, 0, 1, -4, -1, 3, 1, -1, 0, 0, 3, 0, -1, 1, 0], [1, -4, 0, -1, 0, -2, -1, 3, -5, 1, 2, 0, 1, 6, -6, -5, -2, 0], [-8, 5, 0, 6, -4, 2, -4, 4, -4, 0, 0, 5, -4, -5, 0, 1, 0, 1], [14, 0, -2, 2, 2, -2, 2, 0, 2, 0, 0, -2, 2, 2, 0, 0, 0, 0], [-8, 2, 0, 0, 0, -4, 8, -4, 4, 2, 4, 0, 0, 5, -2, -2, -4, 0], [9, -2, 0, 1, 0, 0, -9, 1, -3, -3, 0, 0, -1, 6, -2, 0, 0, 0], [-6, 3, 0, 2, 0, -2, 6, -2, 2, 2, 2, 0, -2, 3, 1, 1, -2, 0], [-8, 4, 4, -4, 7, -1, -8, 0, 0, 0, -1, 4, 3, 0, -3, 0, 0, 3], [3, -5, 0, -3, 1, -3, 3, 0, 0, 0, -3, -1, -2, 0, 8, 0, 2, -1], [2, 0, 0, 0, 2, 0, -2, 2, -2, 4, 4, -4, -2, -4, 8, 0, 0, 0], [-7, 5, 2, 4, -1, 2, -1, -1, -1, -3, 0, 4, -1, -4, 0, 0, 3, 0], [-10, -1, 0, -6, 0, -3, 10, -5, 3, -1, 3, 0, 6, 3, -4, 3, -3, 0], [8, 7, 2, -2, 0, -2, 0, -2, 0, 0, 0, -1, 0, 1, 0, -3, 0, -3], [-4, 0, -2, 0, -4, 0, 12, -6, 4, 0, 1, -4, 0, -4, -4, 2, -1, -2], [-5, 0, 2, 0, -5, 0, 5, -3, 5, -1, -2, 1, -5, 1, 1, -1, 1, 1], [2, -2, -3, 7, 0, 6, 0, 3, 0, 3, 0, -1, 0, 1, 0, 0, -3, 0], [-6, 2, 0, 0, 0, 2, 6, 2, -6, -6, -2, 0, 0, 1, 4, -5, 2, 0], [5, -4, 0, 3, 0, 8, -5, 4, 0, -2, -8, 0, -3, -4, 4, 0, 8, 0], [-2, 0, 3, 0, -2, 0, -20, 1, 2, 0, 0, 1, -4, 1, -3, 3, 0, -3], [1, 1, 6, -3, 0, -3, 1, 0, 0, 0, -3, -4, -3, 0, 2, 0, 7, 2], [0, 0, 1, 0, 0, 0, -6, 1, 0, -1, 0, 3, 4, 3, -8, -4, -1, 4], [-3, 0, 3, 0, -3, 0, 1, 0, 3, -1, -6, 0, -4, 0, -4, 3, 5, -3], [10, 2, -8, -1, 7, 0, 10, 0, 0, 0, 0, 2, 6, 0, -2, 0, -5, 3], [1, 0, -3, 0, 1, 0, 19, -2, -1, 2, 4, -7, 2, -7, 4, 2, -2, -2], [8, -5, -5, 2, -1, 7, 8, 0, 0, 0, 7, 0, 1, 0, -2, 0, -2, -7], [-6, -5, 2, -3, 1, 0, -6, 0, 0, 0, 0, -8, -2, 0, 5, 0, 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[0, 0, 1, 0, 0, 0, 2, 1, 0, -2, -6, -3, 0, -3, 5, 1, 4, -1], [0, -6, 0, 8, 0, 10, 0, 8, 4, -6, -10, 0, -8, -7, 4, -3, 10, 0], [-4, 2, 4, -8, 8, -4, -4, 0, 0, 0, -4, -5, 0, 0, 2, 0, -4, -4], [9, 3, 8, 1, -1, -3, 9, 0, 0, 0, -3, 13, 0, 0, 0, 0, -6, 5], [-1, 4, -2, 2, -4, -4, -1, 0, 0, 0, -4, -10, -2, 0, 0, 0, 2, 0], [0, 0, -6, 0, 0, 0, 4, -6, 0, -4, -4, -3, 6, -3, -12, 5, 0, -5], [4, 8, 14, -3, -4, -5, 4, 0, 0, 0, -5, 7, -7, 0, -3, 0, 9, 10], [6, 0, -2, 0, 0, -2, 6, 0, 0, 0, -2, -7, 0, 0, 2, 0, -10, -3], [-8, 1, -12, 1, 1, 6, -8, 0, 0, 0, 6, 0, 2, 0, -7, 0, -11, 2], [-5, 10, 0, 1, 0, 0, 5, -7, -5, -5, 0, 0, -1, -2, 10, -5, 0, 0], [0, 0, -5, 0, 0, 0, -2, -5, 0, -6, 4, -4, 11, -4, -4, 1, -10, -1], [23, -2, -5, 8, -5, -2, -5, 10, -5, 4, 0, -4, -5, 4, 0, 3, -4, 3], [-14, -4, 0, -2, 0, 2, 14, -10, 14, 6, -2, 0, 2, -3, -2, 0, 2, 0], [-8, -18, 4, -11, -2, 6, -2, -2, -2, -10, 0, -2, -2, 2, 0, -6, 10, -6], [13, 1, 0, -6, 5, -13, 5, -5, 5, 3, 0, 3, 5, -3, 0, 3, -3, 3], [3, -1, 0, -3, 0, -2, -3, 3, 7, 7, 2, 0, 3, 6, -3, 4, -2, 0], [-1, 3, -4, 0, 1, 5, -1, 0, 0, 0, 5, -11, 1, 0, -8, 0, -6, 1], [4, 0, -3, 0, 4, 0, 14, 1, -4, 2, 2, 1, 0, 1, 19, -7, 0, 7], [20, 13, 2, 8, 0, 7, 0, -2, 0, 3, 0, 8, 0, -8, 0, 2, -3, 2], [9, 3, 12, -1, 2, -5, 9, 0, 0, 0, -5, 0, 1, 0, 2, 0, 7, 4], [-9, 2, 0, -11, 0, -4, 9, -11, 7, -7, 4, 0, 11, 12, -2, 1, -4, 0], [12, -1, -4, -4, 4, 12, 4, 0, 4, 0, 0, 1, 4, -1, 0, -1, 0, -1], [12, 1, 0, -3, 0, 3, -12, 1, 7, 3, -3, 0, 3, 3, 4, -7, 3, 0], [-24, 1, -2, -4, 8, 11, 8, -6, 8, -1, 0, 6, 8, -6, 0, 4, 1, 4], [-3, -4, 0, 4, 0, -4, 3, -2, -1, 5, 4, 0, -4, 2, -8, -1, -4, 0], [-12, 3, 0, -10, 0, 0, 12, -12, 10, 4, 0, 0, 10, 3, 3, -1, 0, 0], [-11, 3, 0, 1, 1, -3, -11, 0, 0, 0, -3, 3, 2, 0, 0, 0, 0, -5], [1, 0, -4, 0, 1, 0, -9, -3, -1, -1, 4, 1, 3, 1, 0, -5, -5, 5], [3, 11, 0, 2, 5, 4, 5, -5, 5, 3, 0, 12, 5, -12, 0, 6, -3, 6], [7, 0, 2, 0, 7, 0, -33, 9, -7, 2, -1, 5, 5, 5, 8, -3, 3, 3], [-10, -6, 0, -7, 0, 6, 10, -13, 1, -3, -6, 0, 7, -7, 0, 3, 6, 0], [2, 5, 4, 2, -6, 2, -6, 2, -6, 0, 0, -3, -6, 3, 0, 5, 0, 5], [11, -1, 0, 8, 0, -1, -11, 14, -11, 7, 1, 0, -8, 0, -2, 3, -1, 0], [-3, 8, 5, 4, -7, 8, -7, 2, -7, -6, 0, 5, -7, -5, 0, -2, 6, -2], [-3, 9, 0, 2, 0, 5, 3, -2, 6, -2, -5, 0, -2, -12, 14, 4, 5, 0], [6, 4, 6, -2, -2, -4, 6, 0, 0, 0, -4, 7, -4, 0, 0, 0, 10, 4], [1, 0, -4, 0, 1, 0, 11, -3, -1, -7, 2, 5, 5, 5, -7, 1, -9, -1], [13, 10, -8, 10, -3, 2, 13, 0, 0, 0, 2, -3, 7, 0, -12, 0, -13, -3], [-1, 0, 5, 0, -1, 0, 7, 4, 1, 7, -4, 10, -4, 10, 0, -1, 11, 1], [-10, 6, 2, 4, -6, 4, -10, 0, 0, 0, 4, 7, -2, 0, -10, 0, 2, -5], [15, -2, 0, -5, 0, -6, -15, -3, 1, -5, 6, 0, 5, 8, -8, -9, -6, 0], [8, -8, -3, 4, 2, 4, 2, 1, 2, -7, 0, 2, 2, -2, 0, -4, 7, -4], [-9, 6, 0, -4, 0, -1, 9, 3, 7, 5, 1, 0, 4, 4, 5, 1, -1, 0], [-15, -6, 6, -3, -5, -2, -15, 0, 0, 0, -2, -9, -8, 0, 8, 0, 7, -7], [6, 0, -5, 0, 6, 0, -8, 1, -6, -4, -6, -3, 10, -3, -3, 5, 2, -5], [3, 0, -4, 0, 3, 0, 1, -1, -3, -1, 8, -8, 1, -8, -16, 1, -9, -1], [0, -9, -3, 3, 1, 11, 0, 0, 0, 0, 11, 8, 4, 0, -2, 0, -8, -1], [15, -10, -5, 3, 1, -6, 1, 4, 1, -3, 0, -1, 1, 1, 0, 2, 3, 2], [8, 0, 1, 0, 8, 0, -2, 9, -8, 1, 2, -3, -6, -3, -2, 2, -1, -2], [6, 0, -7, 0, 6, 0, 4, -1, -6, -2, 0, 1, 3, 1, -8, 4, -2, -4], [-12, -15, 0, -1, 0, -1, 12, -8, 6, -1, 1, 0, 1, 14, -16, 0, -1, 0], [-12, 3, -5, -2, 6, -8, 6, -1, 6, 6, 0, 0, 6, 0, 0, 2, -6, 2], [-3, -1, 0, -3, 0, -1, 3, -2, 3, 3, 1, 0, 3, -12, -2, 9, -1, 0], [7, -2, 0, -1, 0, -2, -7, 3, 7, -1, 2, 0, 1, 0, -4, -6, -2, 0], [28, -11, 3, -6, 0, 11, 0, -3, 0, -8, 0, 1, 0, -1, 0, -4, 8, -4], [-12, 14, 2, 1, 4, -6, -12, 0, 0, 0, -6, 0, 5, 0, -8, 0, -9, 2], [2, 0, -8, 0, 2, 0, 14, -6, -2, 2, 8, 9, 2, 9, -8, -4, -6, 4], [4, 0, -1, 0, 4, 0, 12, 3, -4, -3, 1, -11, 2, -11, 12, -4, -4, 4], [4, 1, 0, 5, -7, 2, 4, 0, 0, 0, 2, 2, -2, 0, -3, 0, 3, -2], [-2, -1, -4, 2, 6, -6, 6, -2, 6, 6, 0, 3, 6, -3, 0, 7, -6, 7], [-23, 7, 0, -10, 0, -9, 23, -10, 8, 6, 9, 0, 10, 8, -2, -2, -9, 0], [3, 0, -4, 0, 3, 0, -21, -1, -3, -8, 1, 3, 9, 3, -12, 1, -9, -1], [-10, 12, -2, 6, 2, -6, 2, 0, 2, 6, 0, 0, 2, 0, 0, 6, -6, 6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1600_j_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1600_2_j_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_1600_2_j_d(:prec:=18)
    chi := MakeCharacter_1600_j();
    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_1600_2_j_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_1600_2_j_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1600_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![256, 0, 2704, 0, 10624, 0, 19744, 0, 18624, 0, 9464, 0, 2656, 0, 408, 0, 32, 0, 1]>],Snew);
    return Vf;
end function;