
// Make newform 1776.2.q.q in Magma, downloaded from the LMFDB on 30 April 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1776_q();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1776_q_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_1776_2_q_q();".
// 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_1776_2_q_q();".
// 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 := [9, -42, 199, -40, 127, -12, 67, -2, 9, 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, 2, 0, 0, 0, 0, 0, 0, 0, 0], [6161881, -398034, 1876446, -162941, 994032, -71604, 111041, -6318, 14742, -777], [6197315, -1331505, 6277095, -1686958, 3325240, -239530, 450451, -21135, 49315, 1232], [-330792, 20979, -98901, 790705, -52392, 3774, -3116, 333, -777, -1601], [6080223, -1226610, 5782590, 729699, 3063280, -220660, 434871, -19470, 45430, -6773], [4568151, -21645833, 2038445, -13846352, 1311846, -7378340, 219529, -990045, 4803, -110264], [-18259962, 86477695, -22256023, 54335863, -5040492, 27160594, -816530, 3616977, 13161, 395017], [-213039, -85389230, 7135046, -54896585, 2265288, -29298548, 544993, -3941226, -25014, -438725], [-117819, -27043556, 1577650, -19195081, -1272056, -10288740, -124643, -1405052, -51426, -154917]];
    Rf_basisdens := [1, 1, 1536868, 1536868, 768434, 1536868, 4610604, 4610604, 4610604, 1536868];
    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_1776_q();"
function MakeCharacter_1776_q()
    N := 1776;
    order := 3;
    char_gens := [223, 1333, 593, 1297];
    v := [3, 3, 3, 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_1776_q_Hecke();"
function MakeCharacter_1776_q_Hecke( : Kf := false)
    N := 1776;
    order := 3;
    char_gens := [223, 1333, 593, 1297];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 1, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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], [1, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 1, 0, 0, 0, 1, 0, 0], [1, 0, -1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, -1, 0, 0, 1], [0, 0, -1, 0, 0, 1, 1, 0, -1, 1], [1, -1, 0, 0, 0, 0, -1, 0, -1, 0], [1, 0, 0, 1, -1, 1, 0, 0, 0, 0], [-1, 0, 1, 0, -1, -1, 0, 0, 0, 0], [2, 0, -1, 0, -1, -1, 0, 0, 0, 0], [3, 1, 0, 1, 0, 0, -1, 0, 1, 1], [-4, 0, 0, 1, 0, 0, 4, 1, 1, -2], [-1, 0, 2, 0, 0, -1, 0, 0, 0, 0], [2, 0, -1, 1, -2, -1, 0, 0, 0, 0], [0, 0, -1, 0, 0, 1, -4, -1, -1, 1], [0, 0, 1, 0, 0, -2, -1, 0, 1, -2], [1, 0, 0, 1, 0, 0, -1, 1, 2, 2], [1, 1, 0, 1, 0, 0, -1, 1, 1, 4], [2, 1, 0, 2, 0, 0, -2, 2, 0, 2], [4, 0, 1, 0, -1, -1, 0, 0, 0, 0], [1, -1, 0, -1, 0, 0, -1, -1, 3, -2], [0, -1, 2, 0, -1, 0, 0, -2, 2, 0], [0, 1, -2, 0, 1, 0, 7, 1, -2, 0], [-2, 0, 1, 0, -1, 0, 0, 0, 0, 0], [6, 0, 1, 1, 1, 0, 0, 0, 0, 0], [-3, 0, 0, -1, 0, 1, 0, 0, 0, 0], [-2, -2, 0, -2, 0, 0, 2, -2, 0, 0], [0, 1, -1, 0, 1, 0, 0, 2, -1, 0], [0, 1, -2, 0, 1, -2, -8, -2, -2, -2], [0, 1, 1, 0, 1, -1, 2, 2, 1, -1], [0, 2, 0, 0, 2, 2, -2, 2, 0, 2], [8, 0, -1, 1, 1, 2, 0, 0, 0, 0], [0, 3, -1, 0, 3, 1, -6, 0, -1, 1], [-2, 0, 1, -1, -2, -2, 0, 0, 0, 0], [4, 2, 0, 1, 0, 0, -4, 1, -2, 0], [0, -2, 0, 0, -2, 3, -4, -1, 0, 3], [0, -1, -3, 0, -1, 5, 0, -1, -3, 5], [8, -2, 0, -1, 0, 0, -8, -1, -3, 1], [0, -1, 2, 0, -1, 1, -8, 0, 2, 1], [3, 0, 3, 0, 1, 2, 0, 0, 0, 0], [-5, 3, 0, 1, 0, 0, 5, 1, -1, 1], [4, 0, 1, -1, 1, -1, 0, 0, 0, 0], [-6, 0, 0, 1, 0, 1, 0, 0, 0, 0], [0, -1, -1, 0, -1, -3, 1, 0, -1, -3], [-3, 0, 1, -3, -1, -2, 0, 0, 0, 0], [0, 0, 0, -1, -4, 0, 0, 0, 0, 0], [5, 0, -3, 1, 1, 0, 0, 0, 0, 0], [-6, -1, 0, 3, 0, 0, 6, 3, -3, -1], [-8, 0, 0, -1, 0, 0, 8, -1, 4, -3], [5, 0, -1, 4, -2, 3, 0, 0, 0, 0], [0, -3, -2, 0, -3, -1, 11, -3, -2, -1], [2, 0, 0, -3, 0, 0, -2, -3, -2, 0], [-7, 0, -2, 3, 0, 3, 0, 0, 0, 0], [0, 0, -2, 0, 0, -1, 6, 2, -2, -1], [7, -1, 0, -2, 0, 0, -7, -2, -3, -2], [-1, 0, 5, -2, 0, 0, 0, 0, 0, 0], [0, 2, 4, 0, 2, -3, 7, 0, 4, -3], [2, -2, 0, -4, 0, 0, -2, -4, 2, -5], [0, 0, 3, 0, 0, -3, 1, -4, 3, -3], [-7, 1, 0, 3, 0, 0, 7, 3, -3, 4], [-3, 2, 0, 3, 0, 0, 3, 3, 0, 8], [-10, 0, 0, 1, 2, -2, 0, 0, 0, 0], [0, 4, 2, 0, 4, -2, -2, 0, 2, -2], [0, 4, 2, 0, 4, 2, 1, -2, 2, 2], [0, 1, -2, 0, 1, -1, 2, 0, -2, -1], [0, 1, 1, 0, 1, -4, 5, 5, 1, -4], [-7, -2, 0, 2, 0, 0, 7, 2, -4, -4], [0, 0, 4, -2, -2, 0, 0, 0, 0, 0], [0, 1, 3, 0, 1, -5, -9, 4, 3, -5], [0, -2, 2, 0, -2, 0, 5, -1, 2, 0], [-1, 0, -2, 3, 0, 7, 0, 0, 0, 0], [13, -2, 0, 0, 0, 0, -13, 0, 2, 1], [-1, 1, 0, 1, 0, 0, 1, 1, -3, 7], [0, 6, 0, 0, 6, 2, -4, 0, 0, 2], [-15, -1, 0, -1, 0, 0, 15, -1, -2, -1], [7, 2, 0, -2, 0, 0, -7, -2, 3, 1], [8, 0, -1, 2, -1, -6, 0, 0, 0, 0], [11, 0, 0, 5, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3, -2, 0, 0], [0, 1, -3, 0, 1, -7, 0, -3, -3, -7], [4, 0, 2, -2, 4, -3, 0, 0, 0, 0], [-17, 4, 0, 2, 0, 0, 17, 2, -1, 0], [6, 0, -1, -3, 7, 0, 0, 0, 0, 0], [-8, 5, 0, -2, 0, 0, 8, -2, 1, 5], [-7, 0, 3, -4, -1, 0, 0, 0, 0, 0], [-21, -1, 0, 0, 0, 0, 21, 0, -3, -4], [5, 1, 0, 0, 0, 0, -5, 0, 7, -1], [0, 2, -3, 0, 2, 2, 9, 0, -3, 2], [7, 0, 0, -4, 0, 0, -7, -4, 0, -5], [7, 0, 0, -5, 3, 3, 0, 0, 0, 0], [0, -3, -3, 0, -3, 3, 0, -3, -3, 3], [-22, 0, 1, -1, 3, 1, 0, 0, 0, 0], [-8, 0, -1, -3, 1, -1, 0, 0, 0, 0], [-3, -3, 0, 2, 0, 0, 3, 2, 5, -6], [0, -2, 4, 0, -2, 1, 17, 3, 4, 1], [0, 3, 1, 0, 3, 5, -1, -6, 1, 5], [0, -3, 0, 3, 0, 0, 0, 3, -3, -3], [-2, 0, 0, -5, 0, 0, 2, -5, -2, -6], [-18, 0, 7, 2, -1, 4, 0, 0, 0, 0], [13, 0, -1, 5, -5, 0, 0, 0, 0, 0], [0, 5, 0, 0, 5, 3, -20, 2, 0, 3], [-20, 0, -6, -2, -3, 4, 0, 0, 0, 0], [-7, 0, 1, -6, 3, 3, 0, 0, 0, 0], [0, 1, -1, 0, 1, 3, -8, 2, -1, 3], [0, -3, -9, 0, -3, -3, 10, -3, -9, -3], [-6, -2, 0, -1, 0, 0, 6, -1, 1, -5], [-14, 0, -3, -5, 1, 6, 0, 0, 0, 0], [-8, -3, 0, -2, 0, 0, 8, -2, 0, 4], [0, -2, -2, 0, -2, -2, 13, 2, -2, -2], [0, -1, 1, 0, -1, -1, 6, 1, 1, -1], [0, 5, 3, 0, 5, -1, -5, 0, 3, -1], [0, -1, -5, 0, -1, 8, -5, -2, -5, 8], [12, 0, 4, 3, 2, -8, 0, 0, 0, 0], [0, 1, 3, 0, 1, 2, -15, -7, 3, 2], [0, 3, 4, 0, 3, -6, 3, -1, 4, -6], [22, 0, 0, 3, -2, -6, 0, 0, 0, 0], [-1, 2, 0, 1, 0, 0, 1, 1, -2, -9], [1, 3, 0, -2, 0, 0, -1, -2, 3, 1], [18, 2, 0, 5, 0, 0, -18, 5, 1, -1], [-3, 1, 0, 3, 0, 0, 3, 3, 5, 5], [-9, 2, 0, 3, 0, 0, 9, 3, -4, -3], [7, 0, -1, 0, 2, -13, 0, 0, 0, 0], [0, -5, 4, 0, -5, 5, 3, -1, 4, 5], [0, 0, 4, 0, 0, -3, -5, -6, 4, -3], [0, 0, 6, 0, 0, -2, -2, -4, 6, -2], [14, 0, 2, -1, 6, -4, 0, 0, 0, 0], [0, -4, -1, 0, -4, -11, 2, -1, -1, -11], [-15, 0, 0, 2, 0, 0, 15, 2, 8, -5], [9, 1, 0, -1, 0, 0, -9, -1, 9, 0], [-8, 0, -5, -5, 1, 9, 0, 0, 0, 0], [17, 4, 0, 1, 0, 0, -17, 1, 10, 5], [3, 0, -2, 0, 4, -1, 0, 0, 0, 0], [0, -5, 1, 0, -5, 3, 11, -5, 1, 3], [0, -2, 0, 0, -2, 4, -9, 3, 0, 4], [8, 0, -6, 3, 0, -2, 0, 0, 0, 0], [13, -1, 0, 0, 0, 0, -13, 0, 1, -1], [16, 0, -4, 5, -2, -2, 0, 0, 0, 0], [0, 0, 7, 0, 0, -10, -1, 6, 7, -10], [0, 1, 0, -3, 0, 0, 0, -3, -3, 9], [-4, 4, 0, 0, 0, 0, 4, 0, 0, -4], [-18, -6, 0, -4, 0, 0, 18, -4, -8, -4], [0, 1, 7, 0, 1, -5, 4, 6, 7, -5], [-2, 7, 0, 2, 0, 0, 2, 2, 2, 10], [0, 2, -6, 0, 2, 1, 29, 2, -6, 1], [0, -3, -7, 0, -3, 5, -14, 3, -7, 5], [0, -6, 2, 0, -6, 7, -4, -6, 2, 7], [-6, 0, 0, 0, -4, -1, 0, 0, 0, 0], [18, 0, -3, -4, -1, -3, 0, 0, 0, 0], [0, 2, -5, 0, 2, 4, -13, 0, -5, 4], [26, 0, 3, 0, 1, 4, 0, 0, 0, 0], [7, 3, 0, 1, 0, 0, -7, 1, 12, 6], [-26, -1, 0, -1, 0, 0, 26, -1, 3, -1], [3, 0, 0, 3, -2, -7, 0, 0, 0, 0], [-8, -3, 0, 4, 0, 0, 8, 4, 1, -5], [-18, 0, 5, 5, -6, -1, 0, 0, 0, 0], [-13, 0, 11, 3, -5, -8, 0, 0, 0, 0], [22, 4, 0, 6, 0, 0, -22, 6, -2, 1], [0, -3, 1, 0, -3, -4, -8, 2, 1, -4], [0, 1, -8, 0, 1, -8, 3, -1, -8, -8], [0, 3, -5, 0, 3, 5, 20, -3, -5, 5], [0, -4, -2, 0, -4, 2, -2, -2, -2, 2], [2, 3, 0, 4, 0, 0, -2, 4, -1, -7], [0, 4, 3, 0, 4, 9, -6, -3, 3, 9], [0, 0, -2, 0, 0, 1, 23, 3, -2, 1], [0, 0, 9, 0, 0, 1, -22, 1, 9, 1], [-10, 0, -3, -5, 3, 3, 0, 0, 0, 0], [0, -3, -7, 0, -3, -1, 18, -1, -7, -1], [-15, 0, 7, -1, 5, 11, 0, 0, 0, 0], [7, 0, -7, 0, -9, -9, 0, 0, 0, 0], [0, 5, -3, 0, 5, 3, -18, 3, -3, 3], [0, -5, -7, 0, -5, 1, -15, -1, -7, 1], [-6, 2, 0, 5, 0, 0, 6, 5, -5, 1], [1, 4, 0, -1, 0, 0, -1, -1, 4, -6], [5, -3, 0, -2, 0, 0, -5, -2, 5, 2], [10, 3, 0, -3, 0, 0, -10, -3, 7, -3], [0, -5, -1, 0, -5, 5, 14, -2, -1, 5], [0, 3, -3, 0, 3, 4, -28, 7, -3, 4], [-2, 0, 2, -4, 2, 0, 0, 0, 0, 0], [8, -5, 0, -1, 0, 0, -8, -1, 5, -2], [7, 0, -4, 5, -8, -7, 0, 0, 0, 0], [-2, 4, 0, 3, 0, 0, 2, 3, 7, 1], [0, -3, 5, 0, -3, -9, -11, -8, 5, -9], [-19, 1, 0, 5, 0, 0, 19, 5, 0, -12], [-6, 2, 0, 1, 0, 0, 6, 1, -1, 7], [-23, 0, -6, 1, -9, 6, 0, 0, 0, 0], [-27, 3, 0, 2, 0, 0, 27, 2, -3, -1], [6, -2, 0, 7, 0, 0, -6, 7, 6, -10], [14, 6, 0, 0, 0, 0, -14, 0, -2, -3], [-6, -8, 0, -3, 0, 0, 6, -3, 3, -1], [-11, 0, 5, 4, 7, 7, 0, 0, 0, 0], [0, 7, -4, 0, 7, 3, -1, -1, -4, 3], [-13, -12, 0, -6, 0, 0, 13, -6, 0, 1], [6, 2, 0, -3, 0, 0, -6, -3, 3, 8], [28, 8, 0, 12, 0, 0, -28, 12, 4, 2], [0, 2, -5, 0, 2, 1, 25, -2, -5, 1], [0, -5, 1, 0, -5, 5, 7, 5, 1, 5], [7, 0, -5, -6, 1, 0, 0, 0, 0, 0], [11, 10, 0, 5, 0, 0, -11, 5, 6, 3], [0, 4, -5, 0, 4, 9, 2, 3, -5, 9], [-6, 0, -2, -8, 3, -4, 0, 0, 0, 0], [-12, 0, 5, 4, -9, -11, 0, 0, 0, 0], [0, 3, 7, 0, 3, -7, 21, -6, 7, -7], [0, -1, 9, 0, -1, 0, -8, 0, 9, 0], [17, 0, 12, -7, -2, 3, 0, 0, 0, 0], [11, -12, 0, -2, 0, 0, -11, -2, 1, -1], [0, 6, 2, 0, 6, 4, 16, -7, 2, 4], [0, -6, 5, 0, -6, -14, -25, 2, 5, -14], [-11, 0, 1, 2, 6, -3, 0, 0, 0, 0], [2, -4, 0, 3, 0, 0, -2, 3, 6, 2], [0, 3, 7, 0, 3, 7, -10, 4, 7, 7], [-11, 0, 4, -7, 1, 4, 0, 0, 0, 0], [12, 0, -2, -3, 2, -2, 0, 0, 0, 0], [0, -5, 2, 0, -5, 0, -2, 0, 2, 0], [-37, 1, 0, -5, 0, 0, 37, -5, 4, -3], [-14, 0, 3, 2, -7, -17, 0, 0, 0, 0], [-6, -2, 0, 5, 0, 0, 6, 5, -10, 2], [-16, 0, -2, 0, 3, -3, 0, 0, 0, 0], [0, 1, 6, 0, 1, -10, 6, -6, 6, -10], [6, 0, 0, -7, 0, 0, -6, -7, -5, -1], [0, 0, 2, 0, 0, 2, -2, -2, 2, 2], [-10, -1, 0, -12, 0, 0, 10, -12, 9, -7], [15, 6, 0, 2, 0, 0, -15, 2, 7, 7], [0, 7, 5, 0, 7, 7, -6, -3, 5, 7], [0, 2, -11, 0, 2, -3, -14, -5, -11, -3], [-9, 0, -2, -1, 2, -6, 0, 0, 0, 0], [4, 0, -1, -9, 1, 11, 0, 0, 0, 0], [-21, 5, 0, -2, 0, 0, 21, -2, -7, 4], [22, 2, 0, -8, 0, 0, -22, -8, -2, 8], [-29, 0, 7, -10, 8, 6, 0, 0, 0, 0], [-6, 0, -5, 2, -1, 8, 0, 0, 0, 0], [0, -4, -10, 0, -4, -7, 19, -4, -10, -7], [0, 1, -3, 0, 1, -15, 10, 4, -3, -15], [-6, 0, 12, -6, 4, -1, 0, 0, 0, 0], [0, -11, 0, -7, 0, 0, 0, -7, -1, 3], [-16, 4, 0, 7, 0, 0, 16, 7, 1, 11], [0, -1, 7, 0, -1, -10, 5, 4, 7, -10], [-10, -3, 0, 3, 0, 0, 10, 3, 5, 4], [-6, 2, 0, 6, 0, 0, 6, 6, 12, -2], [-9, 0, -4, 5, 2, 11, 0, 0, 0, 0], [-25, 0, 6, -7, 6, 11, 0, 0, 0, 0], [14, 0, 2, 3, 0, 6, 0, 0, 0, 0], [6, 0, -16, 1, 2, 10, 0, 0, 0, 0], [-37, -5, 0, -11, 0, 0, 37, -11, -3, -5], [-13, 0, 8, 1, -1, -4, 0, 0, 0, 0], [-5, 4, 0, 3, 0, 0, 5, 3, -2, -3], [-33, 6, 0, 6, 0, 0, 33, 6, -8, 13], [0, 4, -2, 0, 4, 4, 18, 8, -2, 4], [0, 5, 7, 0, 5, 9, -32, -3, 7, 9], [21, 0, -1, -4, -2, 8, 0, 0, 0, 0], [29, 0, -9, 3, -1, 9, 0, 0, 0, 0], [-6, 0, 0, 6, -8, -1, 0, 0, 0, 0], [0, 2, 6, 0, 2, 1, -21, -3, 6, 1], [-15, 5, 0, 5, 0, 0, 15, 5, 3, 6], [0, -6, -17, 0, -6, -1, -2, -5, -17, -1], [0, 3, 8, 0, 3, 4, -28, -4, 8, 4], [3, 1, 0, -3, 0, 0, -3, -3, -7, -6], [-26, 0, 11, 5, -3, 1, 0, 0, 0, 0], [0, -10, -8, 0, -10, 0, 5, -3, -8, 0], [3, 0, -1, -9, 3, -3, 0, 0, 0, 0], [0, 5, -1, 0, 5, -3, -32, -4, -1, -3], [0, -8, -3, 0, -8, -9, -4, 1, -3, -9], [17, 9, 0, 8, 0, 0, -17, 8, -7, 6], [0, 1, 5, 0, 1, 1, -2, -2, 5, 1], [16, 1, 0, 9, 0, 0, -16, 9, 7, -1], [-7, -5, 0, -11, 0, 0, 7, -11, -1, -14], [-38, -5, 0, -6, 0, 0, 38, -6, -2, 5], [2, -1, 0, 13, 0, 0, -2, 13, 1, -6], [0, -2, -2, 0, -2, 1, -21, 10, -2, 1], [-11, 0, 1, 2, 0, 10, 0, 0, 0, 0], [0, -1, 11, 0, -1, -5, 22, 4, 11, -5], [8, 0, 13, -4, 7, -3, 0, 0, 0, 0], [37, 0, -2, 6, -2, 4, 0, 0, 0, 0], [0, -6, 8, 0, -6, -3, -19, -11, 8, -3], [16, 0, -3, 2, -1, -14, 0, 0, 0, 0], [-9, 2, 0, -1, 0, 0, 9, -1, 14, -13], [27, 0, -6, -3, 3, -1, 0, 0, 0, 0], [7, 3, 0, 1, 0, 0, -7, 1, -13, 8], [0, 1, -1, 0, 1, -12, -28, -3, -1, -12], [0, -9, 0, 0, -9, 2, 2, -8, 0, 2], [12, 0, 7, -5, 10, -3, 0, 0, 0, 0], [14, 0, 0, 0, 0, 0, -14, 0, 6, 4], [1, 4, 0, 2, 0, 0, -1, 2, -7, 10], [12, 0, -1, -3, 1, -9, 0, 0, 0, 0], [0, 4, 10, 0, 4, -10, 12, 7, 10, -10], [0, 7, -3, 0, 7, 0, -15, -6, -3, 0], [-39, 0, 2, -6, 6, -4, 0, 0, 0, 0], [54, 0, 1, 1, 1, 3, 0, 0, 0, 0], [-38, 0, -1, -10, 1, 9, 0, 0, 0, 0], [0, -7, 1, 0, -7, -3, -35, -2, 1, -3], [-27, 0, 9, 0, 10, 14, 0, 0, 0, 0], [0, 0, -4, 0, 0, -7, 45, 7, -4, -7], [-4, 0, 2, 4, -1, 5, 0, 0, 0, 0], [-26, 3, 0, -5, 0, 0, 26, -5, 3, -11], [0, -6, 0, 0, -6, -20, -29, -6, 0, -20], [0, 1, -4, 0, 1, 6, -11, -3, -4, 6], [35, 0, 1, -1, -7, -2, 0, 0, 0, 0], [0, 4, -10, 0, 4, 4, 27, 7, -10, 4], [-9, -4, 0, 9, 0, 0, 9, 9, 6, 3], [18, 0, -1, 10, -7, -3, 0, 0, 0, 0], [-10, -5, 0, 8, 0, 0, 10, 8, -1, -2], [-10, 0, 11, 7, 4, -1, 0, 0, 0, 0], [-11, 0, 1, 4, 5, -6, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1776_q_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1776_2_q_q();".
// 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_1776_2_q_q(:prec:=10)
    chi := MakeCharacter_1776_q();
    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_1776_2_q_q();".
// 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_1776_2_q_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1776_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![144, -336, 1036, 252, 809, -194, 210, -14, 18, -2, 1]>,<11,R![48, 88, 26, -17, -3, 1]>],Snew);
    return Vf;
end function;