
// Make newform 1386.2.c.b in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1386_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1386_c_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_1386_2_c_b();".
// 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_1386_2_c_b();".
// 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 := [729, 0, 324, -216, 153, -48, 88, -16, 17, -8, 4, 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], [567, 13878, -3393, 3810, -4414, 1684, -944, 1004, -301, 42, -97, 6], [-502281, 190674, -206631, 165492, -120456, 47456, -44516, 16600, -7417, 3122, -1767, 20], [-145476, 21627, -46692, 32661, -22400, 6450, -9806, 2100, -1156, 655, -206, -89], [364257, -127413, 22599, -117567, 47184, -12382, 22222, -8648, -535, -2341, 489, 335], [-328293, 33291, -71307, 51597, -41190, -5602, -14324, 1168, -97, 35, -495, -421], [0, 1863, -648, 927, -456, 448, -172, 170, -50, 37, -12, 7], [-324648, -46737, -37800, 29169, -36216, -14636, -14530, -1486, 808, -263, -510, -707], [-398763, -56619, -68607, 35775, -36960, -16160, -20518, -3706, 109, -641, -489, -761], [155925, -126819, 94437, -72417, 61168, -29782, 19430, -10598, 4981, -1869, 1015, -267], [-157221, -332667, 161433, -76689, 99360, -77362, 18958, -24722, 12179, -3481, 2067, -1471], [-194643, 20439, -35091, 34377, -22962, -1634, -9547, 533, -47, 208, -264, -305]];
    Rf_basisdens := [1, 11502, 34506, 11502, 34506, 34506, 486, 34506, 34506, 11502, 34506, 5751];
    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_1386_c();"
function MakeCharacter_1386_c()
    N := 1386;
    order := 2;
    char_gens := [155, 199, 1135];
    v := [1, 2, 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_1386_c_Hecke();"
function MakeCharacter_1386_c_Hecke(Kf)
    N := 1386;
    order := 2;
    char_gens := [155, 199, 1135];
    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]];
    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 := [[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, 1, 0, 0, 0, -1, 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, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1], [1, 0, 0, 0, 1, 0, 1, 0, -1, -1, 1, 0], [0, 1, 0, -1, 0, -1, 0, 0, 0, 0, 0, 1], [0, 0, 0, 2, 0, 0, 0, -1, 0, 0, 0, -1], [-1, 0, 1, 0, 0, 0, 3, 0, 1, 0, -2, 0], [1, 0, 0, 0, 0, 0, -1, 0, -2, -1, 1, 0], [2, 0, -2, 0, 0, 0, 0, 0, 2, 0, 0, 0], [3, 0, 0, 0, 0, 0, -1, 0, -2, 0, 2, 0], [0, 2, 0, 0, 0, -2, 0, 1, 0, 2, -2, 1], [0, -1, 0, 1, 0, 1, 0, 0, 0, -2, 2, -3], [0, 4, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0], [0, -2, 0, -2, 0, -2, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, -2, 0, 1, 0, 0, 0, -1], [-3, 0, -1, 0, -2, 0, -1, 0, 1, 1, -1, 0], [0, -2, 0, 2, 0, 2, 0, -1, 0, 0, 0, -1], [0, 1, 0, 1, 0, -1, 0, 1, 0, 0, 0, 2], [0, -5, 0, -1, 0, 1, 0, 0, 0, 1, -1, 2], [-1, 0, 4, 0, -1, 0, 1, 0, 1, 2, -4, 0], [0, 0, 0, -4, 0, 0, 0, 1, 0, -2, 2, 1], [4, 0, 0, 0, -2, 0, -2, 0, 2, 1, -3, 0], [6, 0, 0, 0, 2, 0, -4, 0, 2, 2, 2, 0], [1, 0, -2, 0, 3, 0, -5, 0, -1, -2, 4, 0], [-2, 0, 4, 0, 0, 0, 2, 0, 0, 2, -2, 0], [0, 9, 0, -1, 0, -1, 0, 1, 0, 1, -1, 3], [0, 2, 0, 0, 0, 2, 0, -1, 0, -2, 2, 1], [0, 5, 0, -3, 0, -1, 0, 0, 0, 3, -3, 6], [-1, 0, 0, 0, 1, 0, 1, 0, -1, -3, -1, 0], [0, 2, 0, 0, 0, 0, 0, 1, 0, -1, 1, -3], [0, -3, 0, -2, 0, 3, 0, 3, 0, 0, 0, -1], [4, 0, 0, 0, -2, 0, -4, 0, -2, 1, 1, 0], [0, 2, 0, 0, 0, 4, 0, 0, 0, -1, 1, 0], [-1, 0, 2, 0, -4, 0, -1, 0, 0, 2, -4, 0], [4, 0, -2, 0, -4, 0, -6, 0, 2, 1, -3, 0], [-4, 0, -2, 0, 2, 0, 4, 0, 4, 0, 0, 0], [0, 0, 2, 0, 0, 0, 2, 0, 2, 1, -3, 0], [0, 3, 0, 1, 0, 5, 0, 2, 0, -1, 1, -6], [3, 0, -2, 0, 0, 0, -1, 0, 0, 0, 2, 0], [0, -6, 0, -2, 0, -2, 0, 3, 0, 2, -2, 1], [0, 4, 0, 0, 0, 4, 0, -2, 0, 0, 0, -2], [-1, 0, 5, 0, -6, 0, -1, 0, -1, 5, -5, 0], [-1, 0, 0, 0, -2, 0, 1, 0, 0, -2, -4, 0], [0, -8, 0, 0, 0, 0, 0, 3, 0, 0, 0, -3], [-11, 0, 4, 0, -2, 0, 3, 0, 0, 2, -4, 0], [-3, 0, 2, 0, 2, 0, -5, 0, -2, 2, 4, 0], [-5, 0, 6, 0, 1, 0, 3, 0, -3, -1, -3, 0], [2, 0, -4, 0, 4, 0, -4, 0, 0, -2, 6, 0], [-4, 0, -6, 0, 0, 0, 2, 0, 2, -5, -1, 0], [0, -5, 0, 2, 0, 5, 0, -2, 0, -6, 6, -6], [0, 2, 0, -2, 0, 6, 0, 0, 0, 0, 0, -2], [0, -12, 0, 2, 0, 0, 0, -1, 0, 2, -2, 3], [-9, 0, 1, 0, -4, 0, 1, 0, 1, 3, -3, 0], [0, -9, 0, -1, 0, -3, 0, 3, 0, 4, -4, 4], [0, -8, 0, -2, 0, 0, 0, -2, 0, -2, 2, -2], [0, -9, 0, 1, 0, 1, 0, 1, 0, -1, 1, -5], [-6, 0, -4, 0, -2, 0, 0, 0, 2, -4, -4, 0], [0, -9, 0, -2, 0, 5, 0, 3, 0, -2, 2, -1], [4, 0, -8, 0, -2, 0, -2, 0, 2, -1, 3, 0], [0, 9, 0, 3, 0, 3, 0, 0, 0, 2, -2, 1], [0, 5, 0, -4, 0, -9, 0, 5, 0, 4, -4, 5], [2, 0, 4, 0, 4, 0, 0, 0, 0, 1, 1, 0], [0, 10, 0, 0, 0, -2, 0, 4, 0, 0, 0, -4], [-5, 0, 9, 0, -2, 0, 1, 0, -5, 1, -5, 0], [0, -2, 0, -4, 0, -2, 0, -2, 0, -2, 2, -2], [8, 0, 2, 0, -6, 0, -4, 0, 0, 3, -5, 0], [0, 4, 0, 0, 0, 0, 0, -1, 0, -2, 2, -3], [0, 4, 0, 0, 0, -4, 0, 3, 0, 4, -4, -3], [-9, 0, -3, 0, 4, 0, 1, 0, 5, -3, -1, 0], [-3, 0, 2, 0, 5, 0, 7, 0, 5, -1, -3, 0], [0, 2, 0, -4, 0, 2, 0, -1, 0, -6, 6, -3], [4, 0, -2, 0, 0, 0, -6, 0, 6, 3, -1, 0], [0, 5, 0, 4, 0, -5, 0, 3, 0, 2, -2, 3], [0, -10, 0, 0, 0, 10, 0, -4, 0, -2, 2, 2], [15, 0, -2, 0, -3, 0, -5, 0, -7, -3, 3, 0], [0, 6, 0, -4, 0, -12, 0, 5, 0, 3, -3, 5], [0, -13, 0, 2, 0, 5, 0, -6, 0, -2, 2, -2], [0, 24, 0, -2, 0, 0, 0, 2, 0, 2, -2, 2], [6, 0, 8, 0, 0, 0, 0, 0, 0, 6, -2, 0], [-3, 0, -5, 0, 0, 0, 3, 0, 7, 3, 1, 0], [8, 0, -2, 0, 0, 0, 2, 0, -2, 3, 7, 0], [0, -10, 0, -6, 0, 6, 0, 0, 0, -4, 4, -4], [0, -11, 0, -5, 0, 3, 0, -4, 0, -3, 3, 0], [0, 6, 0, 4, 0, 2, 0, 3, 0, 0, 0, 1], [0, 8, 0, -2, 0, -4, 0, 2, 0, -2, 2, -4], [-6, 0, -4, 0, -2, 0, 4, 0, -6, -5, 3, 0], [4, 0, 4, 0, -2, 0, 0, 0, -6, -1, -1, 0], [0, -10, 0, -2, 0, 2, 0, -4, 0, -4, 4, -2], [0, 0, 2, 0, 0, 0, -8, 0, -2, 0, 0, 0], [4, 0, -6, 0, 0, 0, 0, 0, -2, -4, 4, 0], [10, 0, 2, 0, -4, 0, 2, 0, 6, 6, -6, 0], [-1, 0, -3, 0, 2, 0, -11, 0, 3, 1, 3, 0], [-4, 0, 6, 0, -8, 0, 0, 0, 2, 10, -6, 0], [0, 1, 0, 1, 0, -5, 0, 1, 0, 4, -4, 0], [0, -16, 0, 0, 0, -4, 0, -3, 0, -2, 2, -1], [0, 13, 0, 2, 0, -13, 0, 7, 0, 8, -8, 7], [0, -2, 0, -4, 0, -6, 0, -5, 0, -4, 4, 1], [0, 2, 0, 2, 0, 10, 0, -1, 0, 2, -2, -1], [-7, 0, -1, 0, 4, 0, 13, 0, 3, -2, 0, 0], [7, 0, -2, 0, 8, 0, 1, 0, -4, -5, 9, 0], [-6, 0, 4, 0, 0, 0, -4, 0, -4, 2, 2, 0], [0, 6, 0, 8, 0, -2, 0, 1, 0, 6, -6, -1], [4, 0, -12, 0, 2, 0, -2, 0, 2, -7, 5, 0], [0, -8, 0, 2, 0, 4, 0, -2, 0, -6, 6, -12], [13, 0, -8, 0, 0, 0, 1, 0, 2, -6, 0, 0], [0, -6, 0, 8, 0, 2, 0, -1, 0, 2, -2, -3], [0, 1, 0, 2, 0, 7, 0, -6, 0, -6, 6, -6], [0, -6, 0, 2, 0, -2, 0, 0, 0, 0, 0, 6], [0, 6, 0, 8, 0, -4, 0, -3, 0, 7, -7, 5], [0, 25, 0, 1, 0, -5, 0, -3, 0, -1, 1, 1], [-2, 0, -2, 0, 0, 0, -10, 0, -2, 0, 4, 0], [-6, 0, -4, 0, 2, 0, 6, 0, 2, 1, 5, 0], [0, -5, 0, -1, 0, 1, 0, -1, 0, 1, -1, 3], [-8, 0, -2, 0, -2, 0, 8, 0, 8, 4, -4, 0], [0, 1, 0, 0, 0, -5, 0, 1, 0, -4, 4, -3], [0, -6, 0, 4, 0, -6, 0, 0, 0, 0, 0, 2], [-10, 0, 12, 0, -2, 0, 2, 0, -2, 7, -5, 0], [9, 0, 6, 0, -5, 0, 1, 0, -5, -1, -7, 0], [0, -6, 0, -2, 0, -6, 0, 0, 0, 6, -6, 6], [-16, 0, 8, 0, 2, 0, 6, 0, -2, 4, 0, 0], [0, 13, 0, -1, 0, 3, 0, 2, 0, 1, -1, -2], [-22, 0, 4, 0, -2, 0, 2, 0, -2, 2, -2, 0], [14, 0, -12, 0, 2, 0, 0, 0, -2, -11, 5, 0], [2, 0, 6, 0, 2, 0, -4, 0, 4, 8, 0, 0], [0, -15, 0, -8, 0, -1, 0, 5, 0, -2, 2, 1], [-1, 0, 0, 0, -6, 0, 1, 0, -4, -2, -4, 0], [0, -10, 0, -4, 0, 10, 0, -7, 0, -8, 8, -5], [0, 4, 0, -2, 0, 4, 0, 1, 0, 2, -2, 7], [-8, 0, -8, 0, 0, 0, 0, 0, 0, -6, 2, 0], [18, 0, 0, 0, 4, 0, -6, 0, 0, -3, 1, 0], [0, 0, -8, 0, 2, 0, 2, 0, -2, -8, 4, 0], [9, 0, -8, 0, 0, 0, -3, 0, 10, 4, 2, 0], [0, -1, 0, 3, 0, 9, 0, 3, 0, 0, 0, -6], [0, 7, 0, 3, 0, -11, 0, 3, 0, 4, -4, 4], [0, 7, 0, -11, 0, -3, 0, 2, 0, 2, -2, 9], [0, 5, 0, 1, 0, -1, 0, -5, 0, -4, 4, -6], [16, 0, -12, 0, -2, 0, 2, 0, 6, 0, 4, 0], [0, 29, 0, 2, 0, -1, 0, -1, 0, 2, -2, 3], [-10, 0, -4, 0, 0, 0, -4, 0, -4, -2, 6, 0], [-4, 0, 10, 0, -6, 0, 8, 0, 0, 6, -10, 0], [-20, 0, 4, 0, 4, 0, 8, 0, 0, 1, 1, 0], [7, 0, -6, 0, 0, 0, -1, 0, -4, -2, 8, 0], [0, -7, 0, 8, 0, -5, 0, 7, 0, 8, -8, -1], [0, -6, 0, -2, 0, 2, 0, 3, 0, -8, 8, -3], [3, 0, 12, 0, -1, 0, 7, 0, -3, 7, -3, 0], [2, 0, 2, 0, -6, 0, -6, 0, -4, -2, -6, 0], [0, -14, 0, 4, 0, -6, 0, 1, 0, 0, 0, 1], [0, 1, 0, -1, 0, -5, 0, -5, 0, 3, -3, 9], [0, 20, 0, 10, 0, -8, 0, 1, 0, 10, -10, 5], [8, 0, -10, 0, 4, 0, 6, 0, 2, -7, 5, 0], [18, 0, -8, 0, 6, 0, 6, 0, 2, -4, 8, 0], [5, 0, -1, 0, -2, 0, -9, 0, 1, 1, -1, 0], [0, 22, 0, 12, 0, -2, 0, 3, 0, 4, -4, -1], [0, -22, 0, 0, 0, 4, 0, -8, 0, -1, 1, -4], [0, -16, 0, -2, 0, 4, 0, 3, 0, 0, 0, -3], [0, 11, 0, -1, 0, -11, 0, 1, 0, 8, -8, 12], [4, 0, -8, 0, 8, 0, 2, 0, 4, -4, 8, 0], [0, -4, 0, 0, 0, 0, 0, -4, 0, -2, 2, -4], [22, 0, 2, 0, -4, 0, -16, 0, -14, 2, 10, 0], [0, 0, 0, 0, 0, -8, 0, -2, 0, 0, 0, 10], [0, 6, 0, 0, 0, 2, 0, -2, 0, -2, 2, 0], [0, -23, 0, 1, 0, 3, 0, -3, 0, -5, 5, -7], [0, 7, 0, -3, 0, -3, 0, 2, 0, 4, -4, 3], [6, 0, -2, 0, -2, 0, -2, 0, -12, -5, 7, 0], [0, 10, 0, 6, 0, -10, 0, 9, 0, 8, -8, 9], [0, -10, 0, -18, 0, 6, 0, 4, 0, -4, 4, 0], [0, 17, 0, -3, 0, 11, 0, -5, 0, 0, 0, -2], [-24, 0, 2, 0, 4, 0, 0, 0, 10, 4, -4, 0], [11, 0, 2, 0, 1, 0, 15, 0, 1, -1, -3, 0], [0, 0, 4, 0, -8, 0, 0, 0, -4, -2, -10, 0], [0, -12, 0, 8, 0, 0, 0, 0, 0, -4, 4, -2], [-19, 0, -2, 0, 9, 0, 7, 0, -3, -5, 9, 0], [0, -8, 0, -16, 0, 0, 0, 1, 0, 0, 0, 7], [0, -2, 0, 10, 0, 6, 0, -5, 0, 6, -6, 3], [0, -4, 0, -12, 0, 0, 0, -6, 0, -2, 2, 14], [0, -16, 0, 6, 0, 12, 0, -10, 0, -2, 2, -8], [0, -2, 0, 8, 0, 2, 0, 5, 0, 2, -2, -7], [-16, 0, 0, 0, 8, 0, 8, 0, 0, -7, 1, 0], [-5, 0, -4, 0, 1, 0, -3, 0, -5, 1, 11, 0], [-10, 0, 6, 0, 4, 0, -4, 0, 6, 0, -8, 0], [19, 0, 4, 0, -4, 0, 7, 0, 6, 6, -8, 0], [0, 2, 0, -2, 0, -2, 0, -1, 0, 4, -4, 15], [0, 23, 0, -4, 0, 9, 0, 0, 0, 0, 0, -4], [0, -30, 0, 0, 0, 4, 0, -3, 0, -7, 7, -11], [0, 0, 0, 0, -2, 0, -8, 0, -2, 0, 0, 0], [0, 4, 0, 2, 0, -8, 0, 8, 0, 8, -8, 4], [-4, 0, -4, 0, -2, 0, -8, 0, 2, -2, -2, 0], [6, 0, 6, 0, -10, 0, -12, 0, -8, 10, 2, 0], [12, 0, -10, 0, -6, 0, -8, 0, 4, -5, -5, 0], [7, 0, 5, 0, 2, 0, -11, 0, -5, 3, 5, 0], [0, 15, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [2, 0, -2, 0, 6, 0, -6, 0, -4, 1, 13, 0], [0, 8, 0, -4, 0, 4, 0, 5, 0, 8, -8, 9], [0, -8, 0, 0, 0, 4, 0, -2, 0, 0, 0, -8], [2, 0, -10, 0, -2, 0, -4, 0, 8, 4, 4, 0], [7, 0, 8, 0, 0, 0, -9, 0, 6, 6, -8, 0], [-30, 0, 0, 0, 2, 0, 10, 0, -2, -6, -2, 0], [-12, 0, 0, 0, -6, 0, -12, 0, 10, 5, -11, 0], [0, -18, 0, -4, 0, 6, 0, -2, 0, -8, 8, 4], [-11, 0, -10, 0, -9, 0, 1, 0, 15, 3, -11, 0], [0, -21, 0, 2, 0, 21, 0, -4, 0, -2, 2, 0], [0, -10, 0, 0, 0, 2, 0, 2, 0, 4, -4, -10], [0, -13, 0, 4, 0, 13, 0, -6, 0, -4, 4, -2], [-15, 0, 10, 0, -1, 0, 3, 0, -1, 6, -4, 0], [-24, 0, 0, 0, 4, 0, 0, 0, 4, -2, -2, 0], [-14, 0, 6, 0, 2, 0, -8, 0, 4, 0, -8, 0], [18, 0, -4, 0, 2, 0, -18, 0, 2, 4, 8, 0], [0, -2, 0, -2, 0, 14, 0, 8, 0, -2, 2, -6], [0, 21, 0, -4, 0, -5, 0, 12, 0, 0, 0, 0], [8, 0, 4, 0, -4, 0, -8, 0, 0, 11, 3, 0], [0, 32, 0, 0, 0, -4, 0, 4, 0, 4, -4, 6], [-2, 0, -6, 0, -6, 0, -2, 0, 12, 6, -6, 0], [4, 0, 0, 0, 2, 0, 6, 0, 6, 0, -4, 0], [0, -5, 0, 11, 0, -7, 0, -2, 0, 9, -9, 12], [-9, 0, 0, 0, -5, 0, -9, 0, -3, 5, 3, 0], [0, 12, 0, -6, 0, 4, 0, 3, 0, -6, 6, 1], [0, -20, 0, -4, 0, 4, 0, 2, 0, 4, -4, 10], [0, -10, 0, -8, 0, -2, 0, 3, 0, -6, 6, -3], [0, 16, 0, 2, 0, 8, 0, 2, 0, -6, 6, -6], [0, 10, 0, 4, 0, 16, 0, -1, 0, -5, 5, -13], [-8, 0, 8, 0, 6, 0, 0, 0, -10, -2, 6, 0], [7, 0, 10, 0, -4, 0, -7, 0, -12, 3, 1, 0], [0, -3, 0, 3, 0, -9, 0, 11, 0, 7, -7, -3], [0, -28, 0, -2, 0, 12, 0, -3, 0, -4, 4, 5], [0, 6, 0, -6, 0, 2, 0, -5, 0, 4, -4, 3], [0, 24, 0, 16, 0, -12, 0, 8, 0, 14, -14, 4], [-16, 0, -2, 0, 6, 0, 8, 0, 8, -2, -2, 0], [-12, 0, 0, 0, 8, 0, 2, 0, -4, -6, 6, 0], [0, -15, 0, 2, 0, -5, 0, 9, 0, 10, -10, 1], [0, -2, 0, 4, 0, 16, 0, -6, 0, -5, 5, -6], [-6, 0, 6, 0, 4, 0, -4, 0, -10, -6, 2, 0], [14, 0, -10, 0, 2, 0, -2, 0, -8, -16, 4, 0], [-4, 0, 2, 0, -2, 0, 20, 0, 4, -2, -10, 0], [-10, 0, 2, 0, -2, 0, -12, 0, 4, 3, -5, 0], [-10, 0, 0, 0, 2, 0, -4, 0, -6, -3, 5, 0], [0, -28, 0, -4, 0, 24, 0, -2, 0, -10, 10, -18], [0, -1, 0, 1, 0, 5, 0, -16, 0, 0, 0, 5], [0, -7, 0, -9, 0, -1, 0, 6, 0, 5, -5, 2], [0, -3, 0, 2, 0, 3, 0, 5, 0, 4, -4, 5], [-11, 0, 6, 0, 7, 0, 7, 0, 3, 2, 0, 0], [-32, 0, -2, 0, 6, 0, -6, 0, -4, -6, 6, 0], [-13, 0, 0, 0, 3, 0, 3, 0, 13, 3, -7, 0], [24, 0, 8, 0, -4, 0, -8, 0, -12, 0, 0, 0], [15, 0, 14, 0, -1, 0, -3, 0, -9, 5, -1, 0], [0, -22, 0, 8, 0, -2, 0, -2, 0, -2, 2, -8], [0, -4, 0, -8, 0, 16, 0, -3, 0, -4, 4, -7], [-5, 0, 5, 0, 4, 0, -3, 0, 5, 5, -1, 0], [0, -6, 0, 8, 0, -8, 0, 7, 0, 7, -7, 11], [-45, 0, 0, 0, 7, 0, 15, 0, 1, -5, 1, 0], [0, 0, 0, 2, 0, -12, 0, 3, 0, 4, -4, 1], [-20, 0, -10, 0, 2, 0, 6, 0, 8, -11, -7, 0], [-36, 0, 6, 0, -2, 0, 18, 0, 12, 10, -10, 0], [-8, 0, -6, 0, -12, 0, 4, 0, 10, 3, -13, 0], [2, 0, 6, 0, -2, 0, 0, 0, 8, 6, -10, 0], [0, -29, 0, 6, 0, -3, 0, -1, 0, 4, -4, -9], [0, -15, 0, -11, 0, 3, 0, -11, 0, -12, 12, -12], [0, -25, 0, -2, 0, 9, 0, -12, 0, -6, 6, -16], [0, -22, 0, 4, 0, 0, 0, -4, 0, -7, 7, 0], [2, 0, 2, 0, 8, 0, -6, 0, -14, -6, 14, 0], [0, 2, 0, 10, 0, 2, 0, 5, 0, 16, -16, 3], [0, 26, 0, 6, 0, 6, 0, -1, 0, 0, 0, -5], [0, 30, 0, 20, 0, 6, 0, 1, 0, 2, -2, -9], [-4, 0, -16, 0, 6, 0, -8, 0, -10, -14, 18, 0], [0, -18, 0, 12, 0, -2, 0, -2, 0, 10, -10, 6], [0, -6, 0, -6, 0, -2, 0, -1, 0, -4, 4, -5], [0, -20, 0, -2, 0, 8, 0, -11, 0, -8, 8, -5], [11, 0, 1, 0, 12, 0, 3, 0, -3, -8, 6, 0], [-1, 0, -6, 0, 4, 0, 7, 0, -8, -10, 8, 0], [28, 0, -8, 0, -4, 0, -8, 0, 4, 2, 2, 0], [8, 0, -8, 0, -8, 0, 14, 0, 12, -2, -14, 0], [0, -13, 0, 11, 0, -7, 0, -6, 0, 1, -1, -4], [0, 23, 0, 2, 0, -23, 0, 8, 0, 14, -14, 12], [-11, 0, 12, 0, 10, 0, 19, 0, 4, 2, -4, 0], [0, 24, 0, -4, 0, -12, 0, 0, 0, -2, 2, 4], [0, -10, 0, 8, 0, -2, 0, 11, 0, 6, -6, 1], [0, 24, 0, 26, 0, -4, 0, -2, 0, 6, -6, -4], [41, 0, -2, 0, -4, 0, -1, 0, -4, -1, 1, 0], [12, 0, -2, 0, -10, 0, 0, 0, 0, -2, -10, 0], [0, 0, -8, 0, 2, 0, 16, 0, 2, -10, -2, 0], [8, 0, -10, 0, 12, 0, 16, 0, -2, -14, 10, 0], [0, -20, 0, 0, 0, 0, 0, -1, 0, -4, 4, 11], [0, 15, 0, -3, 0, -3, 0, -12, 0, -6, 6, -5], [0, 0, 0, -12, 0, -4, 0, -5, 0, -10, 10, -7], [-7, 0, 7, 0, 2, 0, 9, 0, 1, 2, -4, 0], [28, 0, 8, 0, 12, 0, 4, 0, 4, 2, 2, 0], [17, 0, 14, 0, 1, 0, 3, 0, -3, 11, 1, 0], [-9, 0, -4, 0, 11, 0, 15, 0, 1, -9, 5, 0], [0, 10, 0, 4, 0, -6, 0, 2, 0, -2, 2, -14], [0, -6, 0, -6, 0, 14, 0, -12, 0, -10, 10, 8], [-4, 0, -6, 0, -6, 0, 6, 0, 4, 2, -2, 0], [-21, 0, 2, 0, 4, 0, 13, 0, 12, 1, -9, 0], [0, -7, 0, -1, 0, -5, 0, -11, 0, -1, 1, -1], [2, 0, 10, 0, -16, 0, 4, 0, 6, 16, -16, 0], [45, 0, -2, 0, -3, 0, -9, 0, 1, 7, 5, 0], [0, 27, 0, -9, 0, 1, 0, -1, 0, 0, 0, 0], [15, 0, -2, 0, 12, 0, 9, 0, 0, -11, 3, 0], [0, -5, 0, -18, 0, -7, 0, 1, 0, -14, 14, 1], [0, 9, 0, -3, 0, -17, 0, 5, 0, 8, -8, 6], [20, 0, -12, 0, -4, 0, 0, 0, 0, 4, 12, 0], [0, 23, 0, 11, 0, 13, 0, 2, 0, -7, 7, -16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1386_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1386_2_c_b();".
// 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_1386_2_c_b(:prec:=12)
    chi := MakeCharacter_1386_c();
    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_1386_2_c_b();".
// 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_1386_2_c_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1386_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<17,R![-376, -136, 306, 136, -20, -8, 1]>],Snew);
    return Vf;
end function;