
// Make newform 1386.2.g.a in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1386_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1386_g_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_g_a();".
// 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_g_a();".
// 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 := [324, 0, 1080, 0, 1800, 0, -2028, 0, 1189, 0, 80, 0, 8, 0, -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, 0, 0, 0, 0], [-51358644, 0, -229027878, 0, 264348225, 0, -110537567, 0, -7435564, 0, -939814, 0, 438755, 0, -96575, 0], [-7232976, 0, 2396970, 0, 3917118, 0, -4851221, 0, -332152, 0, -16306, 0, 11990, 0, -3833, 0], [0, -5487054570, 0, -4822618578, 0, 5354283043, 0, -3735647454, 0, -248772494, 0, -22413962, 0, 11597979, 0, -3079634], [0, 3202119162, 0, 38419343964, 0, -34397644005, 0, 14397572539, 0, 952830506, 0, 129110888, 0, -57579817, 0, 12591283], [-9136826568, 25498762434, 2325320910, 22083943080, 3800759034, -25836351497, -4702927103, 17514628229, -333984496, 1189121858, -14001178, 105758504, 11685650, -55196585, -3765719, 14545077], [-6275124, 0, -27991926, 0, 23130228, 0, -13578607, 0, -893276, 0, -96938, 0, 45868, 0, -11395, 0], [3932968608, 25498762434, -2325320910, 22083943080, -3800759034, -25836351497, 4702927103, 17514628229, 333984496, 1189121858, 14001178, 105758504, -11685650, -55196585, 3765719, 14545077], [-13615971048, 31120049628, 881758260, 25614327990, 1441445244, -35425496144, -1782429418, 22114467473, -147925256, 1512241796, -4803548, 136369838, 4446220, -72148880, -1441114, 18506049], [18819829008, 31120049628, -881758260, 25614327990, -1441445244, -35425496144, 1782429418, 22114467473, 147925256, 1512241796, 4803548, 136369838, -4446220, -72148880, 1441114, 18506049], [-18889526916, 14747147118, -84253473054, 178659278442, 75040099632, -161200544481, -40803930363, 69289811864, -2703731964, 4576799302, -309169302, 617221546, 143608152, -274567721, -34694355, 60462680], [-9644970996, -14747147118, -43028455014, -178659278442, 27457419132, 161200544481, -20907168303, -69289811864, -1365330444, -4576799302, -140002782, -617221546, 64632252, 274567721, -17310855, -60462680], [25206770376, -17473193856, 112411745244, -218620670994, -121172690412, 202259564982, 54294184518, -94072523573, 3640498104, -6207754144, 450833772, -785602642, -210264972, 357230942, 47163030, -81296765], [-25206770376, -17473193856, -112411745244, -218620670994, 121172690412, 202259564982, -54294184518, -94072523573, -3640498104, -6207754144, -450833772, -785602642, 210264972, 357230942, -47163030, -81296765], [0, 37162486386, 0, 28420002066, 0, -48610532639, 0, 27613656994, 0, 1897812810, 0, 176425470, 0, -93544963, 0, 23298266], [0, -2860280478, 0, -37432034028, 0, 35778301767, 0, -18241907237, 0, -1206672370, 0, -141046900, 0, 65944679, 0, -15573353]];
    Rf_basisdens := [1, 86730966, 1867860, 2601928980, 7805786940, 5203857960, 2256660, 5203857960, 5203857960, 5203857960, 15611573880, 15611573880, 15611573880, 15611573880, 2601928980, 1115112420];
    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_g();"
function MakeCharacter_1386_g()
    N := 1386;
    order := 2;
    char_gens := [155, 199, 1135];
    v := [1, 1, 2];
    // 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_g_Hecke();"
function MakeCharacter_1386_g_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, 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, 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, -1, 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, -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, -1, 0, 0, -1, 0, 0, 0, 0, 0, 1, -1, 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, 1, -1, 0, 0], [0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, -1, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 1], [3, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 2, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 1], [0, -5, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, -1, 0, 0], [0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 0, -4, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, -1, 0, 0], [-1, 0, 0, 2, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0], [-1, 0, 1, 0, 0, 2, 0, -2, -1, 1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, -1, 0, 0, 0, 0, 0, -1, 1, 1, 1, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 2], [0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0], [0, 2, 0, 0, 2, 0, 0, 0, 0, 0, -2, 2, 1, 1, 0, 0], [0, 0, 0, 1, 0, -1, 0, -1, -1, -1, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, -1, 1, 0, 0], [-6, 0, -1, 0, 0, -3, 0, 3, -1, 1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 1, 1, 1, -1, 0, 0], [3, 0, 1, 0, 0, -2, 0, 2, -1, 1, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 3, 0, 0, 0, 0, 0, -3, 3, 0, 0, 0, -2], [0, -5, 0, 0, 0, 0, -2, 0, 0, 0, 1, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 2, 0, 2, 2, 2, 0, 0, 0, 0, -2, 0], [0, -2, 0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 1, -1, 0, 0], [-8, 0, -2, 0, 0, 1, 0, -1, 1, -1, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 4, 0, -1, 0, -1, 1, 1, 0, 0, 0, 0, 1, 0], [5, 0, -2, 0, 0, -2, 0, 2, 1, -1, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, -1, -1, 0, 0], [0, -2, 0, 0, 2, 0, 0, 0, 0, 0, 2, -2, 1, 1, 0, -2], [0, -5, 0, 0, 0, 0, -1, 0, 0, 0, 1, 1, 0, 0, 0, 0], [-3, 0, 0, -4, 0, -3, 0, -3, 0, 0, 0, 0, 0, 0, -1, 0], [0, -4, 0, 0, 0, 0, -2, 0, 0, 0, -4, -4, -1, 1, 0, 0], [-4, 0, -2, 0, 0, 2, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 1, 1, -4, 4, 0, 0], [0, 0, 0, -4, 0, -1, 0, -1, -1, -1, 0, 0, 0, 0, -1, 0], [3, 0, 2, 0, 0, -1, 0, 1, -2, 2, 0, 0, 0, 0, 0, 0], [4, 0, 0, -4, 0, 3, 0, 3, -1, -1, 0, 0, 0, 0, 1, 0], [0, -1, 0, 0, 4, 0, 0, 0, 0, 0, 1, -1, -1, -1, 0, -1], [0, 0, 0, -1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 2, 0], [0, 6, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -1, 1, 0, 0], [-1, 0, 0, 3, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, -2, 0], [0, -2, 0, 0, 2, 0, 0, 0, 0, 0, 2, -2, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, -2], [0, 3, 0, 0, 0, 0, -3, 0, 0, 0, -3, -3, -2, 2, 0, 0], [0, -2, 0, 0, 4, 0, 0, 0, 0, 0, 2, -2, 2, 2, 0, -1], [-5, 0, 0, 2, 0, -5, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 0, -3, 0, 0, -1, 0, 1, -1, 1, 0, 0, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 0, -5, 0, -2, 0, -2, 0, 0, 0, 0, 0, 0, 2, 0], [0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, -2], [-4, 0, 0, -4, 0, -6, 0, -6, -2, -2, 0, 0, 0, 0, 1, 0], [0, 1, 0, 0, -3, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0], [1, 0, 0, -2, 0, 2, 0, 2, 1, 1, 0, 0, 0, 0, 2, 0], [0, -13, 0, 0, 0, 0, 2, 0, 0, 0, 5, 5, -1, 1, 0, 0], [-7, 0, 1, 0, 0, 3, 0, -3, -2, 2, 0, 0, 0, 0, 0, 0], [2, 0, 0, 0, 0, -2, 0, 2, 2, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, -1, 1, 0, 0], [-2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 0], [0, -1, 0, 0, 8, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 2], [0, -7, 0, 0, 0, 0, 1, 0, 0, 0, -1, -1, -1, 1, 0, 0], [2, 0, 0, -3, 0, 1, 0, 1, -1, -1, 0, 0, 0, 0, 0, 0], [8, 0, 0, 0, 0, -2, 0, 2, -2, 2, 0, 0, 0, 0, 0, 0], [1, 0, -2, 0, 0, -2, 0, 2, 1, -1, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, -3, 0, 0, 0, 0, 0, 3, -3, -2, -2, 0, 4], [0, 5, 0, 0, 0, 0, -2, 0, 0, 0, -1, -1, 1, -1, 0, 0], [-5, 0, 0, 3, 0, -5, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, 2, 0, 0, 0, 3, 3, -2, 2, 0, 0], [3, 0, 0, -3, 0, 1, 0, 1, -2, -2, 0, 0, 0, 0, 2, 0], [0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2], [13, 0, 4, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0], [0, -11, 0, 0, 0, 0, -1, 0, 0, 0, -5, -5, 2, -2, 0, 0], [-3, 0, 0, 4, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, -2, 0], [3, 0, 0, -6, 0, 1, 0, 1, -2, -2, 0, 0, 0, 0, -4, 0], [0, -7, 0, 0, 0, 0, 3, 0, 0, 0, -1, -1, 4, -4, 0, 0], [0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 3, 3, 3, -3, 0, 0], [-2, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, -1, -1, 0, -2], [14, 0, 0, 0, 0, 2, 0, -2, -4, 4, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 4, 0, 0, 0, 0, 0, 2, -2, 2, 2, 0, -4], [0, 8, 0, 0, -6, 0, 0, 0, 0, 0, -8, 8, -1, -1, 0, 0], [-10, 0, 2, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 3, 0, 0], [-5, 0, -3, 0, 0, 1, 0, -1, -2, 2, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, -4, 0, 0, 0, 0, 0, 2, -2, 1, 1, 0, 0], [0, -2, 0, 0, 8, 0, 0, 0, 0, 0, 2, -2, 2, 2, 0, -1], [0, 1, 0, 0, 10, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0], [2, 0, 0, -1, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, -2, 0], [18, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0], [9, 0, -2, 0, 0, 1, 0, -1, 2, -2, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 1, 1, -4, 4, 0, 0], [0, 3, 0, 0, -4, 0, 0, 0, 0, 0, -3, 3, 0, 0, 0, 3], [0, -6, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-11, 0, 6, 0, 0, 3, 0, -3, -4, 4, 0, 0, 0, 0, 0, 0], [-4, 0, 0, -2, 0, -2, 0, -2, 2, 2, 0, 0, 0, 0, 0, 0], [0, 8, 0, 0, 0, 0, 0, 0, 0, 0, -8, 8, 2, 2, 0, -2], [0, 3, 0, 0, 0, 0, 0, 0, 0, 0, -3, 3, -2, -2, 0, 3], [0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, -3, 0, 0], [-1, 0, 0, 11, 0, -3, 0, -3, -2, -2, 0, 0, 0, 0, 2, 0], [3, 0, 0, -6, 0, 1, 0, 1, -2, -2, 0, 0, 0, 0, -2, 0], [-9, 0, -8, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 3, 0, 0, 0, 4, 4, 0, 0, 0, 0], [4, 0, 0, -2, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0, -2, 0], [4, 0, 4, 0, 0, 0, 0, 0, -4, 4, 0, 0, 0, 0, 0, 0], [0, 8, 0, 0, 0, 0, -1, 0, 0, 0, -4, -4, 0, 0, 0, 0], [2, 0, 0, 2, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 4, 0], [0, -5, 0, 0, -5, 0, 0, 0, 0, 0, 5, -5, -2, -2, 0, 4], [0, -13, 0, 0, 0, 0, -2, 0, 0, 0, 5, 5, 3, -3, 0, 0], [0, -6, 0, 0, 0, 0, 2, 0, 0, 0, -6, -6, 4, -4, 0, 0], [6, 0, 0, -1, 0, 5, 0, 5, -1, -1, 0, 0, 0, 0, -2, 0], [-6, 0, 8, 0, 0, 4, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, -3, -3, 0, 6], [0, -9, 0, 0, 0, 0, 3, 0, 0, 0, -3, -3, 4, -4, 0, 0], [2, 0, 0, -12, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0], [0, 20, 0, 0, 0, 0, 4, 0, 0, 0, 2, 2, 2, -2, 0, 0], [-4, 0, -4, 0, 0, 4, 0, -4, 2, -2, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 7, 0, 0, 0, 0, 0, -1, 1, 4, 4, 0, 0], [4, 0, 0, 0, 0, 3, 0, 3, -1, -1, 0, 0, 0, 0, -5, 0], [4, 0, 0, -4, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0], [5, 0, -2, 0, 0, 3, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 10, 0, 0, 0, 0, 4, 0, 0, 0, -2, -2, -1, 1, 0, 0], [12, 0, -4, 0, 0, 0, 0, 0, -4, 4, 0, 0, 0, 0, 0, 0], [5, 0, -2, 0, 0, -2, 0, 2, 5, -5, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, -8, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 1], [5, 0, 0, 6, 0, 5, 0, 5, 0, 0, 0, 0, 0, 0, -3, 0], [0, 2, 0, 0, 4, 0, 0, 0, 0, 0, -2, 2, 2, 2, 0, -1], [-2, 0, 0, 4, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, -3, 0], [0, -2, 0, 0, 8, 0, 0, 0, 0, 0, 2, -2, -2, -2, 0, 1], [0, -8, 0, 0, 0, 0, -2, 0, 0, 0, -2, -2, 4, -4, 0, 0], [-2, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, -2, 0], [0, -16, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, -2, 2, 0, 0], [20, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 12, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 1, -1, 0, 0], [0, 0, 0, 12, 0, -1, 0, -1, -1, -1, 0, 0, 0, 0, 1, 0], [0, 5, 0, 0, -3, 0, 0, 0, 0, 0, -5, 5, -2, -2, 0, 4], [0, 0, 0, 6, 0, 2, 0, 2, 2, 2, 0, 0, 0, 0, -6, 0], [0, 1, 0, 0, -13, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, -4], [10, 0, 0, 8, 0, 12, 0, 12, 2, 2, 0, 0, 0, 0, -2, 0], [0, -24, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 1, -1, 0, 0], [2, 0, 1, 0, 0, 1, 0, -1, 7, -7, 0, 0, 0, 0, 0, 0], [0, 7, 0, 0, -4, 0, 0, 0, 0, 0, -7, 7, 4, 4, 0, -4], [9, 0, 2, 0, 0, 6, 0, -6, 5, -5, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 6, 0, 0, 0, 0, 0, 4, -4, -2, -2, 0, -2], [-7, 0, 3, 0, 0, -5, 0, 5, 2, -2, 0, 0, 0, 0, 0, 0], [0, -14, 0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 1, -1, 0, 0], [-12, 0, -2, 0, 0, -1, 0, 1, -5, 5, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 10, 0, 0, 0, 0, 0, 3, -3, 0, 0, 0, 0], [3, 0, 0, 2, 0, 3, 0, 3, 0, 0, 0, 0, 0, 0, 3, 0], [0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2], [0, 4, 0, 0, 0, 0, -6, 0, 0, 0, -2, -2, -5, 5, 0, 0], [0, 26, 0, 0, 0, 0, 0, 0, 0, 0, -4, -4, 0, 0, 0, 0], [-11, 0, -4, 0, 0, -3, 0, 3, -4, 4, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 6, 0, 0, 0, 0, 0, 2, -2, 4, 4, 0, -6], [0, 28, 0, 0, 0, 0, 3, 0, 0, 0, 4, 4, 0, 0, 0, 0], [0, 3, 0, 0, 2, 0, 0, 0, 0, 0, -3, 3, 2, 2, 0, -1], [-6, 0, 2, 0, 0, -6, 0, 6, -4, 4, 0, 0, 0, 0, 0, 0], [-2, 0, 0, -10, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0], [4, 0, 2, 0, 0, -4, 0, 4, -2, 2, 0, 0, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 6, -6, 0, 0, 0, -3], [0, 34, 0, 0, 0, 0, -2, 0, 0, 0, -2, -2, -2, 2, 0, 0], [3, 0, 0, 3, 0, 7, 0, 7, 4, 4, 0, 0, 0, 0, 0, 0], [0, -14, 0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 5, -5, 0, 0], [14, 0, 0, 0, 0, 2, 0, -2, 2, -2, 0, 0, 0, 0, 0, 0], [-6, 0, 0, 1, 0, -3, 0, -3, 3, 3, 0, 0, 0, 0, -4, 0], [0, -11, 0, 0, 0, 0, 0, 0, 0, 0, 11, -11, 0, 0, 0, 4], [27, 0, -2, 0, 0, -7, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0], [0, -5, 0, 0, 0, 0, -2, 0, 0, 0, 1, 1, -7, 7, 0, 0], [-7, 0, 0, 6, 0, -5, 0, -5, 2, 2, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 4, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, -2, 0], [-10, 0, -6, 0, 0, -2, 0, 2, 4, -4, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 5, 0, 0, 0, 0, 0, 3, -3, 2, 2, 0, 2], [-3, 0, 6, 0, 0, -2, 0, 2, 5, -5, 0, 0, 0, 0, 0, 0], [0, -9, 0, 0, 4, 0, 0, 0, 0, 0, 9, -9, -4, -4, 0, 3], [0, 20, 0, 0, 0, 0, 2, 0, 0, 0, 8, 8, -1, 1, 0, 0], [0, -4, 0, 0, 7, 0, 0, 0, 0, 0, 4, -4, 2, 2, 0, 4], [-11, 0, 0, 0, 0, 2, 0, -2, 5, -5, 0, 0, 0, 0, 0, 0], [2, 0, 0, 6, 0, 0, 0, 0, -2, -2, 0, 0, 0, 0, 6, 0], [16, 0, -8, 0, 0, -2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, 0, 0, 0, 0, -6, 6, -1, -1, 0, -4], [1, 0, 0, -10, 0, 2, 0, 2, 1, 1, 0, 0, 0, 0, 2, 0], [0, -18, 0, 0, 0, 0, 2, 0, 0, 0, 6, 6, 1, -1, 0, 0], [9, 0, -7, 0, 0, -7, 0, 7, 4, -4, 0, 0, 0, 0, 0, 0], [0, 8, 0, 0, 7, 0, 0, 0, 0, 0, -8, 8, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, 2, 0, 0, 0, 6, 6, -2, 2, 0, 0], [0, -5, 0, 0, -10, 0, 0, 0, 0, 0, 5, -5, -4, -4, 0, -4], [20, 0, -2, 0, 0, -2, 0, 2, -4, 4, 0, 0, 0, 0, 0, 0], [-35, 0, -4, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 4, 0, 0, 0, 0, 0, -1, 1, 2, 2, 0, -7], [0, 2, 0, 0, 12, 0, 0, 0, 0, 0, -2, 2, -2, -2, 0, 6], [0, -26, 0, 0, 0, 0, -4, 0, 0, 0, -2, -2, 1, -1, 0, 0], [3, 0, 0, -4, 0, 5, 0, 5, 2, 2, 0, 0, 0, 0, -2, 0], [-9, 0, 0, -7, 0, -11, 0, -11, -2, -2, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 1, 0, 1, 0, 1, 2, 2, 0, 0, 0, 0, -6, 0], [0, -4, 0, 0, -8, 0, 0, 0, 0, 0, 4, -4, 2, 2, 0, -4], [0, 6, 0, 0, 13, 0, 0, 0, 0, 0, -6, 6, 0, 0, 0, -4], [6, 0, 0, 7, 0, 5, 0, 5, -1, -1, 0, 0, 0, 0, -4, 0], [0, -18, 0, 0, 0, 0, 2, 0, 0, 0, 6, 6, -4, 4, 0, 0], [0, 22, 0, 0, 0, 0, -10, 0, 0, 0, 4, 4, -2, 2, 0, 0], [2, 0, 0, -12, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [10, 0, 2, 0, 0, -10, 0, 10, -2, 2, 0, 0, 0, 0, 0, 0], [0, -6, 0, 0, -5, 0, 0, 0, 0, 0, 6, -6, -4, -4, 0, 0], [4, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 4, -4, -2, -2, 0, 2], [0, 2, 0, 0, 6, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 8], [5, 0, 0, -12, 0, 6, 0, 6, 1, 1, 0, 0, 0, 0, -4, 0], [13, 0, 9, 0, 0, -4, 0, 4, -1, 1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, -5, 0, 0, 0, 0, 0, -1, 1, 2, 2, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, -1, 1, 0, 0], [0, 11, 0, 0, 0, 0, -4, 0, 0, 0, 5, 5, 1, -1, 0, 0], [-10, 0, 4, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0], [1, 0, 0, 16, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, -2, 0], [0, -17, 0, 0, 0, 0, -2, 0, 0, 0, 1, 1, 6, -6, 0, 0], [-12, 0, -10, 0, 0, 6, 0, -6, 2, -2, 0, 0, 0, 0, 0, 0], [0, -9, 0, 0, -8, 0, 0, 0, 0, 0, 9, -9, -4, -4, 0, -1], [-6, 0, 5, 0, 0, 9, 0, -9, -1, 1, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, -6, 0, 0, 0, 0, 0, -3, 3, 0, 0, 0, -6], [0, 12, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 5, -5, 0, 0], [-9, 0, 0, -6, 0, -11, 0, -11, -2, -2, 0, 0, 0, 0, 4, 0], [0, -20, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, -1, 1, 0, 0], [14, 0, -10, 0, 0, -2, 0, 2, 2, -2, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 1, 0, -4, 0, -4, -2, -2, 0, 0, 0, 0, 6, 0], [-10, 0, -2, 0, 0, 3, 0, -3, 1, -1, 0, 0, 0, 0, 0, 0], [0, 14, 0, 0, 0, 0, 2, 0, 0, 0, 8, 8, 0, 0, 0, 0], [2, 0, 0, 8, 0, 0, 0, 0, -2, -2, 0, 0, 0, 0, 4, 0], [0, -2, 0, 0, 8, 0, 0, 0, 0, 0, 2, -2, -2, -2, 0, -2], [7, 0, 0, -10, 0, 4, 0, 4, -3, -3, 0, 0, 0, 0, 4, 0], [0, 40, 0, 0, 0, 0, 0, 0, 0, 0, -2, -2, -4, 4, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, -6, -6, 7, -7, 0, 0], [0, -1, 0, 0, -2, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 7], [0, -3, 0, 0, 0, 0, 7, 0, 0, 0, 3, 3, -4, 4, 0, 0], [6, 0, 0, 3, 0, 6, 0, 6, 0, 0, 0, 0, 0, 0, 6, 0], [2, 0, 0, 9, 0, 1, 0, 1, -1, -1, 0, 0, 0, 0, 4, 0], [26, 0, -6, 0, 0, 6, 0, -6, -2, 2, 0, 0, 0, 0, 0, 0], [0, 5, 0, 0, 11, 0, 0, 0, 0, 0, -5, 5, 2, 2, 0, 0], [0, 10, 0, 0, -6, 0, 0, 0, 0, 0, -10, 10, 4, 4, 0, -4], [-2, 0, 0, -5, 0, 1, 0, 1, 3, 3, 0, 0, 0, 0, -8, 0], [0, -2, 0, 0, 2, 0, 0, 0, 0, 0, 2, -2, 2, 2, 0, -8], [7, 0, -2, 0, 0, -11, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0], [0, 17, 0, 0, 0, 0, -7, 0, 0, 0, -1, -1, 1, -1, 0, 0], [-35, 0, 0, 0, 0, -2, 0, 2, 1, -1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, -11, 0, 0, 0, 0, 0, -1, 1, -2, -2, 0, 4], [0, -6, 0, 0, 0, 0, -4, 0, 0, 0, 6, 6, -5, 5, 0, 0], [-10, 0, 0, 0, 0, -8, 0, -8, 2, 2, 0, 0, 0, 0, -4, 0], [0, -6, 0, 0, 12, 0, 0, 0, 0, 0, 6, -6, -1, -1, 0, 4], [0, 6, 0, 0, 0, 0, 6, 0, 0, 0, 6, 6, -3, 3, 0, 0], [8, 0, 8, 0, 0, 0, 0, 0, -6, 6, 0, 0, 0, 0, 0, 0], [2, 0, 0, -13, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 0, 7], [-3, 0, 0, 5, 0, 3, 0, 3, 6, 6, 0, 0, 0, 0, -6, 0], [8, 0, -10, 0, 0, -1, 0, 1, -1, 1, 0, 0, 0, 0, 0, 0], [0, -28, 0, 0, 0, 0, -4, 0, 0, 0, -4, -4, -2, 2, 0, 0], [6, 0, 0, 0, 0, 4, 0, 4, -2, -2, 0, 0, 0, 0, -4, 0], [-2, 0, 0, -2, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, -4, 0], [0, -3, 0, 0, 16, 0, 0, 0, 0, 0, 3, -3, 0, 0, 0, -4], [4, 0, 0, -4, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0, 2, 0], [0, 33, 0, 0, 0, 0, 2, 0, 0, 0, 3, 3, 3, -3, 0, 0], [0, 9, 0, 0, -2, 0, 0, 0, 0, 0, -9, 9, 0, 0, 0, -2], [-21, 0, -2, 0, 0, -7, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0], [0, -5, 0, 0, 0, 0, 7, 0, 0, 0, 7, 7, 2, -2, 0, 0], [-7, 0, 0, 8, 0, -5, 0, -5, 2, 2, 0, 0, 0, 0, 1, 0], [26, 0, 12, 0, 0, 4, 0, -4, 2, -2, 0, 0, 0, 0, 0, 0], [1, 0, 0, -20, 0, 2, 0, 2, 1, 1, 0, 0, 0, 0, 0, 0], [35, 0, -1, 0, 0, 10, 0, -10, -1, 1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 7, 0, 3, 0, 3, 2, 2, 0, 0, 0, 0, 2, 0], [-12, 0, 0, 12, 0, -9, 0, -9, 3, 3, 0, 0, 0, 0, -1, 0], [0, -4, 0, 0, 0, 0, -6, 0, 0, 0, -10, -10, 1, -1, 0, 0], [-24, 0, 0, 0, 0, 2, 0, -2, -2, 2, 0, 0, 0, 0, 0, 0], [2, 0, 6, 0, 0, 10, 0, -10, 2, -2, 0, 0, 0, 0, 0, 0], [0, -11, 0, 0, 8, 0, 0, 0, 0, 0, 11, -11, -4, -4, 0, 5], [0, 4, 0, 0, -6, 0, 0, 0, 0, 0, -4, 4, 3, 3, 0, -4], [28, 0, 2, 0, 0, 8, 0, -8, 4, -4, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, -4, 0, 0, 0, 0, 0, 4, -4, 2, 2, 0, 0], [2, 0, 0, -2, 0, 0, 0, 0, -2, -2, 0, 0, 0, 0, 4, 0], [0, 0, 0, -8, 0, 2, 0, 2, 2, 2, 0, 0, 0, 0, 3, 0], [0, 28, 0, 0, 0, 0, 4, 0, 0, 0, 4, 4, 3, -3, 0, 0], [17, 0, 1, 0, 0, 5, 0, -5, -6, 6, 0, 0, 0, 0, 0, 0], [0, 10, 0, 0, 0, 0, -8, 0, 0, 0, -2, -2, 5, -5, 0, 0], [6, 0, 0, -5, 0, 5, 0, 5, -1, -1, 0, 0, 0, 0, 0, 0], [0, 7, 0, 0, 5, 0, 0, 0, 0, 0, -7, 7, 2, 2, 0, -4], [0, -5, 0, 0, 0, 0, 4, 0, 0, 0, 1, 1, 5, -5, 0, 0], [0, -6, 0, 0, 2, 0, 0, 0, 0, 0, 6, -6, 4, 4, 0, 6], [0, -14, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 4, -4, 0, 0], [0, -5, 0, 0, -12, 0, 0, 0, 0, 0, 5, -5, -2, -2, 0, -1], [28, 0, 1, 0, 0, -7, 0, 7, 1, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 8], [2, 0, 0, 7, 0, 5, 0, 5, 3, 3, 0, 0, 0, 0, -8, 0], [0, 8, 0, 0, -12, 0, 0, 0, 0, 0, -8, 8, 0, 0, 0, -1], [0, -11, 0, 0, -4, 0, 0, 0, 0, 0, 11, -11, 1, 1, 0, 5], [-2, 0, 0, 1, 0, 4, 0, 4, 6, 6, 0, 0, 0, 0, 2, 0], [-1, 0, 0, 2, 0, 1, 0, 1, 2, 2, 0, 0, 0, 0, -3, 0], [0, 4, 0, 0, 0, 0, -8, 0, 0, 0, 4, 4, 3, -3, 0, 0], [-13, 0, -9, 0, 0, -4, 0, 4, -7, 7, 0, 0, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1386_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1386_2_g_a();".
// 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_g_a(:prec:=16)
    chi := MakeCharacter_1386_g();
    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_g_a();".
// 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_g_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1386_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![72, 0, -168, 0, 118, 0, -24, 0, 1]>],Snew);
    return Vf;
end function;