
// Make newform 2394.2.e.b in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2394_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2394_e_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_2394_2_e_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_2394_2_e_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 := [1, 4, 8, -4, 9, 46, 120, -114, 54, 0, 2, -2, 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], [-921139278, -500214763, -209312110, 5588841687, -21490137717, 15848202401, -5903566419, 241501496, -368064334, 281608184, -108477401, 1204841], [17917471, 97570007, -25486381, 227964888, 174081471, 523983773, -490503010, 249994355, -720282, 8749533, -8559599, 4581636], [-17917471, 28258239, 25486381, -227964888, -174081471, -523983773, 490503010, -249994355, 720282, -8749533, 8559599, -4581636], [37014000, 168826149, -299634772, -12824232, 300219893, 2640599530, -2292586049, 968263716, -8749533, 44394541, -40416578, 17917471], [230084943, 5943769, -59683238, -465780605, 6432847185, -1878067783, -644294087, 1072759908, 93036208, -35893384, -9775985, 20522401], [218567625, -290379430, -297472207, 420288932, 6933652460, -12542798654, 7928723841, -2297873802, 149544607, -216580592, 142257941, -41974538], [393280907, 353056424, 201117637, -1908017636, 8072273018, 7225262552, -8597681653, 4388653824, 74172527, 117210810, -150483059, 82307490], [-225912062, -1015647344, -308770920, 296518457, -1805551174, -16819275411, 14299745714, -6111978542, 56703661, -283529283, 252452879, -113238774], [377640732, 1375723819, 230288873, -1829381190, 5896142065, 18053195120, -16680815036, 7435401174, -5867022, 301705425, -293957423, 138418019], [184817236, 1128179249, 93793995, 724989514, -1445773037, 25001588644, -19980557836, 8124335370, -140795074, 423842543, -353497855, 149885313], [-760408626, -3146649797, 66395006, 5017252753, -5583891755, -69558395473, 58815486947, -24326352374, 251195486, -1173084792, 1040017769, -451390185]];
    Rf_basisdens := [1, 62914123, 62914123, 62914123, 62914123, 62914123, 62914123, 62914123, 62914123, 62914123, 62914123, 62914123];
    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_2394_e();"
function MakeCharacter_2394_e()
    N := 2394;
    order := 2;
    char_gens := [533, 1711, 1009];
    v := [2, 1, 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_2394_e_Hecke();"
function MakeCharacter_2394_e_Hecke(Kf)
    N := 2394;
    order := 2;
    char_gens := [533, 1711, 1009];
    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 := [[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, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [2, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 1], [0, 0, 1, -1, 0, -1, -1, 0, 0, -1, 0, 0], [1, 1, 0, -1, 0, 1, -1, -1, 0, 1, -1, 0], [0, 0, 0, 0, 0, 0, -1, 1, -2, 0, 0, 0], [2, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1], [1, -1, 0, -2, 0, 1, 0, 0, 0, 1, -1, 0], [-1, -1, 0, 0, 0, -1, 0, 0, 0, 1, -1, 0], [0, 0, 1, 0, 0, 0, -1, 1, -2, 0, 0, 1], [0, 0, 2, 0, -2, 0, -1, 1, 0, -2, -2, 0], [-3, -1, 0, 2, 0, -1, 2, 2, 0, -2, 2, 0], [0, 0, 2, 0, 1, 0, 0, 0, -1, -1, -1, 1], [0, 0, -2, 0, -3, 0, 0, 0, -1, 2, 2, 0], [0, 0, -2, 0, -2, 0, 0, 0, -2, 1, 1, 0], [0, 0, 0, 0, -3, 0, -2, 2, -3, -1, -1, 1], [0, 0, -3, 0, -5, 0, 0, 0, 1, 0, 0, 1], [0, 0, 0, 0, 2, 0, 0, 0, 0, -2, -2, 2], [2, 0, 0, 0, 0, -4, -1, -1, 0, -1, 1, 0], [-4, -1, 0, 0, 0, 0, -1, -1, 0, -2, 2, 0], [0, 0, 1, 0, -4, 0, -2, 2, -4, 0, 0, 0], [-2, 1, 0, -3, 0, 2, -2, -2, 0, 3, -3, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2], [0, 0, -3, 0, -4, 0, -1, 1, -2, -1, -1, -1], [0, 0, 0, 0, 3, 0, -1, 1, -1, 1, 1, -1], [0, 0, 3, 0, -1, 0, 2, -2, 1, -2, -2, 1], [0, 0, -2, 0, 0, 0, 3, -3, 0, 1, 1, 0], [2, -2, 0, 0, 0, 2, -1, -1, 0, 1, -1, 0], [0, 0, -2, 0, -7, 0, -1, 1, 1, -1, -1, 1], [6, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -5, 0, 1, 0, 0, 0, -1, 2, 2, 1], [0, 0, -2, 0, 1, 0, -2, 2, -1, 4, 4, 1], [4, -2, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0], [4, 2, 0, 2, 0, -2, -1, -1, 0, 0, 0, 0], [-3, 1, 0, -6, 0, -1, 0, 0, 0, -1, 1, 0], [0, 0, -2, 0, 3, 0, 3, -3, 5, 0, 0, -1], [-4, 0, 0, 1, 0, -2, 0, 0, 0, 2, -2, 0], [-7, -1, 0, 3, 0, 1, 1, 1, 0, 1, -1, 0], [0, 0, 0, 0, 4, 0, -1, 1, -2, 1, 1, 2], [0, -1, 0, -1, 0, 0, 3, 3, 0, -5, 5, 0], [0, 0, -2, 0, 2, 0, -5, 5, -4, -1, -1, 2], [0, 0, 0, 0, -5, 0, -2, 2, 5, -2, -2, 0], [6, 1, 0, -3, 0, 4, -1, -1, 0, 0, 0, 0], [2, 2, 0, 0, 0, 4, 0, 0, 0, 2, -2, 0], [0, 0, -2, 0, -3, 0, 0, 0, 3, -2, -2, 1], [6, 0, 0, 4, 0, 0, 0, 0, 0, -4, 4, 0], [3, -1, 0, -1, 0, -1, -1, -1, 0, -3, 3, 0], [-6, -1, 0, 0, 0, 0, 2, 2, 0, -3, 3, 0], [0, 0, -2, 0, -6, 0, 1, -1, 2, 3, 3, -2], [0, -2, 0, 0, 0, -6, 1, 1, 0, -3, 3, 0], [-5, 1, 0, 1, 0, 3, -1, -1, 0, 5, -5, 0], [13, -1, 0, 2, 0, -1, -1, -1, 0, -1, 1, 0], [0, 0, -2, 0, -6, 0, 3, -3, 2, 4, 4, 0], [-4, 4, 0, 2, 0, 2, 0, 0, 0, 4, -4, 0], [0, 0, -4, 0, 1, 0, -3, 3, -3, -1, -1, 1], [0, 0, -2, 0, 3, 0, 1, -1, -5, 3, 3, -1], [9, -1, 0, -2, 0, 3, 1, 1, 0, -1, 1, 0], [4, -2, 0, 0, 0, -2, 1, 1, 0, -1, 1, 0], [0, 0, -1, 0, 6, 0, 2, -2, 2, -1, -1, -3], [0, 0, 0, 0, 9, 0, 2, -2, 1, -3, -3, -1], [0, 0, 2, 0, -12, 0, -2, 2, -2, 2, 2, 0], [0, 0, 2, 0, 7, 0, 5, -5, 3, 3, 3, -2], [0, 0, 2, 0, 10, 0, 3, -3, 0, -3, -3, -2], [3, 2, 0, -4, 0, 1, -2, -2, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0, 1, 4, 4, 3], [0, 0, -6, 0, -2, 0, -2, 2, -4, -1, -1, 1], [1, 1, 0, 3, 0, 1, 5, 5, 0, -1, 1, 0], [0, 0, -2, 0, 16, 0, -1, 1, 0, 2, 2, -2], [0, 0, 3, 0, -4, 0, 5, -5, 2, 3, 3, -3], [0, 0, 2, 0, -11, 0, 2, -2, 3, 0, 0, -4], [-10, 0, 0, -2, 0, 2, -4, -4, 0, 2, -2, 0], [6, -3, 0, -9, 0, -4, 0, 0, 0, -2, 2, 0], [0, 0, 0, 0, 3, 0, 2, -2, 5, 3, 3, 3], [0, 0, 2, 0, -1, 0, 0, 0, -5, -4, -4, 1], [4, -1, 0, 4, 0, 0, 1, 1, 0, 0, 0, 0], [0, 0, 0, 0, -8, 0, 1, -1, 0, 5, 5, -2], [0, 0, 1, 0, 8, 0, 5, -5, 6, 1, 1, -1], [0, 0, 2, 0, 8, 0, -6, 6, -4, -4, -4, 4], [-8, -1, 0, 0, 0, 0, 3, 3, 0, -6, 6, 0], [10, 1, 0, -3, 0, -2, -6, -6, 0, -3, 3, 0], [-3, -2, 0, -4, 0, 5, -3, -3, 0, 1, -1, 0], [0, 0, 4, 0, -4, 0, -2, 2, -6, 0, 0, -2], [2, 0, 0, -2, 0, -4, 1, 1, 0, -5, 5, 0], [0, 0, -1, 0, -12, 0, -2, 2, 0, -4, -4, 0], [-6, -2, 0, 10, 0, 2, 3, 3, 0, 2, -2, 0], [0, 0, 0, 0, 12, 0, 0, 0, 6, -2, -2, 0], [0, 0, -1, 0, -10, 0, -2, 2, -6, 3, 3, 1], [0, 0, 1, 0, -3, 0, 0, 0, -5, 0, 0, 1], [3, 2, 0, -2, 0, -5, -1, -1, 0, -3, 3, 0], [-3, 3, 0, 0, 0, -3, -2, -2, 0, 3, -3, 0], [0, 0, -3, 0, 16, 0, 3, -3, 2, 4, 4, 1], [15, -3, 0, -4, 0, -3, -1, -1, 0, -7, 7, 0], [-2, -2, 0, -8, 0, 0, 3, 3, 0, 1, -1, 0], [2, 0, 0, -4, 0, -4, -1, -1, 0, 1, -1, 0], [-10, 0, 0, -4, 0, -4, 0, 0, 0, -5, 5, 0], [0, 0, 6, 0, 9, 0, -1, 1, -3, -7, -7, 2], [2, -1, 0, 1, 0, -4, -4, -4, 0, -2, 2, 0], [-5, 1, 0, 6, 0, -3, 0, 0, 0, 2, -2, 0], [0, 0, 2, 0, -10, 0, -3, 3, 4, -1, -1, -2], [3, -1, 0, 4, 0, -5, 4, 4, 0, -7, 7, 0], [0, 0, -4, 0, -18, 0, 2, -2, -2, 6, 6, -4], [0, 0, 2, 0, 6, 0, 4, -4, -4, -4, -4, 0], [0, 0, 6, 0, 16, 0, 3, -3, 4, 4, 4, 1], [0, 0, 0, 0, 6, 0, 2, -2, 2, 4, 4, -2], [-6, -1, 0, 6, 0, -6, 3, 3, 0, -2, 2, 0], [6, -3, 0, -7, 0, 2, -4, -4, 0, 3, -3, 0], [-4, 0, 0, 6, 0, 2, 0, 0, 0, 4, -4, 0], [4, 2, 0, 8, 0, 6, 0, 0, 0, 4, -4, 0], [0, 0, -6, 0, -14, 0, -3, 3, -4, 5, 5, 0], [10, 2, 0, 8, 0, 2, -1, -1, 0, 0, 0, 0], [0, 0, 0, 0, -2, 0, 2, -2, 4, 0, 0, -4], [0, 0, -6, 0, 5, 0, -1, 1, -5, 1, 1, 3], [0, 0, -7, 0, 10, 0, 4, -4, 6, 7, 7, -1], [4, -1, 0, -1, 0, 4, 5, 5, 0, 1, -1, 0], [0, 0, -2, 0, 5, 0, 3, -3, 3, 0, 0, -1], [10, 0, 0, 3, 0, 0, 2, 2, 0, -6, 6, 0], [0, 0, -4, 0, -8, 0, -7, 7, -6, 3, 3, 2], [1, 5, 0, -6, 0, -5, -2, -2, 0, 5, -5, 0], [0, 0, 2, 0, 21, 0, 1, -1, -1, 2, 2, 3], [0, 0, 6, 0, 26, 0, 3, -3, 6, 1, 1, -2], [14, 1, 0, 7, 0, -4, 2, 2, 0, -2, 2, 0], [4, -2, 0, -8, 0, 2, -6, -6, 0, 1, -1, 0], [0, 0, 7, 0, 10, 0, 2, -2, -2, -1, -1, -3], [0, 0, -10, 0, 0, 0, -1, 1, 0, 6, 6, -2], [0, 0, 0, 0, -15, 0, 2, -2, 3, 0, 0, 1], [6, 4, 0, 4, 0, 6, 0, 0, 0, 4, -4, 0], [0, 0, 0, 0, -6, 0, 6, -6, 6, -1, -1, 0], [0, 0, -7, 0, 21, 0, 6, -6, -1, 2, 2, -1], [18, 2, 0, -6, 0, 0, -2, -2, 0, 2, -2, 0], [0, 0, -8, 0, 12, 0, 3, -3, 0, 8, 8, 1], [0, 0, 8, 0, -7, 0, 0, 0, -3, -1, -1, -1], [23, 5, 0, 0, 0, 5, 5, 5, 0, 1, -1, 0], [10, -2, 0, -4, 0, 2, 0, 0, 0, 0, 0, 0], [5, 5, 0, 4, 0, -1, 0, 0, 0, 3, -3, 0], [0, 4, 0, -8, 0, 0, -5, -5, 0, 1, -1, 0], [22, 0, 0, 2, 0, 2, 2, 2, 0, 0, 0, 0], [-4, -1, 0, -1, 0, 4, 3, 3, 0, 3, -3, 0], [-12, 0, 0, -14, 0, -4, -6, -6, 0, 4, -4, 0], [0, 0, -6, 0, 7, 0, 3, -3, 9, 4, 4, -5], [8, -2, 0, 7, 0, 6, 4, 4, 0, 6, -6, 0], [-2, 6, 0, -8, 0, 4, -5, -5, 0, 2, -2, 0], [0, 0, -12, 0, -7, 0, 2, -2, -5, 5, 5, -3], [-4, 2, 0, 16, 0, 6, 3, 3, 0, -1, 1, 0], [0, 0, 2, 0, -14, 0, 7, -7, 8, -1, -1, 0], [0, 0, -8, 0, -2, 0, 2, -2, 2, 5, 5, 4], [0, 0, -3, 0, -12, 0, 0, 0, -8, 6, 6, -1], [0, 0, -2, 0, 18, 0, 10, -10, 4, 3, 3, -3], [-1, 1, 0, 0, 0, 7, 4, 4, 0, -3, 3, 0], [-6, 0, 0, -2, 0, 10, -4, -4, 0, 2, -2, 0], [0, 0, -2, 0, -5, 0, 4, -4, 9, -2, -2, 4], [0, 0, -8, 0, -2, 0, 6, -6, -2, 0, 0, 2], [2, -2, 0, -2, 0, 2, 4, 4, 0, 4, -4, 0], [0, 0, 4, 0, 10, 0, -6, 6, -4, -3, -3, 7], [0, 0, -4, 0, -15, 0, -4, 4, -3, 3, 3, -3], [-11, -1, 0, 4, 0, 3, -5, -5, 0, 3, -3, 0], [1, -2, 0, -2, 0, -5, 2, 2, 0, -2, 2, 0], [0, 0, 0, 0, 7, 0, 1, -1, 11, -1, -1, 3], [-6, -2, 0, -6, 0, 2, 4, 4, 0, 0, 0, 0], [-6, 2, 0, -4, 0, 0, 2, 2, 0, 4, -4, 0], [0, 0, -2, 0, 17, 0, 2, -2, -7, 2, 2, -1], [-4, 4, 0, 4, 0, 6, -3, -3, 0, -6, 6, 0], [0, 0, 3, 0, 13, 0, 4, -4, 7, 6, 6, -3], [0, 0, -6, 0, -7, 0, 2, -2, -1, 0, 0, 1], [0, 0, 0, 0, -12, 0, -3, 3, -6, -5, -5, 2], [0, 0, -7, 0, -14, 0, -6, 6, -2, -2, -2, -2], [0, 0, -2, 0, -13, 0, -7, 7, -3, 0, 0, 5], [10, -2, 0, 5, 0, -8, 6, 6, 0, -12, 12, 0], [15, 3, 0, -3, 0, -1, -3, -3, 0, 1, -1, 0], [-9, -3, 0, -2, 0, -7, -1, -1, 0, -1, 1, 0], [-30, 5, 0, 9, 0, -4, 3, 3, 0, 0, 0, 0], [0, 0, -6, 0, 8, 0, -2, 2, -6, 1, 1, 3], [4, -2, 0, 12, 0, 8, 2, 2, 0, 4, -4, 0], [-2, 1, 0, -1, 0, 8, 2, 2, 0, 2, -2, 0], [2, 3, 0, 3, 0, 2, -6, -6, 0, -3, 3, 0], [0, 0, 0, 0, -21, 0, -2, 2, 1, -1, -1, -1], [-14, 2, 0, 16, 0, 2, 8, 8, 0, -2, 2, 0], [17, -1, 0, -6, 0, -3, -4, -4, 0, 4, -4, 0], [0, 0, -15, 0, -4, 0, 5, -5, -2, 7, 7, -1], [-13, -1, 0, 2, 0, -5, -6, -6, 0, -3, 3, 0], [5, -7, 0, 9, 0, 1, 5, 5, 0, 1, -1, 0], [0, 0, -3, 0, -6, 0, 2, -2, -2, -4, -4, 2], [0, 0, -1, 0, 4, 0, 4, -4, 8, -8, -8, 1], [-8, 8, 0, 0, 0, 8, -6, -6, 0, 2, -2, 0], [0, 0, -2, 0, -12, 0, 2, -2, 8, 2, 2, -2], [0, 0, 1, 0, 14, 0, 0, 0, -2, -3, -3, -5], [18, 7, 0, -6, 0, 4, -10, -10, 0, 5, -5, 0], [-15, 0, 0, -14, 0, 1, -7, -7, 0, 5, -5, 0], [0, 0, 2, 0, -5, 0, -7, 7, -1, 5, 5, 4], [-17, -1, 0, 0, 0, -3, 0, 0, 0, -3, 3, 0], [-1, 4, 0, 4, 0, -1, 9, 9, 0, -1, 1, 0], [-19, 1, 0, -6, 0, 13, 2, 2, 0, 7, -7, 0], [7, -7, 0, -6, 0, -7, 5, 5, 0, -7, 7, 0], [-20, -2, 0, 4, 0, 10, -2, -2, 0, 2, -2, 0], [0, 0, -2, 0, -20, 0, 3, -3, 0, 0, 0, 5], [0, 0, 7, 0, -28, 0, -1, 1, 6, -4, -4, -1], [0, 0, -2, 0, 10, 0, 5, -5, -4, -2, -2, 0], [-26, 5, 0, -3, 0, 2, -4, -4, 0, -3, 3, 0], [10, 4, 0, -7, 0, 4, 4, 4, 0, 0, 0, 0], [-2, -1, 0, -6, 0, 2, 1, 1, 0, -4, 4, 0], [0, 0, -12, 0, -20, 0, -1, 1, 0, 3, 3, 2], [0, 0, -3, 0, -24, 0, 2, -2, 8, 2, 2, -4], [0, 0, -10, 0, 16, 0, 1, -1, 4, -4, -4, 6], [0, 0, -14, 0, -19, 0, -1, 1, 3, 6, 6, 3], [12, -2, 0, -4, 0, 6, -2, -2, 0, 10, -10, 0], [8, -4, 0, 10, 0, -6, -1, -1, 0, -1, 1, 0], [2, -8, 0, 2, 0, -8, 5, 5, 0, -7, 7, 0], [0, 0, 7, 0, 16, 0, -4, 4, -8, 2, 2, 0], [-6, -6, 0, 2, 0, 2, 1, 1, 0, -6, 6, 0], [5, -1, 0, 14, 0, 1, 0, 0, 0, 0, 0, 0], [-28, 4, 0, -12, 0, 6, -3, -3, 0, 6, -6, 0], [4, -1, 0, 4, 0, 6, 8, 8, 0, -1, 1, 0], [-4, 4, 0, -12, 0, -4, -6, -6, 0, 2, -2, 0], [-9, -5, 0, 10, 0, 7, 4, 4, 0, -1, 1, 0], [0, 0, 2, 0, -8, 0, 10, -10, 8, 4, 4, 4], [-20, -7, 0, 1, 0, 4, -1, -1, 0, 1, -1, 0], [0, 0, 7, 0, -16, 0, -2, 2, -4, -4, -4, -7], [-6, -3, 0, -7, 0, -2, -6, -6, 0, 5, -5, 0], [0, 0, -6, 0, -10, 0, -1, 1, -12, 7, 7, -6], [-4, 4, 0, -2, 0, 12, -2, -2, 0, 12, -12, 0], [-5, 1, 0, 6, 0, -5, 2, 2, 0, -12, 12, 0], [-2, -2, 0, 2, 0, -8, -2, -2, 0, -6, 6, 0], [-10, 6, 0, 4, 0, 8, 1, 1, 0, 15, -15, 0], [0, 0, 0, 0, 4, 0, -8, 8, 4, -5, -5, 6], [2, 3, 0, 7, 0, 10, -2, -2, 0, 13, -13, 0], [15, 0, 0, 4, 0, -1, 3, 3, 0, -3, 3, 0], [-10, -2, 0, -2, 0, 12, -4, -4, 0, -1, 1, 0], [18, -2, 0, 0, 0, -2, 4, 4, 0, -8, 8, 0], [0, 0, -1, 0, -27, 0, 0, 0, 3, 2, 2, -5], [0, 0, -2, 0, 18, 0, -1, 1, 0, -3, -3, -2], [0, 0, -6, 0, 9, 0, 5, -5, 5, 5, 5, 1], [0, 0, 3, 0, 2, 0, -8, 8, -6, -1, -1, 1], [0, 0, 0, 0, 31, 0, 4, -4, -1, 3, 3, 3], [8, -3, 0, 1, 0, -8, 3, 3, 0, -5, 5, 0], [13, -6, 0, 6, 0, -3, 5, 5, 0, -5, 5, 0], [30, 4, 0, 2, 0, -2, -6, -6, 0, 4, -4, 0], [0, 0, 12, 0, -16, 0, -10, 10, -8, -2, -2, 0], [0, 0, -2, 0, -20, 0, -5, 5, 0, 3, 3, 0], [0, 0, 10, 0, 0, 0, -3, 3, -10, -3, -3, 4], [0, 0, 7, 0, 44, 0, -1, 1, -2, -3, -3, 1], [-12, 4, 0, -4, 0, -10, -5, -5, 0, -7, 7, 0], [0, 0, -3, 0, 20, 0, 5, -5, 6, -4, -4, -3], [0, 0, 8, 0, -2, 0, -1, 1, 2, 0, 0, 6], [41, -5, 0, -14, 0, -7, 2, 2, 0, -8, 8, 0], [0, 0, 0, 0, 23, 0, -9, 9, -5, -7, -7, 0], [-1, -1, 0, -1, 0, -13, -5, -5, 0, -3, 3, 0], [0, 4, 0, -10, 0, -2, -8, -8, 0, 3, -3, 0], [0, 0, 6, 0, 20, 0, -1, 1, 8, -4, -4, 1], [-23, 1, 0, 13, 0, 5, 9, 9, 0, 5, -5, 0], [14, -1, 0, -6, 0, 2, -11, -11, 0, -4, 4, 0], [0, 0, 5, 0, -10, 0, -8, 8, -14, -2, -2, 0], [19, -2, 0, 0, 0, -7, 2, 2, 0, -8, 8, 0], [8, 6, 0, -4, 0, 2, -2, -2, 0, 5, -5, 0], [2, -4, 0, 6, 0, -6, -8, -8, 0, -6, 6, 0], [-23, -7, 0, 4, 0, -9, 0, 0, 0, -1, 1, 0], [0, 0, 16, 0, -5, 0, -6, 6, -5, -3, -3, -7], [0, 0, -7, 0, -17, 0, -2, 2, -3, 2, 2, 7], [0, 0, 4, 0, 12, 0, -6, 6, -8, -2, -2, 2], [0, 0, 4, 0, -11, 0, -10, 10, -1, -4, -4, 5], [2, 8, 0, -5, 0, 4, -2, -2, 0, 0, 0, 0], [0, 0, 8, 0, 6, 0, -9, 9, 6, -2, -2, 7], [-28, -4, 0, 18, 0, 2, 13, 13, 0, 1, -1, 0], [0, 0, 10, 0, -14, 0, -1, 1, 4, -4, -4, 0], [0, 0, 2, 0, -24, 0, 10, -10, 10, 1, 1, 3], [0, 0, 12, 0, -1, 0, -9, 9, -5, -5, -5, -6], [36, -1, 0, -5, 0, -8, 3, 3, 0, -1, 1, 0], [-6, -8, 0, 1, 0, -4, 2, 2, 0, -6, 6, 0], [0, 0, 0, 0, -13, 0, 1, -1, 11, -3, -3, -2], [0, 0, 0, 0, 7, 0, -4, 4, -1, 3, 3, 1], [22, 6, 0, -2, 0, 14, 0, 0, 0, 8, -8, 0], [2, -3, 0, -10, 0, 0, 2, 2, 0, -1, 1, 0], [0, 0, -8, 0, -30, 0, 1, -1, 0, 3, 3, 4], [-19, 2, 0, -2, 0, -3, -3, -3, 0, -5, 5, 0], [0, 0, -7, 0, -8, 0, 2, -2, -12, 6, 6, -1], [0, 0, -2, 0, -4, 0, -5, 5, 2, -3, -3, 0], [0, 0, 12, 0, -26, 0, -9, 9, -6, -5, -5, 4], [-10, 0, 0, 14, 0, 0, 5, 5, 0, -2, 2, 0], [16, 0, 0, -12, 0, -8, 3, 3, 0, -10, 10, 0], [0, 0, -5, 0, 14, 0, 8, -8, 10, 1, 1, 1], [22, -10, 0, -13, 0, -8, 0, 0, 0, -8, 8, 0], [0, 0, 14, 0, -19, 0, -5, 5, -3, -7, -7, -3], [-5, 5, 0, -3, 0, -9, -1, -1, 0, 7, -7, 0], [-18, 4, 0, 6, 0, 8, 3, 3, 0, -1, 1, 0], [0, 0, -8, 0, 4, 0, 1, -1, -6, -2, -2, -4], [0, 0, 0, 0, 12, 0, 15, -15, 14, 1, 1, -10], [31, -7, 0, -6, 0, -17, 6, 6, 0, -15, 15, 0], [24, 9, 0, -3, 0, 4, 1, 1, 0, 1, -1, 0], [0, 0, -10, 0, -4, 0, -3, 3, -4, 3, 3, 8], [0, 0, -4, 0, -29, 0, 7, -7, -1, -3, -3, 3], [-4, -8, 0, 0, 0, -8, 4, 4, 0, -2, 2, 0], [0, 0, -15, 0, 0, 0, 3, -3, 2, -3, -3, 3], [0, 0, 7, 0, 30, 0, 0, 0, 14, 0, 0, 2], [26, -1, 0, 11, 0, 8, 7, 7, 0, -2, 2, 0], [0, 0, 1, 0, -42, 0, 6, -6, 6, -2, -2, -2], [-29, -7, 0, 2, 0, -11, 0, 0, 0, 0, 0, 0], [0, 0, 6, 0, 33, 0, 7, -7, 3, -3, -3, -5], [0, 0, 8, 0, 4, 0, -8, 8, -4, -6, -6, 6], [0, 0, -16, 0, 10, 0, 9, -9, 8, 6, 6, -10], [6, 2, 0, 10, 0, -10, -3, -3, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2394_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2394_2_e_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_2394_2_e_b(:prec:=12)
    chi := MakeCharacter_2394_e();
    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_2394_2_e_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_2394_2_e_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2394_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![64, 0, 576, 0, 1680, 0, 1744, 0, 424, 0, 36, 0, 1]>,<13,R![40, 32, -92, 16, 24, -10, 1]>],Snew);
    return Vf;
end function;