
// Make newform 3610.2.a.bi in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3610_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_3610_2_a_bi();".
// 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_3610_2_a_bi();".
// 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 := [-8, 228, -168, -455, 78, 183, -6, -24, 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, 1, 0, 0, 0, 0, 0, 0, 0], [6504, -15744, 1765, 24062, -8957, -5902, 2204, 360, -131], [3744, -2187, -17246, -4033, 9106, 1556, -1444, -111, 70], [-1580, 2832, 933, -2466, 415, 542, -140, -32, 9], [560, -12216, 9573, 14604, -9493, -3704, 1976, 236, -111], [-11726, 17771, 45620, -9371, -21732, 842, 3078, -9, -134], [-10888, 106136, -49539, -113178, 49091, 26298, -10032, -1580, 557], [1084, -30402, 28449, 43524, -26633, -11212, 5320, 706, -291]];
    Rf_basisdens := [1, 1, 3876, 1938, 204, 1938, 1938, 3876, 1938];
    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_3610_a();"
function MakeCharacter_3610_a()
    N := 3610;
    order := 1;
    char_gens := [2167, 3251];
    v := [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;

function MakeCharacter_3610_a_Hecke(Kf)
    return MakeCharacter_3610_a();
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, 1, 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], [1, 0, 0, 1, 1, 1, 0, -1, -1], [-1, 1, 1, 0, 0, 0, 0, -1, -1], [1, 1, -1, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 1, 1, -1, 0, -1, 0, 0, 0], [0, 0, 1, 0, 0, -1, 0, -1, 0], [1, 0, 2, -1, 0, 0, 1, 0, 0], [-1, -1, -1, 1, 1, 1, 0, -1, 1], [0, -1, 2, 0, 1, 1, 1, -1, 1], [2, 1, 0, 2, 0, -1, 0, 0, -1], [0, -1, 1, 2, -1, 0, 0, 0, 0], [0, 0, 0, -2, 0, 0, 0, -1, 0], [-2, 0, -3, 3, 0, 1, 1, -1, -1], [2, 2, 4, -3, -1, 0, -1, 1, 1], [0, 0, 0, 0, 0, -1, 0, 2, 2], [-2, 0, 4, -5, -1, 0, -1, 1, 1], [-4, 1, 1, 2, 0, 0, 0, 1, 0], [0, -2, -2, 2, 0, 2, -2, 0, 0], [0, -2, 2, -4, -2, -1, 0, 2, 2], [3, 2, 1, 2, -2, -2, 1, 0, -1], [-2, 0, -4, 3, 1, -1, 1, -1, -1], [-1, 2, 2, -4, 1, 2, 0, -1, -1], [-1, 2, -2, 1, 0, -2, 1, 1, 2], [2, -1, 0, 0, 0, -1, -2, 0, 1], [-2, -2, 5, -3, -1, -1, 1, 0, 1], [-2, -1, -3, 4, 0, 2, 0, 1, -2], [0, -3, 6, -3, 0, 0, -2, 1, 0], [7, 0, 5, 0, 0, -1, 1, 0, 2], [2, 1, -1, 1, 1, -1, -1, -2, 1], [5, 1, 1, -5, -1, 1, 1, 0, 0], [-1, 0, 3, 1, 0, -1, -1, 1, 2], [-5, 0, 0, -2, -1, 0, 0, 1, 1], [-1, -1, 5, 3, 1, 3, 0, -1, -1], [-2, 1, -2, 2, 2, -2, 0, 0, 2], [1, -1, -1, 3, 1, -1, 2, 0, -1], [0, 1, 4, -3, 0, 2, -2, 0, 0], [7, 1, 0, 1, -1, -1, 0, 0, -1], [-3, -2, 1, 0, -1, 1, 0, 0, -1], [4, -2, -4, -2, 2, -2, 0, 0, 2], [-7, 3, -1, -4, -1, -2, 0, 2, 0], [4, -1, 6, -2, 1, 0, -1, 1, 1], [6, 0, 0, 1, -1, 0, 1, -1, -1], [3, -4, 4, -2, -2, 1, 1, 0, -1], [3, -3, 1, 5, -1, 3, -2, 4, 1], [-2, -2, -6, 6, -2, -3, -2, 2, 2], [2, -2, -1, 0, 0, -3, 0, 3, 2], [2, -1, -5, 7, -1, 3, -1, 0, -3], [7, 0, 6, -7, 0, -2, 1, -2, 0], [-5, -4, 1, -3, -1, 0, 1, -2, 2], [7, 0, 2, -2, 0, 1, 4, -2, 3], [-2, -1, -5, -1, 1, 1, -1, 0, 1], [7, -1, -7, 7, -1, 3, -2, 0, 3], [10, 0, 3, 1, -1, 3, -1, 0, -5], [-1, 0, 7, 1, 0, 3, -3, 1, -2], [-7, -1, 1, -5, 1, 0, 1, -2, -2], [2, -5, -1, -1, 1, 1, -1, -3, -3], [2, -5, 2, -6, -2, 1, 0, 0, 0], [-6, -2, 5, 0, 0, -1, -2, 2, -2], [0, -3, -4, 0, 2, 1, 2, -2, 0], [1, 2, 1, -2, -1, 1, -2, 0, -7], [-10, 3, 3, -5, -1, -1, 1, 0, 1], [4, 1, -2, 1, 0, -2, 2, 2, 4], [-1, -1, 1, 3, 2, 0, -1, -2, -2], [-6, 1, -3, 5, 1, -1, -3, 0, -3], [14, 0, -6, 0, 0, -1, 2, 0, 0], [7, -4, -1, 6, -3, 1, 0, 4, 1], [-2, 0, 2, 1, -1, 1, -1, -1, -1], [0, -4, 7, -5, -1, -3, 3, -2, 3], [3, -1, -1, 3, 1, 2, 1, 0, -2], [7, -2, 1, 1, -2, -1, -1, -4, 0], [8, 1, -6, -6, 3, 1, 0, -1, -2], [11, 1, 9, -8, 2, 2, 4, -3, 1], [13, 2, -3, -9, -2, -3, -1, 1, 0], [2, -3, -5, 2, 1, 3, 1, 2, 1], [6, 3, 2, -3, 5, -1, 1, -1, 3], [-1, 3, -4, 15, 0, -1, -3, 1, -2], [15, -3, 3, 0, -3, 2, 0, 1, 0], [-9, 2, -1, 2, 1, 3, 2, -6, -1], [3, -6, 0, -1, -2, 0, -1, 0, -2], [10, -4, -2, -6, -2, -4, 2, 2, 6], [0, 2, -9, 12, -2, -1, -2, -1, 2], [2, -1, -2, -8, -2, -4, 0, 4, 4], [-2, -2, 4, -5, 4, 2, -2, 0, -4], [-6, -4, 4, 5, 5, 1, -3, -1, 1], [3, 0, -3, 5, 0, -3, 3, -3, 2], [-6, -5, 8, 1, 4, 0, 0, -2, 2], [0, -2, 6, 0, 0, -1, 4, 4, 2], [-5, 0, 1, -3, 0, -1, -5, -1, 2], [-7, 0, 5, -1, 4, -1, -1, -4, 0], [0, -2, -6, 2, 3, -3, 1, 0, -2], [-1, -1, -1, -9, 2, 0, -1, 2, -2], [1, 1, 9, -4, 4, 3, 1, -1, 4], [5, 2, -2, -3, 0, -6, 1, 4, 2], [9, 1, 1, -5, 0, -7, 1, 1, 2], [6, 0, 4, -2, 2, 0, -2, 0, -4], [-4, -4, 11, -4, 2, -1, 2, 3, 6], [-2, 1, -2, 4, -2, -1, -2, -4, -7], [-10, 2, 5, 6, -2, -1, 0, -4, -6], [0, 1, 6, 4, -2, -1, 0, 0, 4], [4, -5, -10, 2, 0, 5, 2, 1, 1], [-7, 0, -4, -2, -4, -5, 0, 4, 1], [-2, 1, 1, -6, -6, 0, 0, 1, 0], [-4, -3, 14, -2, -2, -1, -2, 0, 1], [-14, 1, -11, 2, 0, 2, 2, 1, 2], [1, -2, -3, 1, -4, 1, -3, 7, 2], [-9, 6, 3, -3, 5, 2, -2, 0, 1], [-5, -2, 10, -8, -3, 2, 2, -4, -3], [-14, -3, 3, -3, -2, -3, -2, 1, 4], [2, -1, -1, 0, 0, -4, 4, 1, 6], [7, -3, -5, 4, 1, 4, 0, -1, 2], [1, 10, -4, -6, 3, 2, 2, -3, -1], [1, 3, 7, -10, 6, 5, -2, -1, 1], [-16, 1, -4, 4, 0, -2, 0, 0, -2], [9, -5, 1, -2, 2, -3, -1, 2, 2], [9, -2, 5, -2, 3, 1, -2, 3, 5], [9, -2, 6, 2, 0, 1, 1, 0, 5], [8, -4, 2, -2, -4, -4, 0, 4, 4], [-1, 3, 1, -2, -5, -6, 4, 2, 0], [-11, -1, 3, 7, -1, -5, 4, 3, 3], [4, 3, 4, 8, -2, -1, 0, -4, 3], [0, 2, -16, 2, -3, -5, 3, 6, 4], [1, 2, 8, 8, 3, 6, 0, -7, -7], [9, -6, -8, 12, 1, -2, -4, 1, 7], [9, -2, -2, -2, 1, 2, 0, -1, -7], [-11, 0, 15, -11, -6, -7, 1, 5, 2], [-19, -6, 5, 0, -3, 1, 0, -1, 1], [-5, 1, 2, -1, -1, -7, 0, 0, 5], [17, -6, 6, -12, -5, -4, -2, 6, 5], [4, 2, -2, -1, -3, -4, -3, 1, -1], [-9, -7, -1, -7, 1, 5, 4, -2, 1], [-10, -2, 10, -3, 6, 2, 0, -4, 6], [-9, 6, 12, -12, 0, 2, 0, -4, -4], [-1, -1, -7, 8, 6, 1, 1, -4, 2], [2, 1, 6, -8, -2, 3, 2, -2, -4], [-14, 1, 5, 10, 1, -1, 1, 0, 1], [8, 4, 8, -7, -6, 2, 3, 0, -7], [5, 2, -6, -2, 3, 4, 2, 4, -5], [10, -2, -13, -2, -4, -9, 0, 9, 4], [-3, -1, -1, -3, 1, 5, 2, -1, 7], [-4, 1, -2, 0, -4, -3, -2, 4, 4], [-3, -4, 9, -26, -3, -1, 2, 0, -1], [12, 10, 13, -1, -3, -1, 1, -2, -9], [-5, 1, -4, 6, 0, -7, 2, -2, 3], [-6, 3, 3, -15, -3, -3, 3, 0, -9], [5, -3, -1, -1, 0, 0, -1, 2, -2], [-21, 1, 8, -17, 5, 4, -4, -7, -1], [-13, 2, -15, 7, 0, -1, 1, -2, -2], [4, -2, 1, 10, 6, 2, 2, -8, -2], [-4, 10, -2, -6, -2, -3, 2, -4, -2], [12, 3, -10, 7, -2, 5, -1, -2, -5], [6, -3, 16, -2, 2, 5, 2, -2, 0], [-3, 2, 8, -6, -5, -8, 4, 1, 3], [-3, -8, -6, 14, 3, 6, 4, -5, -3], [5, -7, 8, -9, -2, 1, -3, 7, -2], [-4, 5, -13, 3, 1, 1, -3, 6, 1], [-12, 4, 14, -3, 1, 0, -3, -1, -5], [2, -4, -4, 4, 2, 6, 2, 0, 2], [-14, -5, 7, -12, 0, 2, -4, 1, 0], [-13, -2, -1, 6, 3, 7, 0, -4, -1], [5, 2, -2, -18, -6, -1, 4, -4, 5], [-4, 1, -1, 13, 13, 5, -1, -4, -1], [-4, -3, -9, 3, 2, 1, 0, -8, 4], [-7, 6, 4, 6, -1, 2, 0, 1, -1], [-3, -9, -2, -9, -1, 0, 4, 2, -3], [2, -7, -13, 12, 5, 1, 3, 1, 5], [6, -8, 5, -1, 1, -1, 3, 3, 3], [-5, 2, -1, 6, 8, 3, 1, -4, -2], [-4, 4, -8, 7, -1, 2, 1, -9, 1], [-9, 2, -25, 21, 2, 1, -3, 1, 2], [-4, 3, -1, -9, -5, -3, -5, 10, -3], [-8, 2, 2, -7, -7, -2, 1, 5, -1], [1, 1, 6, -3, 0, -7, -1, 3, 0], [-4, 4, -13, 5, 0, 1, 3, -3, 5], [12, 2, -4, -24, 2, 0, 0, 2, 2], [3, -3, 8, 1, 7, 0, 2, 2, 13], [15, 4, 3, -1, 4, 5, 3, -7, 2], [-1, -3, 13, -1, 5, 1, 4, 2, -1], [-15, 5, -3, -6, 1, 0, -2, -3, -6], [8, -2, -20, 13, 5, 6, 1, -2, 3], [10, -9, -1, -20, -2, 0, 6, 1, 4], [16, 1, -9, 13, 2, 1, 0, -1, -6], [10, -2, -4, 5, -5, -6, 1, 7, -5], [-6, 1, 18, -17, 2, 3, 3, 2, -1], [24, -5, -2, 4, -4, -5, 2, 6, -1], [-11, 9, -3, 0, -6, -9, -4, 5, 3], [-1, -2, 9, 2, -3, -3, 6, -2, -3], [-8, -14, 0, -17, -1, 1, 3, 3, 5], [6, 4, -8, -2, 2, -2, 2, 4, 2], [-5, -7, 2, -12, 3, -2, 0, 7, 1], [7, -4, -6, 23, 6, 2, -3, 0, -2], [14, 5, 0, 2, -2, -4, 6, -2, -6], [-5, -7, -17, 2, -3, -6, 0, 8, 6], [5, -1, 0, -4, 0, -5, -1, -1, 6], [-1, -3, 3, -4, -4, -3, -9, 7, 0], [-4, 3, -3, -4, -6, -8, 2, 5, 12], [0, 13, 14, -6, 1, -2, 3, -2, -5], [-5, -2, 8, 0, 3, 4, -4, 1, 1], [1, 0, 6, 9, 2, 6, -3, 0, 2], [5, -3, 6, 14, 4, 4, 1, -3, 4], [1, -2, -10, 17, 9, 0, -4, 2, 7], [5, -1, -6, -11, -3, -7, 4, 4, 3], [10, -4, 9, -14, -6, 1, 0, 0, -4], [20, -4, -4, 5, -3, 0, -1, -3, -5], [-2, -3, 6, -18, 2, 2, 2, 0, 2], [1, -1, 9, -16, -6, -9, 0, 9, 11], [-16, 0, -6, 1, 0, -3, -4, -3, 0], [-20, -12, -8, 9, 3, 1, 1, -1, 5], [-9, 0, -20, 15, 4, 0, -1, 0, -2], [-5, 17, -5, 2, 0, 0, 6, -3, -1], [4, -4, 0, -14, -10, 1, 4, 2, 2], [-7, 0, -7, -4, -5, -3, 4, 4, -5], [0, 3, -16, 2, -8, 1, 2, 9, -5], [-10, 8, 1, 10, 0, 5, 2, -3, -6], [9, 9, 10, 6, 1, -1, -4, 1, 4], [4, -7, -9, 19, 10, 6, -1, -5, 1], [9, -1, -4, 2, 0, -5, 2, 2, 7], [1, 10, -2, -15, 0, 0, 5, -6, 4], [-8, 8, -13, 10, -2, 1, -4, 5, -4], [9, 2, -7, 7, 3, 4, 2, -4, -3], [-6, 7, 4, 2, 1, 9, 0, -11, -2], [4, 0, -2, -10, -4, 3, -4, 2, -6], [2, 4, 2, 8, -2, 4, -4, 2, -12], [-16, -2, 16, -1, 3, -1, 3, -3, 3], [-27, 4, -1, 0, 1, -3, 0, -2, 3], [13, 1, 8, -6, 2, 2, -5, 0, -2], [-9, -1, -17, 14, -2, -1, -5, 3, 6], [-12, -6, -13, 1, 1, -1, 5, -5, -1], [6, 4, 2, -5, 0, -7, -2, 5, 6], [-13, 2, -12, 10, 9, 0, -4, -1, 7], [9, 2, 8, 7, -1, 0, 6, -6, -11], [-10, -1, 4, 14, 10, 5, -2, 0, 0], [3, 5, -21, 24, 0, -7, -1, -3, 0], [14, 1, 12, -2, -1, -5, 1, -1, -3], [-10, -11, 6, -9, -4, -6, 4, 3, 10], [38, 2, 15, -1, 1, 2, 3, -3, -7], [-14, 8, -2, -1, 7, -6, -1, -1, 5], [10, 7, 0, -4, -4, 2, 0, 0, 0], [-20, 2, -7, 0, 1, 5, -3, -6, -11], [-23, 1, 17, -26, 0, -4, 2, -1, -1], [-7, -10, -6, -4, -7, -8, 2, 7, 5], [-29, 7, -7, 4, -1, 2, 0, 6, 0], [14, -4, 1, 5, -5, 1, -3, 6, -3], [-19, 0, 13, -14, 1, 1, -4, -4, -7], [23, 6, 5, -5, -6, -6, -1, 5, -4], [6, 1, 10, 12, -2, -4, 0, 3, 12], [13, 3, 10, -5, -5, 7, -3, 1, -6], [-7, -2, -26, 25, 4, 0, -1, 1, -2], [7, 2, -5, -2, -8, 0, -1, -2, -15], [16, -7, 10, -19, -4, 6, -2, 0, -10], [9, 12, 9, -3, -3, -8, 4, -6, -9], [8, -5, 14, -25, 2, 2, 4, 1, 4], [0, -4, 11, -6, 1, 5, 4, -6, -3], [19, -4, -6, 16, 1, -4, -4, 5, -1], [18, 0, 18, -18, -2, -1, 2, -2, -6], [-2, 7, 19, 4, -1, -9, 3, 1, 3], [9, 11, 9, -8, -5, -4, -2, -6, -2], [7, 7, 3, -18, -4, -1, 3, -3, -4], [20, 7, 10, -8, -4, -3, -2, 2, -6], [7, -6, -1, 6, 7, 7, 6, -4, 9], [5, 7, 4, -13, -3, 5, -3, -3, -14], [-2, -2, -6, 7, 9, 3, -1, -9, -7], [-4, -2, -19, 12, 5, -5, 1, 6, 7], [-12, -2, -16, 22, 4, 0, -6, 2, 2], [11, 12, -2, 15, -1, 0, 0, -6, -11], [20, 1, 4, -20, 4, -1, -4, 2, 7], [-6, -13, -20, 15, -2, 7, -5, 0, 3], [-7, 4, -23, 11, -8, -3, -1, 3, 4], [26, 7, -12, 2, 2, 1, -4, -2, -5], [10, -2, -6, -7, -1, -1, 1, -1, -5], [-4, -2, -21, 24, 6, 5, 4, -3, -2], [3, 7, -3, -2, -3, -6, 4, 10, 8], [6, 3, 21, -3, 0, -3, 0, -6, -8], [8, 5, -18, -10, 2, 4, -4, -8, -2], [-27, 8, 4, -3, 6, 4, -1, -2, -12], [10, 1, 22, -9, -1, -7, 3, 5, 3], [7, 6, 2, -26, 1, -2, 6, -4, -3], [5, 2, -4, -19, -2, 0, -3, 10, 8], [-1, 0, 15, -31, 4, -1, -1, -7, 2], [0, 0, 20, -6, 6, 14, -6, -2, -2], [-18, 6, -18, -9, -1, -2, 1, 3, 3], [-4, -9, 16, 4, -6, -11, 0, 6, 13], [-6, 6, -19, 0, -2, -9, 2, 11, 8], [-6, -3, 11, -10, 10, 6, -4, -1, 2], [1, 12, -21, 2, -1, 1, -4, 8, -7], [-13, -8, -15, 4, -1, -3, -2, 8, -1], [-29, 2, 14, 10, 2, 14, -2, -4, -2], [6, -10, 12, -12, -6, 4, 8, 2, 4], [14, 9, -14, 16, 2, -1, 0, 8, 3], [4, 2, 6, 7, -5, -9, -1, 9, 9], [22, 6, 3, 10, 7, 9, 0, -12, -7], [21, -2, -1, 14, 7, 5, 0, -12, -11], [-3, -4, 12, -2, 5, 4, 6, -3, 7], [-11, 4, 10, -20, -1, 0, -2, 3, -5], [5, -4, 3, 14, 7, 1, -6, 5, -3], [-16, 2, -4, 0, -1, 1, -7, 4, 6], [8, 0, -16, 10, 2, -9, 6, 6, 2], [-6, 11, -17, 7, 3, -1, -3, -8, -3], [-13, -1, -8, -10, -6, 0, 1, 13, 4], [10, -4, -11, -3, -1, 1, 7, 6, 5], [-10, 10, -24, 18, -2, -3, 2, 4, 6], [19, 1, -16, 21, -3, -5, 2, 4, 5], [2, -11, -7, 9, -3, -11, 3, -2, 13], [28, 1, 0, -2, -6, 9, 2, -2, -6], [0, 14, 6, 3, -7, -4, -3, 5, -5], [22, 0, 1, -15, -3, 1, 1, -4, 1], [8, 9, -1, -19, -6, 2, 5, -1, -15], [21, -4, -12, 6, 5, -4, 0, 4, 11], [13, -4, 8, 3, 0, 0, -3, -6, -8], [2, -6, -3, 9, 0, 20, 2, -8, -8], [20, 2, 0, 20, -4, -1, 2, 4, 0], [-12, -8, -8, 13, 3, 4, -5, -2, -5], [8, 6, 23, -4, 5, 8, 7, -14, -8], [-4, 5, -16, 15, 0, -5, 4, 0, 4], [-4, 4, -15, 24, 6, 9, 0, -7, -16], [-2, -6, 10, -1, -3, -5, 1, 5, 13], [10, 15, -2, -12, -4, -3, 8, -5, -9], [7, -9, 24, 3, -3, -1, -2, 10, 3], [-26, -1, -7, -5, 5, 1, 1, -4, -11], [-8, 8, 0, 2, -12, -8, 6, 10, 4], [2, 4, 5, -6, -8, -1, 2, -3, 0], [23, 3, 15, -14, -9, -2, -4, 10, -2], [4, 13, -4, -2, -5, -3, 4, 5, 2], [2, -4, -1, 9, 7, 6, -3, 1, 7], [26, 7, 8, -2, -2, -8, 2, -2, 4], [-1, -9, -7, 17, 5, -1, 0, -2, 11], [11, -2, 22, -2, 5, 4, -2, -8, -3], [-6, -4, -8, 31, 3, 8, -1, -1, 3], [-2, -9, 5, 1, 4, -2, -3, 3, 13], [4, 16, 10, -1, -1, -4, 7, -7, -9], [10, 0, 2, -4, -10, -8, 2, 4, 8], [38, 2, 9, 1, -1, -6, -3, -3, -3], [6, 9, -2, -2, 10, 5, 2, -2, 5], [2, 8, -2, -5, 7, 4, -3, -15, -3], [-10, -3, 7, 6, 2, 2, 6, 5, 2], [-1, -2, -4, -10, 4, 8, 2, -12, -4], [-4, 15, 22, -9, -2, 3, -5, -2, 1], [18, 15, 3, -10, -1, -5, -1, 0, 3], [-8, 11, 17, 0, 2, -2, 4, -1, -4], [-19, 10, 3, -8, 1, -7, -4, -4, -5], [-18, 0, 5, -12, -2, 9, 0, 3, -10], [-11, -1, -20, 5, 7, 2, -4, 2, 7], [0, 10, 19, -21, 7, 4, 3, -1, 3], [8, -8, 23, -33, -5, 1, 3, 0, -3], [32, 13, -28, 20, -4, 7, -4, -2, -7], [-2, -10, 15, -21, 3, -9, 3, 0, 15], [2, -9, 7, -7, -4, -13, 0, 7, 0], [38, 5, 16, 0, 2, -4, -4, 3, 2], [-28, -1, -3, 6, -6, 0, -8, 1, -2], [-8, -20, -14, -14, -8, 6, -2, 6, 0], [19, 4, 0, -5, -2, -4, -1, 2, -8], [-11, -8, -5, 17, 2, 1, -5, 3, 12], [-14, 15, -5, 0, 0, 2, 4, 3, -6], [-1, -4, -9, 23, 6, 11, -5, 1, 10], [-4, 6, 16, -9, 0, -11, -4, -3, -2], [32, 2, -8, 11, -3, 1, 7, 5, 3], [-6, -5, 6, 8, -1, 1, 2, 5, 8], [-35, 3, 4, -12, -2, 2, 7, -7, -6], [1, -12, -9, -13, -5, 2, 3, 0, 2], [0, -9, -2, 23, 14, 2, -8, -1, 0], [10, 3, -28, 22, 4, -6, 6, -3, 6], [-32, 9, 2, -14, 6, -7, -4, -2, 3], [11, 5, -9, 22, 3, -4, 8, 0, 10], [-16, 1, -22, -3, -4, -2, -2, 3, -12], [-2, -1, 21, -9, 2, -8, 7, 1, 7], [-4, -14, 19, 0, 4, 0, -4, 2, 4], [28, 0, -18, 27, 2, 1, -10, -1, 0], [-13, 11, -1, 12, 4, 15, -3, -11, -20], [-14, 3, 19, 10, -5, -7, 3, 9, 9], [6, -6, -24, 13, 5, -2, 1, -5, -5], [24, -2, 10, 11, -5, -8, 3, 9, 13], [8, 5, 3, -9, -6, -2, -5, -1, 9], [10, 8, 7, -4, -9, -19, 6, 4, 1], [3, 0, -15, 14, -1, -11, 4, 0, 1], [-16, -2, -16, 15, 3, -5, -5, 7, 5], [-28, 10, -6, 9, 8, 2, -5, -4, 5], [-14, -3, -7, -6, 2, -2, 2, 9, 12], [-5, -4, -5, -10, -5, -7, 2, 8, 19], [9, 3, -5, -8, 9, 2, -6, -4, 4], [-40, 9, 9, -1, -4, -3, -2, 3, 0], [12, 10, 16, -11, 5, -5, -1, -5, 5], [19, 12, 24, 7, -1, -1, -4, 1, 5], [17, 7, 3, -18, -4, 3, 1, 4, -10], [41, 2, -5, -2, 3, -3, 10, -2, 1], [-8, 2, -6, -7, -7, -10, -7, 4, 11], [-2, -11, -2, 8, -10, -4, -2, 2, -6], [28, 6, 11, 4, 0, 7, -8, 7, -6], [-14, -10, 3, 25, 2, 2, -8, -2, -6], [-22, -12, -12, -4, 4, 12, -2, -3, -10], [-19, 3, 27, -11, -1, 7, 3, 2, 0], [10, -1, -28, 20, -2, -5, 2, 10, 8], [7, -8, -28, 32, 11, -2, 0, -3, 9], [14, 6, 12, 3, -7, -5, 1, 9, 17], [18, 2, -31, 15, -7, -3, 3, 6, 7], [6, -2, 3, -20, -6, 8, 4, -4, -8], [20, 2, 3, -5, 2, -9, 5, -9, -13], [10, -4, -21, -1, -13, -3, -5, 14, 11], [13, -6, -30, 23, 0, -2, 1, 4, 10], [4, -4, -2, -5, -7, -1, 1, -5, 1], [-5, -15, -25, 18, 10, -4, -2, -1, 11], [-5, -11, -1, 20, -5, 0, -2, 4, -2], [27, 22, 5, 0, -7, -15, 8, -2, -1], [-15, 10, -10, 7, 4, -2, 3, -12, -4], [-18, -11, 10, 7, 8, 7, -1, 8, 1], [-9, -5, 30, 1, 2, 9, -1, -7, 0], [-6, 3, 10, -20, -16, -9, 2, 6, -3], [-9, -7, -15, 9, 9, 13, -1, -2, 0], [-14, 9, -13, 14, 4, 10, -6, 1, 4], [5, 3, -4, 8, 4, 2, 1, -7, -8], [-10, -2, 34, -32, 2, 3, -2, 2, 0], [20, -7, 0, -12, -4, 6, -4, 7, 2], [-6, 6, 0, -21, -11, -11, 3, 5, 1], [10, -2, -24, 34, 12, 4, 2, -6, 6], [-23, -8, -2, 16, 7, 14, -2, -3, 5], [23, -5, -16, 7, 13, -3, -3, -7, 2], [-12, -6, 14, -2, -4, -4, -2, 6, -4], [-16, 3, 5, -13, -2, 3, 4, -2, -2], [20, 12, 20, -16, -4, 0, 6, -4, -2], [-12, -13, 5, 25, -2, -1, -8, 7, 0], [17, 20, -16, 5, 6, 4, 1, 2, -10], [-11, 2, 1, -16, 2, 9, 1, -10, -20], [24, 12, 20, -20, -8, -10, -4, 8, 6], [-5, 0, -12, 11, 4, 4, -9, 4, 8], [-28, 2, -26, 2, 4, 3, -6, 0, -4], [15, 19, 11, 6, -4, -1, 5, -1, -4], [19, -9, -23, 16, 5, -8, -4, 1, 6], [-12, -18, 1, 28, 14, 9, -2, 1, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3610_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3610_2_a_bi();".
// 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_3610_2_a_bi(:prec:=9)
    chi := MakeCharacter_3610_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_3610_2_a_bi();".
// 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_3610_2_a_bi( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3610_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-8, 228, -168, -455, 78, 183, -6, -24, 0, 1]>,<7,R![-136, -5592, 9372, -3540, -1236, 804, 35, -51, 0, 1]>,<13,R![-13544, -27240, -8136, 8288, 3894, -486, -415, -27, 9, 1]>],Snew);
    return Vf;
end function;