
// Make newform 7396.2.a.j in Magma, downloaded from the LMFDB on 11 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7396_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_7396_2_a_j();".
// 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_7396_2_a_j();".
// 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 := [-139, -2, 96, 7, -18, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [-59, 57, 23, -16, -2, 1], [-155, 41, 39, -16, -2, 1], [93, -25, -41, 10, 4, -1], [-321, 239, 165, -84, -18, 7]];
    Rf_basisdens := [1, 1, 16, 16, 8, 16];
    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_7396_a();"
function MakeCharacter_7396_a()
    N := 7396;
    order := 1;
    char_gens := [3699, 3701];
    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_7396_a_Hecke(Kf)
    return MakeCharacter_7396_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]];
    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], [1, 0, 1, 0, -1, 0], [1, -1, 0, 0, 0, 0], [0, 0, 0, 0, -1, 1], [-1, -1, -1, 0, 0, 1], [0, -1, -1, -1, 1, 0], [-1, 1, 2, 0, 0, 0], [1, 1, 1, 0, -2, 1], [0, 2, 2, 0, 1, 0], [1, -2, 1, 0, 1, 0], [3, 0, 3, 0, -2, -1], [-3, 0, -1, -1, 2, -1], [-1, -1, 1, -1, 3, -1], [0, 0, 0, 0, 0, 0], [5, 0, 3, -1, -4, 0], [3, -2, 4, 1, -4, -1], [3, 1, 0, -2, 1, -1], [-1, 0, 3, 0, 3, -2], [3, -1, -5, 0, -2, 1], [2, -1, 2, -1, 1, -1], [6, -1, 1, -3, 0, -1], [3, -1, 5, -2, 1, -3], [2, 2, 2, 3, -1, 1], [4, -1, -2, 0, -5, -2], [-12, -1, -6, -1, 4, 0], [7, 0, -1, -1, 2, -2], [-3, -3, 2, 0, -4, 2], [2, -1, -4, 0, -2, 1], [-3, -4, 1, 1, 1, 1], [-8, 2, -7, 2, 6, 0], [-1, 0, 7, 0, 1, 0], [5, -1, -1, 1, -2, -2], [2, 2, -1, 0, 2, -2], [3, 2, 6, -1, -2, 1], [7, 2, -1, -3, 3, -1], [3, -1, -7, 1, 8, 2], [9, 3, 7, 0, -1, 0], [9, 6, 11, 1, -2, -2], [4, -1, 7, 1, 0, 1], [-10, -2, -12, -2, 7, -1], [11, -1, 5, -2, -2, 0], [1, -1, 7, 0, -5, -4], [12, 0, 6, 2, -5, -2], [4, 4, 0, 1, 1, -5], [2, 0, -2, -1, -5, 3], [-12, -1, -11, -3, 11, -2], [2, -6, 0, -2, 4, 1], [1, -3, -8, 0, 10, 4], [10, 2, -6, 1, -1, -3], [2, 4, -2, -1, 7, -1], [0, 0, 0, -3, 2, 2], [-1, 5, -5, 2, -7, 0], [-7, 1, 8, -2, 1, 1], [9, 3, 3, 4, -2, 6], [1, -3, 1, -4, -6, 2], [8, -3, 3, -5, 4, -1], [-17, 3, -11, 2, 6, -1], [5, -2, -2, 5, -5, 2], [4, 2, 4, -2, 4, -7], [2, 4, 18, -2, -4, -3], [-10, -2, -6, -5, 3, -3], [3, -1, 10, -2, -7, -3], [-14, 4, -2, 2, 9, 1], [-5, -1, -4, 0, -1, -1], [-3, -7, -9, -1, -1, -1], [-2, -3, -11, -1, 12, 5], [-4, 6, -7, 0, 6, 0], [11, 2, -3, 3, -9, 2], [-4, 3, 10, -2, -11, -4], [6, 0, 3, 6, 2, 4], [-6, -1, -4, -3, -1, -1], [-12, 1, -8, -1, 12, 4], [14, -4, 4, -4, -2, -5], [14, 0, 10, -1, -6, 0], [-11, 5, -8, 4, -1, -1], [-10, -3, -6, -3, 6, 4], [15, -6, 3, -2, -14, -3], [-10, 1, -3, 3, 0, 1], [5, 0, 9, 6, -8, 3], [-4, 2, -4, 7, 0, 2], [-3, 0, 15, -1, -6, -4], [10, 2, -10, 0, 4, -1], [-7, 2, 1, -5, 8, 1], [5, -5, -1, 1, 1, -1], [10, 2, 4, 2, -12, -1], [-4, 3, -9, 1, -1, 6], [0, -2, -8, -2, 9, 4], [-3, 3, 4, 0, 2, -6], [-17, 4, -13, 4, 13, 0], [9, -6, 0, -3, 4, 1], [3, 6, 11, 5, 2, -3], [-2, -7, -12, -1, 14, 3], [-10, -1, -12, 2, 7, 8], [16, 2, 8, 1, -16, -2], [-6, -4, 1, -4, 7, -5], [-3, -2, -6, -3, 20, -3], [14, -2, -4, -6, -1, -1], [21, 0, -6, 1, -6, 1], [6, 4, -2, 2, 2, 0], [14, -3, 4, 0, -10, -5], [2, 0, 6, 3, -14, 0], [-14, 2, 2, -7, 10, -2], [-4, 2, -2, 0, 11, 0], [5, 1, 0, 2, -8, -2], [-11, 2, -11, -4, 13, 4], [3, -4, -1, 0, 9, 4], [-8, 3, 8, -3, -13, -5], [8, 3, -6, -1, 4, 4], [15, -4, -5, 6, 2, 3], [12, -1, 8, 5, -10, 1], [4, 2, 5, 4, 3, -5], [2, -5, -12, -3, -7, 5], [15, 2, 7, 1, 5, -3], [1, 2, -8, 5, -12, 5], [-3, -10, -3, -4, -6, -5], [6, 10, 6, -4, 4, -1], [15, -3, 1, -3, -2, -6], [11, 3, 17, 3, -8, 6], [-5, 2, 8, -3, 2, -3], [-10, -1, -4, -10, 13, -6], [15, 2, 8, 1, 11, 0], [17, 5, 19, 2, -8, -3], [-2, 2, -6, 8, 10, 9], [-5, 0, 8, -3, -7, 0], [14, -7, 4, -2, -9, -6], [13, -10, -10, 3, -6, -1], [-3, 0, -11, -2, 19, 4], [9, -10, -7, 0, 1, 4], [7, 5, 3, 6, 5, -4], [13, 4, 5, 2, -6, 1], [-3, -4, -6, -1, 12, -1], [21, -1, -1, 1, 11, 1], [-21, -3, -17, -2, 5, 6], [1, -6, 2, -1, -2, -1], [-9, -6, 0, 1, -4, 3], [17, -7, -8, -2, -4, -2], [21, -2, 3, -3, 10, -7], [-12, -3, -1, -9, -1, -8], [-1, 2, -1, 9, 4, 2], [-3, -4, 10, -3, 3, -4], [4, -4, 4, -8, 1, -1], [-7, 0, -11, 1, 1, -3], [20, -1, -10, 11, -5, 2], [3, -4, 2, 1, 8, 5], [-4, -3, 6, 6, -20, 3], [6, 8, 6, 5, -4, 6], [-15, -1, 5, 6, -8, 6], [11, 4, 3, 0, 11, -6], [-25, 3, -27, 2, 20, 6], [3, 2, 11, 0, -9, 0], [9, 0, 6, -3, -3, 2], [0, -1, 2, -4, -1, -2], [0, -1, -16, 1, -1, -3], [-6, 8, 13, 4, -5, -5], [10, 9, 10, 5, -9, -9], [-6, -2, 3, 2, -3, -1], [6, -5, 15, 3, -22, 1], [19, -10, 4, 1, -5, 0], [-2, 3, 0, 5, 3, 3], [11, -9, -6, 2, -13, -1], [23, -6, 15, 1, -24, -5], [12, -9, 8, -3, -8, 3], [-20, -5, -2, -2, 8, -3], [7, 0, 3, -2, 10, -1], [-2, 6, 6, -7, -2, -2], [-15, -8, -3, -3, -11, 6], [6, -8, 6, -5, 6, -12], [-1, 0, 20, -1, -29, -2], [19, 7, 9, 2, -16, 1], [-1, 5, 5, -4, 1, -6], [7, -1, 10, 0, -14, 0], [6, 4, -2, -10, 6, -5], [-6, 7, 16, -3, -14, -11], [20, -5, 12, -7, -11, -10], [-26, -3, 0, -1, -9, -2], [6, -3, -15, 7, -2, 11], [15, -6, 9, 3, -9, -3], [-12, 2, -20, 3, 26, 0], [34, -1, 28, 2, -13, 0], [18, 11, 6, 7, -26, -1], [-4, -6, 8, 3, 6, -4], [-31, -4, -16, 3, 16, -5], [-28, 1, 2, -2, 8, 3], [2, 4, -14, -1, -8, -6], [-24, 1, -28, 11, -9, 13], [11, 6, 15, 1, -20, -6], [-30, 3, -6, -1, 21, 4], [-18, 8, -9, 12, 3, 7], [-22, 4, 6, 6, 9, -4], [2, 2, 9, -2, -24, -4], [25, 3, 20, 4, -2, 2], [-19, -4, -3, -10, 8, -1], [5, 8, 20, 3, -8, -5], [7, -6, -1, -7, -2, 0], [22, -3, -17, 5, -8, -3], [-9, -2, -37, 1, 5, 1], [11, 3, 13, 3, -29, -1], [-10, -2, -6, 4, 13, 10], [-11, 15, 0, 8, 5, 3], [-2, -6, 8, 6, 4, 1], [-2, -1, 12, -2, 20, -1], [-7, 1, -6, 8, 17, 9], [-8, 5, 11, -3, -14, -7], [0, 6, -20, -5, 33, -3], [19, -3, 13, 1, 6, 2], [8, 5, 35, -7, 7, -8], [-14, -2, -8, 2, -2, 1], [-6, 6, -2, 0, 21, 3], [0, -10, -10, 4, 0, 8], [18, 1, 16, -1, -29, 6], [-1, 3, 19, 0, -17, -3], [15, 6, 43, -3, -22, -8], [30, -5, 5, -5, -6, 1], [4, 4, 7, 2, -6, 2], [-7, 3, -13, 1, 25, -5], [5, -10, 1, -8, -11, 4], [-3, 9, -5, 6, 3, -9], [22, 3, -12, 9, -6, -4], [-11, -2, -5, -3, -16, -4], [-6, 5, 28, -12, -12, -3], [-14, 6, 3, -4, -5, -11], [-10, 4, -16, 4, 14, -5], [20, 5, 3, -3, -6, 1], [10, -2, 12, -2, -21, -2], [32, 9, 23, 7, -35, 6], [8, 3, -9, -1, -6, -1], [1, 8, 25, 2, -19, -10], [2, 11, -6, 7, 0, 6], [-12, 1, 0, 7, -10, -2], [5, -1, 1, 10, -17, 4], [44, -7, -6, 1, -3, -3], [8, 2, -12, -3, 0, -2], [-21, 0, -11, -7, 22, -6], [-8, 4, 8, 8, 15, 12], [-21, 3, 1, -3, 3, 11], [9, -6, 17, -2, -20, 5], [2, -4, 23, -2, 13, 1], [-12, -11, -26, -2, 22, 1], [27, 1, 29, -8, -10, -2], [0, 6, -28, -2, 16, -14], [-21, -9, 1, 5, -10, 4], [2, -8, -7, -4, -12, 2], [0, 8, 1, 6, 12, -12], [-38, -3, -6, -7, 13, -2], [36, -1, -13, 11, -18, 3], [-24, -5, -28, -11, 34, 6], [-24, 3, 20, -7, 7, -1], [1, -5, -35, -6, 3, 8], [3, -3, 1, 1, 13, 3], [36, -5, 31, 5, -24, 1], [4, -5, 12, 5, -25, -4], [-10, 5, 5, -1, -3, -8], [-31, -3, -38, 0, 19, 3], [9, -1, -9, 5, -27, 7], [0, 5, -10, 10, -18, 9], [-29, -2, -43, 7, 19, 5], [-14, 5, 1, -9, -1, -10], [-38, 12, -18, 6, 18, 4], [-6, -1, -8, -6, 23, -8], [-10, 0, -14, 6, -4, 5], [8, 6, -4, -5, -6, -12], [-3, -10, 0, -11, -3, 8], [-6, -1, -11, 5, 21, 10], [-13, -4, 16, -5, -6, -5], [18, 11, 28, 5, -17, -1], [5, 3, 31, -13, 3, -7], [-18, -10, -22, -4, -1, 8], [5, 1, 13, 0, -30, -2], [35, 10, 15, -4, -1, -4], [13, -4, 6, -3, 2, -11], [11, 0, 11, 8, 6, 3], [-2, 4, 25, -4, 9, -3], [17, -9, 16, -4, -34, -2], [30, -2, -12, -6, -8, -3], [-43, 0, -3, -2, -3, 6], [-43, -3, -3, -8, 20, -7], [17, -6, 27, 1, -35, 5], [20, 10, 25, 6, 5, 1], [30, -11, 13, -5, -15, -4], [-28, -12, -31, -2, -3, 7], [-52, 0, -33, 2, 13, 3], [9, -2, -19, 6, 10, -5], [-15, -7, 7, -5, 24, -6], [-10, -12, -10, -1, 14, -4], [10, -1, 20, -12, 20, -9], [0, -8, 16, 7, -20, 0], [-8, -2, 8, -2, 31, -2], [-30, -3, -12, 3, -9, 5], [-6, -4, -21, 4, -3, 1], [12, 14, -8, 3, -10, 8], [-5, 10, -5, 13, 24, -2], [10, -4, 4, 8, 4, -1], [26, -4, 6, -1, -16, 6], [1, 1, -3, 8, 24, 11], [11, 1, -19, 1, -8, 0], [-34, -2, -26, -7, 21, 7], [-15, -4, 0, -1, -2, 1], [-5, 2, 16, 1, -39, -6], [15, -15, -23, 0, -11, -5], [-3, 11, 17, -6, 2, 1], [-7, -8, 17, 1, -12, 2], [-31, 4, 5, 1, 3, 3], [-43, -10, -11, 2, 0, 5], [19, 6, 11, 0, -8, 11], [-36, 1, -11, -1, 26, -3], [-28, 2, -24, 14, 2, 11], [-17, -5, -7, 6, 24, 6], [8, 12, 1, 4, 5, -5], [23, -2, -5, -8, -21, -8], [4, 6, 8, -2, -31, -8], [38, 11, 8, 13, -36, -1], [5, 15, 20, 8, 7, -3], [38, -9, -3, -3, -17, -8], [24, 12, 10, 10, -2, 3], [40, 4, 22, 6, -9, 0], [-32, -8, -24, -11, 16, 0], [-20, 11, -17, 13, -2, 15], [17, -6, 23, -3, -1, -5], [-13, -1, -9, -12, 10, -6], [-27, -1, -21, -11, 1, -9], [-16, 2, 10, 12, 1, 2], [-8, -12, 13, -4, -2, 2], [22, 13, 0, 7, 1, 3], [-12, 1, -22, -7, 18, -7], [4, -13, 14, -4, -5, 10], [0, 9, -10, 1, -12, 2], [12, 1, 34, -2, -10, 1], [-37, -16, -18, -7, 9, 8], [-18, -1, 0, -18, -7, -22], [1, 5, 7, 2, 6, 15], [-7, -9, -5, 4, -11, 10], [-4, -8, -6, -6, 35, -17], [7, 12, -5, -2, 36, -7], [8, -8, -40, 4, 9, 11], [-4, 5, 12, -5, -11, -5], [-36, -10, -4, -9, 10, 0], [11, 1, -10, -12, 20, -12], [20, 7, 20, -3, 1, -10], [-12, 11, 10, 16, -10, -1], [-40, 5, 11, -1, -4, 3], [0, 3, -6, -16, 31, -14], [3, 1, -19, 5, 3, 5], [10, 0, 15, -2, -26, 12], [-6, -1, 14, 3, -22, 8], [-7, -13, -1, -4, -16, 7], [27, 8, -14, -7, 13, -12], [-4, -1, 2, 7, -22, 10], [16, 21, 26, 0, -24, 3], [33, 9, -17, -5, -4, 2], [-32, -11, 11, 1, 11, -4], [17, -16, 5, -3, 9, -2], [-19, 17, -9, 11, -4, 2], [-18, -4, -9, -6, 21, -11], [6, -5, 8, 20, -14, 9], [21, 0, 17, 5, -34, -8], [4, -3, 2, -11, 7, -17], [-9, 5, -9, 14, 9, 6], [5, 2, 25, -2, -6, -9], [-31, -11, -33, 2, 46, -5], [10, 5, -14, -11, 12, -6], [18, -6, 6, 9, -7, 7], [41, 5, 36, 2, -30, -4], [-21, -12, -23, -7, -3, 1], [14, -2, -21, 2, 30, -2], [-28, 6, -41, 0, 16, -2], [-49, 4, -13, -4, 35, 2], [-21, -3, -46, -4, 50, 2], [16, 13, 16, 11, -36, 2], [5, 1, -12, 6, -16, 14], [-7, 11, 23, -8, 13, -13], [13, -10, -27, 12, -13, 10], [37, 8, 1, 5, -11, -6], [19, -4, 27, -8, 8, -19], [24, -7, 22, 6, -4, 7], [-46, -1, -8, 5, 2, -4], [38, -5, -8, 1, -1, -12], [2, -3, -18, 1, 26, -14], [17, 3, -4, 4, -10, -10], [20, -9, 2, -4, 19, -6], [-13, -9, -33, 8, 6, 14], [9, -4, 25, 2, 5, 2], [-3, 0, 6, 1, -8, -5], [-23, -6, -13, -11, 13, -7], [17, -7, 19, -18, 10, -25], [-2, 20, 22, -10, -4, -6], [-19, 13, 3, -3, 5, 5], [-1, 2, 8, -9, 25, -10], [6, 22, -2, 0, -3, -2], [-6, 6, 25, -2, -14, 2], [-13, 6, -29, -4, -3, -6], [-10, -11, 4, -1, -4, 16], [-16, 10, -19, -2, -4, 2], [24, -4, -9, -2, 20, -12], [9, 16, 13, -6, 16, -5], [-24, -3, 14, -14, 11, 4], [-4, 22, 8, 2, 19, 2], [5, -1, -10, 6, -13, -3], [59, -8, 19, -9, -2, -6], [2, -3, 8, 14, -5, 14], [13, 10, 17, 7, 4, -12], [33, -11, 3, 7, -3, 5], [-3, -7, 5, 0, -34, 14], [5, 0, -25, 7, 56, 4], [25, -2, 25, 4, -28, -15], [32, -8, 26, -14, -6, -8], [-42, -12, -20, -12, 8, -5], [5, -2, 13, 0, -2, 7], [13, 10, -1, -3, -25, 8], [36, 0, -15, 6, 2, 14], [-41, 4, -3, -7, 7, -5], [-26, 5, 21, -9, 26, -1], [-18, -5, 6, 11, -17, 22], [32, 4, 41, 6, -49, -7], [23, 7, -10, 8, -5, -1], [-9, -4, -17, -5, -9, 1], [21, -4, 25, -7, -12, -2], [-26, -9, 0, -2, -11, 2], [-17, -2, -33, -3, -3, 6], [-15, 7, -20, -8, 18, -2], [42, 1, -4, 2, -17, -10], [5, -6, -43, 3, 14, 22], [3, 13, 11, 6, -10, -4], [-26, 0, 14, 13, -26, 4], [-3, 2, -29, 5, -24, 10], [3, 4, 7, 4, -17, -14], [-14, -14, -29, 4, 8, 16], [-17, 10, -15, -9, 34, 6], [13, 8, -15, -2, -4, 3], [-15, 18, -1, -3, 11, -1], [-24, -20, -14, 2, 22, -3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7396_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7396_2_a_j();".
// 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_7396_2_a_j(:prec:=6)
    chi := MakeCharacter_7396_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_7396_2_a_j();".
// 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_7396_2_a_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7396_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, -2, -1, 1]>],Snew);
    return Vf;
end function;