fig, ax = plt.subplots(figsize=(7, 7))
ax.set_facecolor("#ADBBFE")
psi_levels = [-10, -7.5, -5, 0, 5, 10, 20, 30, 40, 50, 60, 70]
contours = ax.contour(
    X, Y, psi,
    levels=psi_levels,
    colors="black",
    linewidths=1.6
)
ax.clabel(
    contours,
    inline=True,
    fontsize=13,
    fmt=r'$\psi=%g$'
)

# Add one arrow along each streamline
for level_segs in contours.allsegs:
    for seg in level_segs:
        verts = seg
        if len(verts) < 2: continue
        i = len(verts) // 2
        x0, y0 = verts[i]
        x1, y1 = verts[i+1]
        ax.annotate(
            '', xy=(x1, y1), xytext=(x0, y0),
            arrowprops=dict(
                arrowstyle='->',
                color='black',
                lw=1.8
            )
        )

vector_points = [(3.0, 4.0), (1.5, -0.5), (2.0, 0.0), (2.5, 1.0)]

for x0, y0 in vector_points:
    u0 = b
    v0 = -3*a*x0**2 - c
    ax.quiver(x0, y0, u0, v0, angles='xy', scale_units='xy', scale=10, width=0.006, color="#aee2ff")