Spherical Symmetry of Hydrogen Wavefunction

In order to have a spherically symmetric wavefunction, there must be no angular dependence. From the general solution to the wavefunction of the electron in the hydrogen atom , we see that the angular dependence is captured by the spherical harmonics . The only spherical harmonic with no angular dependence is the lowest harmonic $Y^0_0$ , which has quantum numbers $\ell = m = 0$ . Since $\ell = 0$ corresponds to the $s$ orbitals, the 3s orbital is spherically symmetric.

Note that the wording in the problem is plural because for instance $2p$ refers to 6 different electronic states, each with $m = \pm 1$ or $m=0$ and $s = \pm 1/2$ .