We calculate axisymmetric magnetic modes of a neutron star possessing a mixed poloidal and toroidal magnetic field , where the toroidal field is assumed to be proportional to a dimensionless parameter \zeta _ { 0 } .
Here , we assume an isentropic structure for the neutron star and consider no effects of rotation .
Ignoring the equilibrium deformation due to the magnetic field , we employ a polytrope of the index n = 1 as the background model for our modal analyses .
For the mixed poloidal and toroidal magnetic field with \zeta _ { 0 } \not = 0 , axisymmetric spheroidal and toroidal modes are coupled .
We compute axisymmetric spheroidal and toroidal magnetic modes as a function of the parameter \zeta _ { 0 } from 0 to \sim 1 for the surface field strengths B _ { S } = 10 ^ { 14 } G and 10 ^ { 15 } G. We find that the frequency \omega of the magnetic modes decreases with increasing \zeta _ { 0 } .
We also find that the frequency of the spheroidal magnetic modes is almost exactly proportional to B _ { S } for \zeta _ { 0 } \lesssim 1 but that this proportionality holds only when \zeta _ { 0 } \ll 1 for the toroidal magnetic modes .
The wave patterns of the spheroidal magnetic modes and toroidal magnetic modes are not strongly affected by the coupling so long as \zeta _ { 0 } \lesssim 1 .
We find no unstable modes having \omega ^ { 2 } < 0 .