Symmetrized density-matrix-renormalization-group calculations have been carried out, within Pariser-Parr-Pople Hamiltonian, to e_xplore the nature of the ground and low-lying e_xcited states of long polythiophene oligomers. We have e_xploitedC₂ symmetry and spin parity of the system to obtain e_xcited states of experimental interest, and studied the lowest dipole allowed e_xcited state and lowest dipole forbidden two photon state, for different oligomer sizes. In the long system limit, the dipole allowed e_xcited state always lies below the lowest dipole forbidden two-photon state which implies, by Kasha rule, that polythiophene fluoresces strongly. The lowest triplet state lies below two-photon state as usual in conjugated polymers. We have doped the system with a hole and an electron and obtained the charge e_xcitation gap and the binding energy of the 1¹B_u⁻ exciton. We have calculated the charge density of the ground, one-photon and two-photon states for the longer system size of 10 thiophene rings to characterize these states. We have studied bond order in these states to get an idea about the equilibrium e_xcited state geometry of the system. We have also studied the charge density distribution of the singly and doubly doped polarons for longer system size, and observe that polythiophenes do not support bipolarons.