We demonstrate that designing a neural quantum state to be an exact eigenstate of the Hamiltonian's symmetries significantly improves both training speed and final variational energy. For the 2D electron gas, we design TorFormer, a neural network wavefunction which is an exact eigenstate of the total momentum. TorFormer describes both the Fermi liquid and Wigner crystal with no supervision and significantly outperforms Psiformer-based references up to large system sizes. For rₛ = 30.0 and 40.0 at N=91, we compare TorFormer trained for 8K steps against the previous best NQS, which required 100K training steps. Our improvement to the total energy at rₛ = 40.0, excluding the trivial Madelung part, is 0.12%---enormous compared to the tiny differences separating phases. Relative to Slater-Jastrow-backflow diffusion Monte Carlo, TorFormer's energy decrease is roughly 9.8 times that of the previous best NQS. Our work demonstrates that neural quantum states can both accurately and efficiently solve large-scale problems.