Dimensions of paramodular forms and compact twist modular forms with involutions

We give an explicit dimension formula for paramodular forms of degree two of prime level with plus or minus sign of the Atkin--Lehner involution of weight $\det^k\operatorname{Sym}(j)$ with $k\geq 3$, as well as a dimension formula for algebraic modular forms of any weight associated with the binary quaternion hermitian maximal lattices in non-principal genus of prime discriminant with fixed sign of the involution. These two formulas are essentially equivalent by a recent result of N. Dummigan, A. Pacetti. G. Rama and G. Tornar\'ia on correspondence between algebraic modular forms and paramodular forms with signs. So we give the formula by calculating the latter. When $p$ is odd, our formula for the latter is based on a class number formula of some quinary lattices by T. Asai and its interpretation to the type number of quaternion hermitian forms given in our previous works. On paramodular forms, we also give a dimensional bias between plus and minus eigenspaces, some list of palindromic Hilbert series, numerical examples for small $p$ and $k$, and the complete list of primes $p$ such that there is no paramodular cusp form of level $p$ of weight 3 with plus sign. This last result has geometric meaning on moduli of Kummer surface with $(1,p)$ polarization.