In this article, we settle a part of the Conjecture by Bandini and Valentino (\cite{ BV19a}) for $S_{ k,l}(\Gamma_0(T))$ when $\mathrm{ dim}\ S_{ k,l}(\mathrm{ GL}_2(A))\leq 2$. Then, we frame this conjecture for prime, higher levels, and provide some evidence in favour of it. For any square-free level $\mathfrak{ n}$, we define oldforms $S_{ k,l}^{ \mathrm{ old}}(\Gamma_0(\mathfrak{ n}))$, newforms $S_{ k,l}^{ \mathrm{ new}}(\Gamma_0(\mathfrak{ n}))$, and investigate their properties. These properties depend on the commutativity of the (partial) Atkin-Lehner operators with the $U_\mathfrak{ p}$-operators. Finally, we show that the set of all $U_\mathfrak{ p}$-operators are simultaneously diagonalizable on $S_{ k,l}^{ \mathrm{ new}}(\Gamma_0(\mathfrak{ n}))$.
Categories: math.NT