I am referring for instance to this question about coefficients of automorphic forms on $GL(3)$. I know that the Ramanujan on average bound is known and gives $$\sum_{n^2 m < x} |\lambda(n,m)|^2 \ll x^{1+\varepsilon}.$$

Is there anything known (in terms of upper bounds, with explicit dependence in the fixed $m$ or $n$) about the partial sums (except trivially bounding by the above) : $$\sum_{n^2< x} |\lambda(n,m)|^2 \qquad \text{and} \qquad \sum_{m < x} |\lambda(n,m)|^2 ?$$