A two-generator upper bound for level-four parabolic cohomology

Let $n>0$ be an integer, and let $H^1_{\mathrm{par}}(\Gamma_1(4),\operatorname{Sym}^n\mathbf C^2)$ denote the quotient of parabolic one-cocycles by principal cocycles, with the binary-form action used in the MTT mission. Then

The estimate holds in both parities. In odd modular weight $k=n+2$, it supplies the cohomological half of the remaining level-four dimension comparison.