I've recently been learning about the special values of symmetric square $L$-functions of modular forms.

If $f$ is a cuspidal modular eigenform (of some weight $k \ge 2$) then its symmetric square $L$-function is the $L$-function given by $\prod_{\text{$\ell$ prime}} D_\ell(\ell^{-s})^{-1}$, where for almost all $\ell$ we have $D_\ell(X) = (1 - \alpha^2 X)(1 - \alpha \beta X)(1 - \beta^2 X)$, $\alpha, \beta$ being as usual the roots of the Hecke polynomial $X^2 - a_\ell(f) X + \ell^{k-1} \varepsilon_f(\ell)$.

Now, it seems to me that the critical values of $L(\operatorname{Sym}^2 f, s)$ (in the sense of Deligne) should be the odd integers in the range $1 \le s \le k-1$, and the even integers in the range $k \le s \le 2k-2$. If you twist the symmetric square by a Dirichlet character $\chi$ with $\chi(-1) = -1$, then the parity flips and you see even integers in $\{1, \dots, k-1\}$ and odds in $\{k, \dots, 2k-2\}$.

What's puzzling me is an assertion in Hida's paper "P-adic L-functions for base-change lifts from GL(2) to GL(3)" (see link, page 93 onwards). Hida works with a different $L$-function, a shifted version of the adjoint $L$-function; so his $L$-function is $L(\operatorname{Sym}^2(f) \otimes \varepsilon_f^{-1}, s)$ in my notation. Thus the critical values of Hida's $L$-function, by my computation, should be the integers in $\{1, \dots, k-1\}$ with the opposite parity to $k$, and the integers in $\{k, \dots, 2k-2\}$ with the same parity as $k$ (so the "near-central" values $k-1$ and $k$ are always critical).

However, Hida states on p96 of his paper that the critical values of his $L$-function are the odd integers in $\{1, \dots, k-1\}$ and the even ones in $\{k, \dots, 2k-2\}$, regardless of the parity of $k$. Have I got this computation wrong, or is this a mistake in Hida's paper?

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