Harder's Euler-factor conjecture for Siegel modular forms

Let $k$ and $j$ be non-negative integers with $j$ even and $k\geq 3$. Let $f=\sum a(n,f)q^n\in S_{2k+j-2}(SL_2(\mathbb Z))$ be a primitive form, and suppose that a sufficiently large prime ideal $\mathfrak p$ of $\mathbb Q(f)$ divides the normalized critical value $\mathbf L(k+j,f;c_{s(k+j)})$. For a Hecke eigenform $F\in S_{(k+j,k)}(\operatorname{Sp}_2(\mathbb Z))$, let $L_p(X,F,\operatorname{Sp})$ and $L_p(X,f)$ denote its spinor Euler factor and the Euler factor of $f$, respectively. Harder's conjecture. There exists such an $F$ and a prime ideal $\mathfrak p'$ above $\mathfrak p$ in a field containing $\mathbb Q(f)\mathbb Q(F)$ such that, for every prime $p$,

$$L_p(X,F,\operatorname{Sp})\equiv L_p(X,f)(1-p^{k-2}X)(1-p^{j+k-1}X)\pmod{\mathfrak p'}.$$

In particular,

$$\lambda_F(T(p))\equiv a(p,f)+p^{k-2}+p^{j+k-1}\pmod{\mathfrak p'}.$$

This conjecture refines the original eigenvalue congruence by expressing it as a congruence of Euler factors; the source notes that it is trivial when $j=0$, while cases such as $(k,j)=(10,4)$ are known. The precise meaning of “large” and the normalization of the critical value require additional choices in the source.