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In this paper, we determine mod $2$ Galois representations$\bar{\rho}_{\psi,2}:G_K:={\rm Gal}(\bar{K}/K)\longrightarrow {\rmGSp}_4(\mathbb{F}_2)$ associated to the mirror motives of rank 4 with pureweight 3 coming from the Dwork quintic family$$X^5_0+X^5_1+X^5_2+X^5_3+X^5_4-5\psi X_0X_1X_2X_3X_4=0,\ \psi\in K$$ definedover a number field $K$ under the irreducibility condition of the quintictrinomial $f_\psi$ below. Applying this result, when $K=F$ is a totally realfield, for some at most qaudratic totally real extension $M/F$, we prove that$\bar{\rho}_{\psi,2}|_{G_M}$ is associated to a Hilbert-Siegel modular Heckeeigen cusp form for ${\rm GSp}_4(\mathbb{A}_M)$ of parallel weight three.  In the course of the proof, we observe that the image of such a mod $2$representation is governed by reciprocity of the quintic trinomial$$f_\psi(x)=4x^5-5\psi x^4+1,\ \psi\in K$$ whose decomposition field isgenerically of type 5-th symmetric group $S_5$. This enable us to use resultson the modularity of 2-dimensional, totally odd Artin representations of ${\rmGal}(\bar{F}/F)$ due to Shu Sasaki and several Langlands functorial lifts forHilbert cusp forms. Then, it guarantees the existence of a desiredHilbert-Siegel modular cusp form of parallel weight three matching with theHodge type of the compatible system in question.A twisted version is alsodiscussed and it is related to general quintic trinomials.

Automorphy of mod 2 Galois representations associated to the quintic Dwork family and reciprocity of some quintic trinomials