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mineThis database catalogs selected Hadamard products of modified hypergeometric functions $f_{k,r,s}$, defined as:
$$f_{k,r,s}(z) = (1-z)^k {_2F_1}\left( {r,s \atop 1}; z \right).$$We conducted a preliminary investigation into Hadamard products $(k_1,r_1,s_1) \star (k_2,r_2,s_2)$ exhibiting notable supercongruence properties at $z=1$. Denoting the series expansions $f_{k_i,r_i,s_i} = \sum_{n=0}^\infty f_{n,i} z^n$ ($i=1,2$), we observe that for certain parameters, the truncated sum
$$\sum_{n=0}^{p-1} f_{n,1} f_{n,2} \cdot 1^n \equiv a_p(k_1,r_1,s_1,k_2,r_2,s_2) \pmod{p^2}$$corresponds modulo $p^2$ to the $p$-th Fourier coefficient of a modular form.
When $k_1 = k_2 = 0$, the Hadamard product reduces to a generalized hypergeometric function:
$${_2F_1}\left( {r_1,s_1 \atop 1}; z \right) \star {_2F_1}\left( {r_2,s_2 \atop 1}; z \right) = {_4F_3}\left( {r_1,s_1,r_2,s_2 \atop 1,1,1}; z \right).$$Specific parameter choices, such as $(r_1,s_1,r_2,s_2) = (\frac{1}{4},\frac{1}{2},\frac{1}{2},\frac{3}{4})$, correspond to one-parameter families of hypergeometric Calabi-Yau threefolds (e.g., $X_{4,2}(1^6)$, the complete intersection of a quartic and a quadric in $\mathbb{P}^5(1,1,1,1,1,1)$). The conifold fiber at $z=1$ admits smooth rigid Calabi-Yau resolutions, inducing supercongruences via modularity. For the $(\frac{1}{4},\frac{1}{2},\frac{1}{2},\frac{3}{4})$ case, the associated modular form is $\mathsf{16.4.a.a} = \mathsf{8.4.a.a} \otimes \chi_{-4}$ (LMFDB label), yielding
$$a_p\left(0,\tfrac{1}{4},\tfrac{1}{2},0,\tfrac{3}{4},\tfrac{1}{2}\right) \equiv a_p(\mathsf{16.4.a.a}) \pmod{p^2}.$$