First we will just do $GL_2$ as we know it works in this case so it is just verifying the calculation does what we want it to.
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The set $B_1$ contains the matrices from $U_p$.
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The set $B_2$ is the remaining matrix in this case.
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Here we just take a random element of $GL_2(\mathbb{F}_3)$. It is important to note whether it is in $K_0(3)$ to match up the results with our expectations.
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The sets $S_1\subset B_1$ and $S_2\subset B_2$ are the sets that are "hit" by $g$.
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Note we get what is expected here. To get the other element of $B_1$, we would just consider where $B_2$ maps. This shows our expectation is valid for this case in $GL_2$.
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We now want to do the same procedure with $Sp(4)$. In order to agree with Taylor's conventions (which are necessary to use his cosets), we must define $Sp(4)$ in SAGE and not use the built-in definition. The following function builds Taylor's Sp_4(\mathbb{F}_p)$ upon inputting $p$.
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We check this for $Sp_4(\mathbb{F}_2)$ by constructing the group and then checking how many elements it has.
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Next we define the space of 4 by 4 matrices.
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We now define the coset representatives $\mathcal{B}_{i}$ for $i = 1, \dots, 4$ as in Lemma 5.3.
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We now want to consider the image of the injection $\sigma_{g}: \mathcal{B}_1 \rightarrow \mathcal{B}$ induced by some $g \in K_0(1)$. Since our version of $Sp(4)$ is now a list, we cannot pick an element randomly. So just put $G[j]$ for $j$ being any number between $0$ and $719$.
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The following function determines the image of $\sigma_{g}$. In particular, one types in coset_partition(G[j]) for $j$ an integer you pick between $0$ and $719$ and the program will output the image of $\sigma_g$. The output is a list of lists, so the first collection of elements is the image intersected with $\mathcal{B}_1$, etc.
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