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$$frac{9r^4-16 r^2 + 64}{9 r^4}$$

$$\frac{9r^4-16 r^2 + 64}{9 r^4}$$

This is the trace of Frobenius on the cohomology of a certain sheaf on the affine curve $a^2 - b^2 \delta =\delta/4 ( 4-r^2)$, where you are taking $a = \sqrt{\delta} b$. As $r$ varies this curve varies in a family of curves, and you can take the cohomology of the family of sheaves, giving a sheaf on $\mathbb A^1$. How the sum varies with $r$ depends on the monodromy of this sheaf.

$$ \sum_{(a+b\sigma)(a-b\sigma) =\frac{\delta}{4} ( 4-r^2)} \chi \left(a + b \sigma+ (r-1) \sigma\right) \chi^* \left(a - b \sigma - (r-1) \sigma\right) \overline{\chi} \left(a +b \sigma + (r+1) \sigma \right)\overline{\chi}^* \left(a -b \sigma + (r+1) \sigma \right)$$

$$ \sum_{\alpha} \chi \left(\alpha + (r-1) \sigma\right) \chi^* \left(\frac{\frac{\delta}{4} (4-r^2)}{\alpha} - (r-1) \sigma\right) \overline{\chi} \left(\alpha + (r+1) \sigma \right)\overline{\chi}^* \left(\frac{\frac{\delta}{4} (4-r^2)}{\alpha} + (r+1) \sigma \right)$$

$$ \sum_{\alpha} \chi \left(\frac{ \alpha + (r-1) \sigma} {\alpha + (r+1) \sigma} \right) \chi^* \left(\frac{\frac{\frac{\delta}{4} (4-r^2)}{\alpha} - (r-1) \sigma}{\frac{\frac{\delta}{4} (4-r^2)}{\alpha} + (r+1) \sigma} \right) $$

$$\frac{ (8+5r) (8 - 8r + 3 r^2) }{3r (-8r+ 5r^2)}$$

This is the trace of Frobenius on the cohomology of a certain sheaf on the affine curve $a^2 - b^2 \delta =\delta/4 ( 4-r^2)$, where I am taking $z=a+ \sqrt{\delta} b$. As $r$ varies this curve varies in a family of curves, and you can take the cohomology of the family of sheaves, giving a sheaf on $\mathbb A^1$. How the sum varies with $r$ depends on the monodromy of this sheaf.

$$ \sum_{(a+b\sigma)(a-b\sigma) =\frac{\delta}{4} ( 4-r^2)} \chi \left(a + b \sigma+ (r-1) \sigma\right) \chi^* \left(a - b \sigma - (r-1) \sigma\right) \overline{\chi} \left(a +b \sigma + (r+1) \sigma \right)\overline{\chi}^* \left(a -b \sigma - (r+1) \sigma \right)$$

$$ \sum_{\alpha} \chi \left(\alpha + (r-1) \sigma\right) \chi^* \left(\frac{\frac{\delta}{4} (4-r^2)}{\alpha} - (r-1) \sigma\right) \overline{\chi} \left(\alpha + (r+1) \sigma \right)\overline{\chi}^* \left(\frac{\frac{\delta}{4} (4-r^2)}{\alpha} - (r+1) \sigma \right)$$

$$ \sum_{\alpha} \chi \left(\frac{ \alpha + (r-1) \sigma} {\alpha + (r+1) \sigma} \right) \chi^* \left(\frac{\frac{\frac{\delta}{4} (4-r^2)}{\alpha} - (r-1) \sigma}{\frac{\frac{\delta}{4} (4-r^2)}{\alpha} - (r+1) \sigma} \right) $$

$$frac{9r^4-16 r^2 + 64}{9 r^4}$$

Because the monodromy group contains $SL_2$, the sum cannot be expressed in terms of simpler sums that have monodromy groups different from $SL_2$, and it will behave in a seemingly random way, with the absolute value of the expnential sum distributed accoring to the Sato-Tate measure.

Because the monodromy group contains $SL_2$, the sum cannot be expressed in terms of simpler sums that have monodromy groups different from $SL_2$, and it will behave in a seemingly random way, with the absolute value of the exponential sum distributed according to the Sato-Tate measure.