For quadratic characters over F_q[T], the murmuration density is exactly 1/(q-1), and for Kummer cubic characters it tends to the root number of the tested prime.
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Murmurations of quadratic and cubic characters over function fields
For quadratic characters over F_q[T], the murmuration density is exactly 1/(q-1), and for Kummer cubic characters it tends to the root number of the tested prime.