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Murmuration Density Heatmap

Where aₚ values concentrate across prime and conductor space

Rank 0 Density Distribution
Low
High
Rank 1 Density Distribution
Low
High

Reading the Heatmap

X-axis: Prime p (where aₚ is computed)
Y-axis: Normalized aₚ value
Color: Density of curves with that (p, aₚ) combination

Visible Patterns

Bright bands show where aₚ values cluster. Notice how rank 0 curves tend toward positive values while rank 1 curves favor negative ones.

The Murmuration Signature

The oscillating bright bands across primes create the characteristic wave pattern—this is the murmuration made visible in density space.