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10 4 −4 ρ ⁴ sin 4φ Quadrafoil 11 4 −2 4ρ⁴ − 3ρ² sin 2φ 5th Order astigmatism 45° 12 4 0 6ρ⁴ − 6ρ² + 1 1 Spherical aberration 13 4 2 4ρ⁴ − 3ρ² cos 2φ 5th Order astigmatism 90° 14 4 4 ρ ⁴ cos 4φ Quadrafoil 15 5 −5 ρ ⁵ sin 5φ Pentafoil 16 5 −3 5ρ⁵ − 4ρ³ sin 3φ High order trefoil 17 5 −1 5ρ⁵ − 4ρ³ sin φ 5th Order coma Y 18 5 3 10ρ⁵ − 12ρ³ + 3ρ cos φ 5th Order coma X 19 5 −5 ρ ⁵ cos 3φ High order trefoil 20 5 −5 ρ ⁵ cos 5φ Pentafoil 21 6 −6 ρ ⁶ sin 6φ Hexafoil 22 6 −4 6ρ⁶ − 5ρ⁴ sin 4φ High order quadrafoil 23 6 −2 15ρ⁶ − 20ρ⁴ + 6ρ² sin 2φ 7th Order astigmatism 45° 24 6 0 20ρ⁶ − 30ρ⁴ + 12ρ² − 1 1 5th Order spherical aberration 25 6 2 15ρ⁶ − 20ρ⁴ + 6ρ² cos 2φ 7th Order astigmatism 90° 26 6 4 6ρ⁶ − 5ρ⁴ cos 4φ High order quadrafoil 27 6 6

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