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Do we have to check some of our overcard hands here to avoid c-betting too much and is this a good hand to check or should I have just bet flop for maybe 1/3rd pot?

NL Holdem 0.02(BB)
HERO ($7.28) [VPIP: 20% | PFR: 15.2% | AGG: 29.6% | Flop Agg: 33.6% | Turn Agg: 30.7% | River Agg: 19.3% | 3-Bet: 7.1% | Fold to 3-Bet: 31.5% | 4-Bet: 28.3% | Hands: 3461]
UTG ($0.97) [VPIP: 13.6% | PFR: 0% | AGG: 6.7% | Hands: 88]
MP ($2.12) [VPIP: 29.1% | PFR: 5.8% | AGG: 29.6% | Hands: 174]
HJ ($1.21) [VPIP: 30.2% | PFR: 20.9% | AGG: 50% | Hands: 86]
CO ($1.42) [VPIP: 42.7% | PFR: 0% | AGG: 28.6% | Hands: 164]
BTN ($2.47) [VPIP: 50% | PFR: 22.7% | AGG: 57.1% | Flop Agg: 50% | Turn Agg: 50% | River Agg: 76.9% | 3-Bet: 0% | 4-Bet: 0% | Cold Call: 66.7% | Hands: 88]
SB ($2) [VPIP: 9.1% | PFR: 8% | AGG: 18.8% | Hands: 176]

Dealt to Hero: A K

UTG Folds, MP Folds, HJ Raises To $0.06, CO Calls $0.06, BTN Calls $0.06, SB Folds, HERO Raises To $0.30, HJ Folds, CO Folds, BTN Calls $0.24

Hero SPR on Flop: [2.97 effective]
Flop ($0.73): J 5 8
HERO Checks, BTN Bets $0.02 (Rem. Stack: $2.15), HERO Calls $0.02 (Rem. Stack: $6.96)

Turn ($0.77): J 5 8 4
HERO Checks, BTN Bets $0.04 (Rem. Stack: $2.11), HERO Calls $0.04 (Rem. Stack: $6.92)

River ($0.85): J 5 8 4 8
HERO Checks, BTN Bets $0.18 (Rem. Stack: $1.93), HERO Folds

I'd cbet ~38% of the time or maybe a bit less with no spade or bd club.

In theory it's mixed, but vs. a 50% vpip player 1/3 cbet probably prints.