The Operator's Manual is correct - 15W40 oil in your hydraulic system. This is straight from the John Deere Dealer Technical Assistance Center:
The following oils have been qualified to be used in the engine and hydraulic systems for the following product lines:
Model.........................................................Engine Oil....................Hydraulic Oil
F620,F680,F687............................................Turf-Gard 10W30............Plus-50 15W40
M653,M655,M665..........................................Turf-Gard 10W30............Plus-50 15W40
647,657,667.................................................Turf-Gard 10W30............Plus-50 0W40 Synthetic
717,727.......................................................Turf-Gard 10W30............Plus-50 0W40 Synthetic
647A,657A,667A,717A,727A,737,757,777,797......Turf-Gard 10W30...........Plus-50 15W40
997.............................................................Plus-50 15W40...............Hy-Gard J20C
I hope that helps you some! In case you are not familiar - "Turf-Gard" and "Plus-50" are John Deere brands / designations...