Question: New York State Route 40 passes through what village that had a population of 592 at the 2010 census?

Context paragraphs:
U.S. Route 44 in New York: U.S. Route 44 (US 44) in the state of New York is a major east–west thoroughfare in the Hudson Valley region of the state.  Its entire 65.98 mi length is maintained by the New York State Department of Transportation (NYSDOT), with the exception of the Mid-Hudson Bridge, which is maintained by the New York State Bridge Authority.  The portion of the route in New York begins at an intersection with US 209 and New York State Route 55 (NY 55) near the hamlet of Kerhonkson and ends at the Connecticut state line near the village of Millerton.  The road passes through rural parts of Ulster and Dutchess counties before crossing into Litchfield County, Connecticut.

Schaghticoke (village), New York: Schaghticoke is a village in Rensselaer County, New York, United States.  The population was 592 at the 2010 census.

Yonkers Avenue: Yonkers Avenue is an east–west street in the city of Yonkers in Westchester County, New York, in the United States.  It is one of four major east–west through routes in the city.  The western terminus of the street is at Nepperhan Avenue, which connects to U.S. Route 9 (US 9) and New York State Route 9A (NY 9A).  Its eastern terminus is at Bronx River Road near the Bronx River Parkway.  The entirety of Yonkers Avenue is maintained by the New York State Department of Transportation as New York State Route 983C from Nepperhan Avenue to the Saw Mill River Parkway and New York State Route 984E from the Saw Mill Parkway to Bronx River Road.  Both are unsigned reference route designations.

New York State Route 28B: New York State Route 28B (NY 28B) was a state highway in Oneida County, New York, in the United States.  The route served as a connector between NY 287 (now NY 365) in the village of Prospect and NY 12 and NY 28 in the village of Remsen.  It was assigned in 1935 and removed in the mid-1960s, at which time most of the route became New York State Route 920V, an unsigned reference route.  The alignment of NY 28B was originally part of legislative Route 26 and later Route 25 in the early 20th century.

Pennsylvania Route 434: Pennsylvania Route 434 (designated by the Pennsylvania Department of Transportation as SR 0434) is a 12.47 mi state highway located in northeast Pennsylvania.  The western terminus of the route is at Pennsylvania Route 739 in the Blooming Grove Township community of Lords Valley.  The eastern terminus of the route is at the New York-Pennsylvania border in Shohola Township, where PA 434 crosses the Delaware River and enters New York, becoming New York State Route 55 at an intersection with New York State Route 97 in the town of Highland.  State Route 434 used to be part of Pennsylvania Route 37 and Pennsylvania Route 137.

New York State Route 27: New York State Route 27 (NY 27) is an east–west 120.58 mi long state highway extending from Interstate 278 (I-278) in the New York City borough of Brooklyn to Montauk Point State Park on Long Island, New York, in the United States.  Its two most prominent components are Sunrise Highway and Montauk Highway, the latter of which includes the Montauk Point State Parkway.  East of the interchange with the Heckscher State Parkway in Islip Terrace, NY 27 acts as the primary east–west highway on southern Long Island.  The entire route in Suffolk, Nassau, and Queens counties was designated by the New York State Senate as the POW/MIA Memorial Highway.  Every town on the South Shore is accessible through Sunrise Highway.  The service roads that parallel the highway from Farmingdale to Brookhaven are officially designated, but not signed, as New York State Route 906C and New York State Route 906D.

New York State Route 75: New York State Route 75 (NY 75) is a north–south state highway in Erie County, New York, in the United States.  It extends for 20.85 mi from an intersection with NY 39 in the Collins hamlet of Collins Center to an interchange with NY 5 in the town of Hamburg.  The route passes through the village of Hamburg, which serves as the northern terminus of a 2 mi overlap between U.S. Route 62 (US 62) and NY 75.  Past Hamburg, NY 75 connects to the New York State Thruway (Interstate 90 or I-90) northwest of the village before ending a short distance from Lake Erie.  The portion of NY 75 south of Hamburg is a two-lane rural highway; in contrast, the section north of the village is four lanes wide and serves commercial and residential areas.

New York State Route 41: New York State Route 41 (NY 41) is a north–south state highway in Central New York in the United States.  The southern terminus of the route is at an interchange with NY 17 in the town of Sanford.  Its northern terminus is at an intersection with U.S. Route 20 (US 20) in the village of Skaneateles.  The route is almost 100 mi long and passes through Broome, Chenango, Cortland, and Onondaga counties.  NY 41 was assigned as part of the 1930 renumbering of state highways in New York, replacing New York State Route 70 from Homer to Skaneateles.  The route initially extended as far north as Jordan; however, NY 41 was cut back to its current northern terminus  1933 .

New York State Route 376: New York State Route 376 (NY 376) is a state highway located entirely within Dutchess County in the Hudson Valley region of New York in the United States.  The route begins at an intersection with NY 52 in East Fishkill and passes north through Hopewell Junction and Red Oaks Mill on its way to the city of Poughkeepsie.  It ends at a junction with U.S. Route 44 (US 44) and NY 55 east of the city limits in Arlington, a hamlet in the town of Poughkeepsie.  NY 376 was originally designated as part of NY 39 in the mid-1920s.  In the 1930 renumbering of state highways in New York, the East Fishkill–Poughkeepsie portion of NY 39 was renumbered to New York State Route 202.  NY 202 was renumbered to NY 376 in 1934 to avoid numerical duplication with the new US 202.

New York State Route 40: New York State Route 40 (NY 40) is a north–south state highway in eastern New York in the United States.  It is 54.67 mi long and runs from NY 7 in the city of Troy north to NY 22 in the town of Granville.  NY 40 also passes through the villages of Schaghticoke and Argyle and enters the vicinity of the village of Greenwich.  It intersects three east–west highways of note: NY 67 just outside Schaghticoke, NY 29 west of Greenwich, and NY 149 in the hamlet of Hartford.  Incidentally, NY 40 has overlaps with all three routes.

Supporting paragraphs: New York State Route 40 Schaghticoke (village), New York
Answer: Schaghticoke