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Southee picked up a 13th five-year haul in Test cricket and has taken 319 in 80 matches.
The first Test was held in Green Park, Kanpur.
India 345 all out: Iyer 105, Gill 52, Southee 5-69, and Jamieson 3-91.
Young 75 and Latham 50.
The scorecard is a summary of what happened.
New Zealand fought back on the second day of the first Test against India in Kanpur, as seamer Tim Southee took 5-69 and openers Will Young and Tom Latham hit half-centuries.
After lunch, India was all out for 345.
Iyer scored a century on his Test debut, but was dismissed for 105 by Southee.
Young and Latham both scored 70 not out as the tourists ended the day trailing by 216 runs.
Southee, who left the field on the first day with a groin strain, picked up four of the India's last five.
Latham successfully reviewed the on-field decision with replays showing an inside edge on both occasions, despite being given out twice.
India are without the services of their captain, with Ajinkya Rahane leading the side.
The rested players are expected to return for the second Test.
India completed a clean sweep of the Twenty20 series.
