$Version
(* "13.3.1 for Mac OS X ARM (64-bit) (July 24, 2023)" *)
Clear["Global`*"]
nn = -k^2 Sin[t] Exp[-2 k^2 Sin[t/2]^2];
i2k[i_] = 1/2 (i - 1);
data2 = Table[nn /. k -> -i2k[i] /. t -> t - 2 // Evaluate, {i, 3, 10}];
The exact values are
(max = Maximize[{#, 19 < t < 21}, t] & /@ data2 // Simplify)
(* {{-E^(-2 Sin[
2 ArcTan[Sqrt[5 + 2 Sqrt[5] + 2 Sqrt[11 + 5 Sqrt[5]]]]]^2) Sin[
4 ArcTan[Sqrt[5 + 2 Sqrt[5] + 2 Sqrt[11 + 5 Sqrt[5]]]]], {t ->
2 + 4 π +
4 ArcTan[Sqrt[5 + 2 Sqrt[5] + 2 Sqrt[11 + 5 Sqrt[5]]]]}}, {-(9/4)
E^(-(9/2) Sin[
2 ArcTan[Sqrt[10 + Sqrt[85] + 2 Sqrt[46 + 5 Sqrt[85]]]]]^2)
Sin[4 ArcTan[Sqrt[
10 + Sqrt[85] + 2 Sqrt[46 + 5 Sqrt[85]]]]], {t ->
2 + 4 π +
4 ArcTan[Sqrt[
10 + Sqrt[85] + 2 Sqrt[46 + 5 Sqrt[85]]]]}}, {-4 E^(-8 Sin[
2 ArcTan[Sqrt[17 + 2 Sqrt[65] + 2 Sqrt[137 + 17 Sqrt[65]]]]]^2)
Sin[4 ArcTan[Sqrt[
17 + 2 Sqrt[65] + 2 Sqrt[137 + 17 Sqrt[65]]]]], {t ->
2 + 4 π +
4 ArcTan[Sqrt[
17 + 2 Sqrt[65] + 2 Sqrt[137 + 17 Sqrt[65]]]]}}, {-(25/4)
E^(-(25/2) Sin[
2 ArcTan[Sqrt[26 + Sqrt[629] + 2 Sqrt[326 + 13 Sqrt[629]]]]]^2)
Sin[4 ArcTan[Sqrt[
26 + Sqrt[629] + 2 Sqrt[326 + 13 Sqrt[629]]]]], {t ->
2 + 4 π +
4 ArcTan[Sqrt[
26 + Sqrt[629] +
2 Sqrt[326 + 13 Sqrt[629]]]]}}, {-9 E^(-18 Sin[
2 ArcTan[Sqrt[37 + 10 Sqrt[13] + 2 Sqrt[667 + 185 Sqrt[13]]]]]^2)
Sin[4 ArcTan[Sqrt[
37 + 10 Sqrt[13] + 2 Sqrt[667 + 185 Sqrt[13]]]]], {t ->
2 + 4 π +
4 ArcTan[Sqrt[
37 + 10 Sqrt[13] + 2 Sqrt[667 + 185 Sqrt[13]]]]}}, {-(49/4)
E^(-(49/2) Sin[
2 ArcTan[Sqrt[
50 + Sqrt[2405] + 2 Sqrt[1226 + 25 Sqrt[2405]]]]]^2)
Sin[4 ArcTan[Sqrt[
50 + Sqrt[2405] + 2 Sqrt[1226 + 25 Sqrt[2405]]]]], {t ->
2 + 4 π +
4 ArcTan[Sqrt[
50 + Sqrt[2405] +
2 Sqrt[1226 + 25 Sqrt[2405]]]]}}, {-16 E^(-32 Sin[
2 ArcTan[Sqrt[
65 + 10 Sqrt[41] + 2 Sqrt[2081 + 325 Sqrt[41]]]]]^2)
Sin[4 ArcTan[Sqrt[
65 + 10 Sqrt[41] + 2 Sqrt[2081 + 325 Sqrt[41]]]]], {t ->
2 + 4 π +
4 ArcTan[Sqrt[
65 + 10 Sqrt[41] + 2 Sqrt[2081 + 325 Sqrt[41]]]]}}, {-(81/4)
E^(-(81/2) Sin[
2 ArcTan[Sqrt[
82 + Sqrt[6565] + 2 Sqrt[3322 + 41 Sqrt[6565]]]]]^2)
Sin[4 ArcTan[Sqrt[
82 + Sqrt[6565] + 2 Sqrt[3322 + 41 Sqrt[6565]]]]], {t ->
2 + 4 π +
4 ArcTan[Sqrt[82 + Sqrt[6565] + 2 Sqrt[3322 + 41 Sqrt[6565]]]]}}} *)
The approximate values are
max // N
(* {{0.536563, {t -> 19.945}}, {0.860824, {t -> 20.2097}}, {1.17579,
{t -> 20.3605}}, {1.48632, {t -> 20.4551}}, {1.7945,
{t -> 20.5194}}, {2.10131, {t -> 20.5658}}, {2.40724,
{t -> 20.6009}}, {2.71259, {t -> 20.6283}}} *)