This page lists the squares modulo a few powers of 7. See Modular Arithmetic and Quadratic Residue (table) for background.

There are also lists for the squares modulo N and the powers of 2, 3, 5, 11.

Squares mod 7^1 = 7: 0,1,2,4 Squares mod 7^2 = 49: 0,1,2,4,8,9,11,15,16,18,22,23,25,29,30,32,36,37,39,43,44,46 Squares mod 7^3 = 343: 0,1,2,4,8,9,11,15,16,18,22,23,25,29,30,32,36,37,39,43,44,46,49,50,51,53,57,58,60,64,65,67,71,72,74,78,79,81,85,86,88,92,93,95,98,99,100,102,106,107,109,113,114,116,120,121,123,127,128,130,134,135,137,141,142,144,148,149,151,155,156,158,162,163,165,169,170,172,176,177,179,183,184,186,190,191,193,196,197,198,200,204,205,207,211,212,214,218,219,221,225,226,228,232,233,235,239,240,242,246,247,249,253,254,256,260,261,263,267,268,270,274,275,277,281,282,284,288,289,291,295,296,298,302,303,305,309,310,312,316,317,319,323,324,326,330,331,333,337,338,340