If you only care about the first 11 bits, then just apply a mask:
string s = Convert.ToString(value & 2047, 2);
This will restrict the value to at most 11 bits - and for negative numbers, all the "unused" bits will be 1s.
For reference: 2047 is binary 0000 ... 0000 0111 1111 1111, i.e. a mask of the least significant 11 bits.