Step 1: Coordination number:
Six ligands means octahedral geometry.
Step 2: Electron count:
$\text{Co}^{3+}$ has six d electrons. A strong field ligand like $\text{NH}_3$ makes the splitting larger than the pairing energy.
Step 3: Fill orbitals:
All six electrons fill the three $t_{2g}$ orbitals in pairs, leaving none unpaired. This is option (D).
Final Answer:
The complex has 0 unpaired electrons and octahedral shape.
\[ \boxed{\text{(D) }0,\ \text{octahedral}} \]