3.2. Some generalities 346

§4. Constructions of (group) ro(fc)-invariant functions 347

4.1. The direct construction 347

4.2. Averaging

F(fcn,

fc)-invariant functions 349

4.3. Bases for /C(Fo(fc))0 and JC(T0(k))oo 361

4.4. Precomposing with A^ 363

§5. Partition identities 369

§6. Production of constant functions 375

6.1. The Frobenius automorphism 375

6.2. Constant functions 378

6.3. Congruences 380

6.4. Functions

JF/^^/V

for negative N 383

6.5. Functions F^-N °f small degree 386

§7. Averaging operators 388

7.1. Automorphisms of/C(ro(fc)) 388

7.2. Other linear maps 391

§8. Modular equations 392

8.1. k = 2 393

8.2. k = 3 395

8.3. k = 5 396

8.4. k = 7 398

8.5. k = 13 399

§9. The ideal of partition identities 399

§10. Examples: Calculations for small k 405

10.1. fc = 2 405

10.2. k = 3 408

10.3. k = 5 411

10.4. k = 7 413

10.5. k = 11 414

10.6. fc = 13 418

10.7. fc = 4 419

10.8. fc = 6 423

§11. The higher level Ramanujan congruences 424

11.1. The level two and three congruences for small primes 424