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
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