§3. Some special cases

(kf

= 1) 183

3.1. fc = l 183

3.2. k = 2 185

3.3. fc = 3 190

3.4. k = 4 198

3.5. A ; = 5 201

3.6. fc = 6 204

§4. Primitive invariant automorphic forms 213

4.1. An index 4 subgroup of r(fc) for even k 213

4.2. A Hilbert space of modified theta constants 215

4.3. Projective representation of Aut

E2/T(k)

218

4.4. More Hilbert spaces of modified theta constants 223

§5. Orders of automorphic forms at cusps 225

5.1. Calculations via T0(k) 225

5.2. The general case 227

§6. The field of meromorphic functions on

M2/T(k)

228

6.1. Functions of small degree 228

6.2. G(&)-invariant functions 230

6.3. Generators for the function field JC(V(k)) 231

§7. Projective representations 235

§8. Some special cases

(kf

— k) 239

8.1. k = 3 239

8.2. k = 5 242

8.3. The function field for H

2

/r(7) 245

8.4. The projective embedding of

M2/T(7)

246

8.5. fc = l l 248

8.6. k = 13 249

8.7. k = 9 250

§9. The function field of

B.2/T(k)

over H

2

/ r 253

§10. Equations that are satisfied by the embedding 253

10.1. The residue theorem 253

10.2. The algorithm 254

10.3. Three term identities 255

10.4. Examples of equations 256