## Available samples

The following table shows two Galois orbits of weights 12 and 14 on the full modular group $\mathrm{Sp}(6,\mathbb{Z}))$, respectively.

Weight | Galois orbits (number of forms) |
---|---|

12 | 12_Miyawaki (1) |

14 | 14_Miyawaki (1) |

## Dimension table of spaces of degree 3 Siegel modular forms

The table below lists, for each bold value of $k$ in the header, the dimensions of the following subspaces of $M_{k}\left(\textrm{Sp}(6,\mathbb{Z})\right)$:

- Total: The full space.
- Miyawaki lifts I: The subspace of Miyawaki lifts of type I.
- Miyawaki lifts II: The subspace of (conjectured) Miyawaki lifts of type II.
- Other: The subspace of cusp forms which are not Miyawaki lifts of type I or II.

$0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $11$ | $12$ | $13$ | $14$ | $15$ | $16$ | $17$ | $18$ | $19$ | $20$ | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Total | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 2 | 0 | 4 | 0 | 3 | 0 | 7 | 0 | 8 | 0 | 11 |

Miyawaki lifts I | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 2 | 0 | 2 | 0 | 3 |

Miyawaki lifts II (conjectured) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 2 | 0 | 2 |

Other | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |

#### Enter a new range of weights for dimension table:

# The ring of Siegel modular forms of degree 3 with respect to the full modular group

S Tsuyumine [MR:853217, MR:853222] gives 34 generators for the ring $M_{*}({\rm Sp}(6,\mathbb{Z}))$ of Siegel modular forms of degree 3 with with respect to the full modular group. Tsuyumine also gives a formula for the dimensions of the spaces at each weight.

Miyawaki [MR:1194572] computed enough coefficients in weight 12 and 14 to determine the the Euler factor at the prime 2, and conjectured the Miyawaki lifts.

C. Poor, J. Shurman, and D. S. Yuen [arXiv:1604.07216] computed coefficients coefficients in weights 16, 18, 20, 22, and determined the eigenforms.