## Available samples

The following table shows some Galois orbits of weights 8, 10, 12, 14 and 16 on the full modular group $\mathrm{Sp}(8,\mathbb{Z})$.

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

8 | 8_Ikeda (1) |

10 | 10_Ikeda (1) |

12 | 12_Ikeda (1) 12_Miyawaki (1) |

14 | 14_Ikeda (2) 14_Miyawaki (1) |

16 | 16_Ikeda (2) 16_Miyawaki (2) 16_Other_I (1) 16_Other_II (2) |

## Dimension table of spaces of degree 4 Siegel modular forms

The table below lists, for each bold value of $k$ in the header, the dimensions of the following subspaces of $M_k\left(\textrm{Sp}(8,\mathbb{Z})\right)$:

- Total: The subspace of cusp forms.
- Ikeda lifts: The subspace of Ikeda lifts.
- Miyawaki lifts: The subspace of Miyawaki lifts.
- Other: The subspace that are not Ikeda or Miyawaki lifts.

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

Total | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 2 | 0 | 3 | 0 | 7 |

Ikeda lifts | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 2 | 0 | 2 |

Miyawaki lifts | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 2 |

Other | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 |

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

# The ring of Siegel modular forms of degree 4 with respect to the full modular group

The dimensions for degree 4 Siegel modular cusp forms $S_k(\mathrm{Sp}(8,Z))$ for the full modular group for weights $k\le 16$ were proven by C. Poor and D. S. Yuen [MR:2302669] . Poor and Yuen also computed Fourier coefficients and some eigenvalues.

The cusp forms in weights up through 14 are either Duke-Imamoglu-Ikeda lifts or Miyawaki lifts. In weight 16, in addition to these two types of lifts, there are other eigenforms that have been shown by Ibukiyama to instantiate a conjectural lift from vector-valued Siegel modular forms.