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

WeightGalois 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 $M_k\left(\textrm{Sp}(8,\mathbb{Z})\right)$:

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

$k$:     

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.