Defining parameters
Level: | \( N \) | \(=\) | \( 84 = 2^{2} \cdot 3 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 84.a (trivial) |
Character field: | \(\Q\) | ||
Newform subspaces: | \( 4 \) | ||
Sturm bound: | \(128\) | ||
Trace bound: | \(3\) | ||
Distinguishing \(T_p\): | \(5\) |
Dimensions
The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(84))\).
Total | New | Old | |
---|---|---|---|
Modular forms | 118 | 6 | 112 |
Cusp forms | 106 | 6 | 100 |
Eisenstein series | 12 | 0 | 12 |
The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.
\(2\) | \(3\) | \(7\) | Fricke | Dim |
---|---|---|---|---|
\(-\) | \(+\) | \(+\) | \(-\) | \(1\) |
\(-\) | \(+\) | \(-\) | \(+\) | \(2\) |
\(-\) | \(-\) | \(+\) | \(+\) | \(2\) |
\(-\) | \(-\) | \(-\) | \(-\) | \(1\) |
Plus space | \(+\) | \(4\) | ||
Minus space | \(-\) | \(2\) |
Trace form
Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(84))\) into newform subspaces
Label | Dim | $A$ | Field | CM | Traces | A-L signs | $q$-expansion | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
$a_{2}$ | $a_{3}$ | $a_{5}$ | $a_{7}$ | 2 | 3 | 7 | |||||||
84.8.a.a | $1$ | $26.240$ | \(\Q\) | None | \(0\) | \(-27\) | \(100\) | \(-343\) | $-$ | $+$ | $+$ | \(q-3^{3}q^{3}+10^{2}q^{5}-7^{3}q^{7}+3^{6}q^{9}+\cdots\) | |
84.8.a.b | $1$ | $26.240$ | \(\Q\) | None | \(0\) | \(27\) | \(-240\) | \(343\) | $-$ | $-$ | $-$ | \(q+3^{3}q^{3}-240q^{5}+7^{3}q^{7}+3^{6}q^{9}+\cdots\) | |
84.8.a.c | $2$ | $26.240$ | \(\Q(\sqrt{3649}) \) | None | \(0\) | \(-54\) | \(264\) | \(686\) | $-$ | $+$ | $-$ | \(q-3^{3}q^{3}+(132-\beta )q^{5}+7^{3}q^{7}+3^{6}q^{9}+\cdots\) | |
84.8.a.d | $2$ | $26.240$ | \(\Q(\sqrt{21961}) \) | None | \(0\) | \(54\) | \(96\) | \(-686\) | $-$ | $-$ | $+$ | \(q+3^{3}q^{3}+(48-\beta )q^{5}-7^{3}q^{7}+3^{6}q^{9}+\cdots\) |
Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(84))\) into lower level spaces
\( S_{8}^{\mathrm{old}}(\Gamma_0(84)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 2}\)