Defining parameters
Level: | \( N \) | \(=\) | \( 324 = 2^{2} \cdot 3^{4} \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 324.a (trivial) |
Character field: | \(\Q\) | ||
Newform subspaces: | \( 5 \) | ||
Sturm bound: | \(432\) | ||
Trace bound: | \(5\) | ||
Distinguishing \(T_p\): | \(5\) |
Dimensions
The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(324))\).
Total | New | Old | |
---|---|---|---|
Modular forms | 396 | 28 | 368 |
Cusp forms | 360 | 28 | 332 |
Eisenstein series | 36 | 0 | 36 |
The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.
\(2\) | \(3\) | Fricke | Dim |
---|---|---|---|
\(-\) | \(+\) | $-$ | \(13\) |
\(-\) | \(-\) | $+$ | \(15\) |
Plus space | \(+\) | \(15\) | |
Minus space | \(-\) | \(13\) |
Trace form
Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(324))\) into newform subspaces
Label | Dim | $A$ | Field | CM | Traces | A-L signs | $q$-expansion | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
$a_{2}$ | $a_{3}$ | $a_{5}$ | $a_{7}$ | 2 | 3 | |||||||
324.8.a.a | $3$ | $101.213$ | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) | None | \(0\) | \(0\) | \(-57\) | \(114\) | $-$ | $+$ | \(q+(-19+\beta _{1})q^{5}+(38-\beta _{2})q^{7}+(146+\cdots)q^{11}+\cdots\) | |
324.8.a.b | $3$ | $101.213$ | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) | None | \(0\) | \(0\) | \(57\) | \(114\) | $-$ | $+$ | \(q+(19-\beta _{1})q^{5}+(38-\beta _{2})q^{7}+(-146+\cdots)q^{11}+\cdots\) | |
324.8.a.c | $7$ | $101.213$ | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) | None | \(0\) | \(0\) | \(-321\) | \(83\) | $-$ | $+$ | \(q+(-46-\beta _{1})q^{5}+(12-\beta _{2})q^{7}+(2^{4}+\cdots)q^{11}+\cdots\) | |
324.8.a.d | $7$ | $101.213$ | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) | None | \(0\) | \(0\) | \(321\) | \(83\) | $-$ | $-$ | \(q+(46+\beta _{1})q^{5}+(12-\beta _{2})q^{7}+(-2^{4}+\cdots)q^{11}+\cdots\) | |
324.8.a.e | $8$ | $101.213$ | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) | None | \(0\) | \(0\) | \(0\) | \(-560\) | $-$ | $-$ | \(q+\beta _{4}q^{5}+(-70-\beta _{1})q^{7}+(\beta _{4}+\beta _{5}+\cdots)q^{11}+\cdots\) |
Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(324))\) into lower level spaces
\( S_{8}^{\mathrm{old}}(\Gamma_0(324)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 10}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 9}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(162))\)\(^{\oplus 2}\)