Defining parameters
Level: | \( N \) | \(=\) | \( 324 = 2^{2} \cdot 3^{4} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 324.a (trivial) |
Character field: | \(\Q\) | ||
Newform subspaces: | \( 5 \) | ||
Sturm bound: | \(216\) | ||
Trace bound: | \(5\) | ||
Distinguishing \(T_p\): | \(5\) |
Dimensions
The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(324))\).
Total | New | Old | |
---|---|---|---|
Modular forms | 180 | 12 | 168 |
Cusp forms | 144 | 12 | 132 |
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 |
---|---|---|---|
\(-\) | \(+\) | \(-\) | \(5\) |
\(-\) | \(-\) | \(+\) | \(7\) |
Plus space | \(+\) | \(7\) | |
Minus space | \(-\) | \(5\) |
Trace form
Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(324))\) into newform subspaces
Label | Dim | $A$ | Field | CM | Traces | A-L signs | $q$-expansion | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
$a_{2}$ | $a_{3}$ | $a_{5}$ | $a_{7}$ | 2 | 3 | |||||||
324.4.a.a | $1$ | $19.117$ | \(\Q\) | None | \(0\) | \(0\) | \(-3\) | \(-4\) | $-$ | $+$ | \(q-3q^{5}-4q^{7}+24q^{11}-5^{2}q^{13}+\cdots\) | |
324.4.a.b | $1$ | $19.117$ | \(\Q\) | None | \(0\) | \(0\) | \(3\) | \(-4\) | $-$ | $+$ | \(q+3q^{5}-4q^{7}-24q^{11}-5^{2}q^{13}+\cdots\) | |
324.4.a.c | $3$ | $19.117$ | 3.3.1509.1 | None | \(0\) | \(0\) | \(-6\) | \(6\) | $-$ | $+$ | \(q+(-2-\beta _{2})q^{5}+(2-\beta _{1}+\beta _{2})q^{7}+\cdots\) | |
324.4.a.d | $3$ | $19.117$ | 3.3.1509.1 | None | \(0\) | \(0\) | \(6\) | \(6\) | $-$ | $-$ | \(q+(2+\beta _{2})q^{5}+(2-\beta _{1}+\beta _{2})q^{7}+(17+\cdots)q^{11}+\cdots\) | |
324.4.a.e | $4$ | $19.117$ | \(\Q(\sqrt{3}, \sqrt{7})\) | None | \(0\) | \(0\) | \(0\) | \(-16\) | $-$ | $-$ | \(q+\beta _{2}q^{5}+(-4+\beta _{3})q^{7}+(-\beta _{1}+\beta _{2}+\cdots)q^{11}+\cdots\) |
Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(324))\) into lower level spaces
\( S_{4}^{\mathrm{old}}(\Gamma_0(324)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(162))\)\(^{\oplus 2}\)