Defining parameters
Level: | \( N \) | \(=\) | \( 324 = 2^{2} \cdot 3^{4} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 324.a (trivial) |
Character field: | \(\Q\) | ||
Newform subspaces: | \( 4 \) | ||
Sturm bound: | \(108\) | ||
Trace bound: | \(7\) | ||
Distinguishing \(T_p\): | \(5\), \(7\) |
Dimensions
The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(324))\).
Total | New | Old | |
---|---|---|---|
Modular forms | 72 | 4 | 68 |
Cusp forms | 37 | 4 | 33 |
Eisenstein series | 35 | 0 | 35 |
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 |
---|---|---|---|
\(-\) | \(+\) | $-$ | \(3\) |
\(-\) | \(-\) | $+$ | \(1\) |
Plus space | \(+\) | \(1\) | |
Minus space | \(-\) | \(3\) |
Trace form
Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(324))\) into newform subspaces
Label | Dim | $A$ | Field | CM | Traces | A-L signs | $q$-expansion | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
$a_{2}$ | $a_{3}$ | $a_{5}$ | $a_{7}$ | 2 | 3 | |||||||
324.2.a.a | $1$ | $2.587$ | \(\Q\) | None | \(0\) | \(0\) | \(-3\) | \(-1\) | $-$ | $-$ | \(q-3q^{5}-q^{7}-3q^{11}-q^{13}-6q^{17}+\cdots\) | |
324.2.a.b | $1$ | $2.587$ | \(\Q\) | None | \(0\) | \(0\) | \(-3\) | \(2\) | $-$ | $+$ | \(q-3q^{5}+2q^{7}+6q^{11}+5q^{13}+3q^{17}+\cdots\) | |
324.2.a.c | $1$ | $2.587$ | \(\Q\) | None | \(0\) | \(0\) | \(3\) | \(-1\) | $-$ | $+$ | \(q+3q^{5}-q^{7}+3q^{11}-q^{13}+6q^{17}+\cdots\) | |
324.2.a.d | $1$ | $2.587$ | \(\Q\) | None | \(0\) | \(0\) | \(3\) | \(2\) | $-$ | $+$ | \(q+3q^{5}+2q^{7}-6q^{11}+5q^{13}-3q^{17}+\cdots\) |
Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(324))\) into lower level spaces
\( S_{2}^{\mathrm{old}}(\Gamma_0(324)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(162))\)\(^{\oplus 2}\)