Defining parameters
| Level: | \( N \) | \(=\) | \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 360.a (trivial) |
| Character field: | \(\Q\) | ||
| Newform subspaces: | \( 5 \) | ||
| Sturm bound: | \(144\) | ||
| Trace bound: | \(7\) | ||
| Distinguishing \(T_p\): | \(7\), \(11\) |
Dimensions
The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(360))\).
| Total | New | Old | |
|---|---|---|---|
| Modular forms | 88 | 5 | 83 |
| Cusp forms | 57 | 5 | 52 |
| Eisenstein series | 31 | 0 | 31 |
The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.
| \(2\) | \(3\) | \(5\) | Fricke | Dim. |
|---|---|---|---|---|
| \(+\) | \(+\) | \(-\) | \(-\) | \(1\) |
| \(+\) | \(-\) | \(+\) | \(-\) | \(1\) |
| \(-\) | \(+\) | \(+\) | \(-\) | \(1\) |
| \(-\) | \(-\) | \(+\) | \(+\) | \(1\) |
| \(-\) | \(-\) | \(-\) | \(-\) | \(1\) |
| Plus space | \(+\) | \(1\) | ||
| Minus space | \(-\) | \(4\) | ||
Trace form
Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(360))\) into newform subspaces
| Label | Dim. | \(A\) | Field | CM | Traces | A-L signs | $q$-expansion | ||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(a_2\) | \(a_3\) | \(a_5\) | \(a_7\) | 2 | 3 | 5 | |||||||
| 360.2.a.a | \(1\) | \(2.875\) | \(\Q\) | None | \(0\) | \(0\) | \(-1\) | \(-4\) | \(-\) | \(-\) | \(+\) | \(q-q^{5}-4q^{7}-4q^{11}-2q^{13}-2q^{17}+\cdots\) | |
| 360.2.a.b | \(1\) | \(2.875\) | \(\Q\) | None | \(0\) | \(0\) | \(-1\) | \(0\) | \(+\) | \(-\) | \(+\) | \(q-q^{5}+4q^{11}+6q^{13}+6q^{17}-4q^{19}+\cdots\) | |
| 360.2.a.c | \(1\) | \(2.875\) | \(\Q\) | None | \(0\) | \(0\) | \(-1\) | \(2\) | \(-\) | \(+\) | \(+\) | \(q-q^{5}+2q^{7}+2q^{11}+4q^{13}-2q^{17}+\cdots\) | |
| 360.2.a.d | \(1\) | \(2.875\) | \(\Q\) | None | \(0\) | \(0\) | \(1\) | \(2\) | \(+\) | \(+\) | \(-\) | \(q+q^{5}+2q^{7}-2q^{11}+4q^{13}+2q^{17}+\cdots\) | |
| 360.2.a.e | \(1\) | \(2.875\) | \(\Q\) | None | \(0\) | \(0\) | \(1\) | \(4\) | \(-\) | \(-\) | \(-\) | \(q+q^{5}+4q^{7}-6q^{13}+2q^{17}+4q^{19}+\cdots\) | |
Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(360))\) into lower level spaces
\( S_{2}^{\mathrm{old}}(\Gamma_0(360)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 2}\)