Defining parameters
| Level: | \( N \) | \(=\) | \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 504.a (trivial) |
| Character field: | \(\Q\) | ||
| Newform subspaces: | \( 8 \) | ||
| Sturm bound: | \(192\) | ||
| Trace bound: | \(11\) | ||
| Distinguishing \(T_p\): | \(5\), \(11\), \(13\) |
Dimensions
The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(504))\).
| Total | New | Old | |
|---|---|---|---|
| Modular forms | 112 | 8 | 104 |
| Cusp forms | 81 | 8 | 73 |
| 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.
| \(2\) | \(3\) | \(7\) | Fricke | Dim. |
|---|---|---|---|---|
| \(+\) | \(+\) | \(+\) | \(+\) | \(1\) |
| \(+\) | \(+\) | \(-\) | \(-\) | \(1\) |
| \(+\) | \(-\) | \(+\) | \(-\) | \(1\) |
| \(-\) | \(+\) | \(+\) | \(-\) | \(1\) |
| \(-\) | \(+\) | \(-\) | \(+\) | \(1\) |
| \(-\) | \(-\) | \(+\) | \(+\) | \(1\) |
| \(-\) | \(-\) | \(-\) | \(-\) | \(2\) |
| Plus space | \(+\) | \(3\) | ||
| Minus space | \(-\) | \(5\) | ||
Trace form
Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(504))\) 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 | |||||||
| 504.2.a.a | \(1\) | \(4.024\) | \(\Q\) | None | \(0\) | \(0\) | \(-2\) | \(-1\) | \(+\) | \(+\) | \(+\) | \(q-2q^{5}-q^{7}-2q^{11}+2q^{13}-6q^{17}+\cdots\) | |
| 504.2.a.b | \(1\) | \(4.024\) | \(\Q\) | None | \(0\) | \(0\) | \(-2\) | \(-1\) | \(-\) | \(-\) | \(+\) | \(q-2q^{5}-q^{7}-2q^{13}-6q^{17}-4q^{19}+\cdots\) | |
| 504.2.a.c | \(1\) | \(4.024\) | \(\Q\) | None | \(0\) | \(0\) | \(-2\) | \(-1\) | \(+\) | \(-\) | \(+\) | \(q-2q^{5}-q^{7}+4q^{11}+2q^{13}+6q^{17}+\cdots\) | |
| 504.2.a.d | \(1\) | \(4.024\) | \(\Q\) | None | \(0\) | \(0\) | \(-2\) | \(1\) | \(-\) | \(+\) | \(-\) | \(q-2q^{5}+q^{7}-6q^{11}-6q^{13}+2q^{17}+\cdots\) | |
| 504.2.a.e | \(1\) | \(4.024\) | \(\Q\) | None | \(0\) | \(0\) | \(-2\) | \(1\) | \(-\) | \(-\) | \(-\) | \(q-2q^{5}+q^{7}+6q^{13}+2q^{17}+4q^{19}+\cdots\) | |
| 504.2.a.f | \(1\) | \(4.024\) | \(\Q\) | None | \(0\) | \(0\) | \(2\) | \(-1\) | \(-\) | \(+\) | \(+\) | \(q+2q^{5}-q^{7}+2q^{11}+2q^{13}+6q^{17}+\cdots\) | |
| 504.2.a.g | \(1\) | \(4.024\) | \(\Q\) | None | \(0\) | \(0\) | \(2\) | \(1\) | \(+\) | \(+\) | \(-\) | \(q+2q^{5}+q^{7}+6q^{11}-6q^{13}-2q^{17}+\cdots\) | |
| 504.2.a.h | \(1\) | \(4.024\) | \(\Q\) | None | \(0\) | \(0\) | \(4\) | \(1\) | \(-\) | \(-\) | \(-\) | \(q+4q^{5}+q^{7}+2q^{17}-2q^{19}-8q^{23}+\cdots\) | |
Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(504))\) into lower level spaces
\( S_{2}^{\mathrm{old}}(\Gamma_0(504)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(252))\)\(^{\oplus 2}\)