Defining parameters
| Level: | \( N \) | \(=\) | \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 600.a (trivial) |
| Character field: | \(\Q\) | ||
| Newform subspaces: | \( 9 \) | ||
| Sturm bound: | \(240\) | ||
| Trace bound: | \(7\) | ||
| Distinguishing \(T_p\): | \(7\), \(11\) |
Dimensions
The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(600))\).
| Total | New | Old | |
|---|---|---|---|
| Modular forms | 144 | 9 | 135 |
| Cusp forms | 97 | 9 | 88 |
| Eisenstein series | 47 | 0 | 47 |
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. |
|---|---|---|---|---|
| \(+\) | \(+\) | \(+\) | \(+\) | \(2\) |
| \(+\) | \(+\) | \(-\) | \(-\) | \(1\) |
| \(+\) | \(-\) | \(+\) | \(-\) | \(1\) |
| \(+\) | \(-\) | \(-\) | \(+\) | \(1\) |
| \(-\) | \(+\) | \(+\) | \(-\) | \(2\) |
| \(-\) | \(-\) | \(-\) | \(-\) | \(2\) |
| Plus space | \(+\) | \(3\) | ||
| Minus space | \(-\) | \(6\) | ||
Trace form
Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(600))\) 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 | |||||||
| 600.2.a.a | \(1\) | \(4.791\) | \(\Q\) | None | \(0\) | \(-1\) | \(0\) | \(-4\) | \(-\) | \(+\) | \(+\) | \(q-q^{3}-4q^{7}+q^{9}+6q^{13}+2q^{17}+\cdots\) | |
| 600.2.a.b | \(1\) | \(4.791\) | \(\Q\) | None | \(0\) | \(-1\) | \(0\) | \(-3\) | \(+\) | \(+\) | \(+\) | \(q-q^{3}-3q^{7}+q^{9}+2q^{11}+3q^{13}+\cdots\) | |
| 600.2.a.c | \(1\) | \(4.791\) | \(\Q\) | None | \(0\) | \(-1\) | \(0\) | \(0\) | \(+\) | \(+\) | \(+\) | \(q-q^{3}+q^{9}-4q^{11}-6q^{13}+6q^{17}+\cdots\) | |
| 600.2.a.d | \(1\) | \(4.791\) | \(\Q\) | None | \(0\) | \(-1\) | \(0\) | \(2\) | \(+\) | \(+\) | \(-\) | \(q-q^{3}+2q^{7}+q^{9}+2q^{11}-2q^{13}+\cdots\) | |
| 600.2.a.e | \(1\) | \(4.791\) | \(\Q\) | None | \(0\) | \(-1\) | \(0\) | \(5\) | \(-\) | \(+\) | \(+\) | \(q-q^{3}+5q^{7}+q^{9}-6q^{11}+3q^{13}+\cdots\) | |
| 600.2.a.f | \(1\) | \(4.791\) | \(\Q\) | None | \(0\) | \(1\) | \(0\) | \(-5\) | \(+\) | \(-\) | \(-\) | \(q+q^{3}-5q^{7}+q^{9}-6q^{11}-3q^{13}+\cdots\) | |
| 600.2.a.g | \(1\) | \(4.791\) | \(\Q\) | None | \(0\) | \(1\) | \(0\) | \(-2\) | \(-\) | \(-\) | \(-\) | \(q+q^{3}-2q^{7}+q^{9}+2q^{11}+2q^{13}+\cdots\) | |
| 600.2.a.h | \(1\) | \(4.791\) | \(\Q\) | None | \(0\) | \(1\) | \(0\) | \(0\) | \(+\) | \(-\) | \(+\) | \(q+q^{3}+q^{9}+4q^{11}+2q^{13}-2q^{17}+\cdots\) | |
| 600.2.a.i | \(1\) | \(4.791\) | \(\Q\) | None | \(0\) | \(1\) | \(0\) | \(3\) | \(-\) | \(-\) | \(-\) | \(q+q^{3}+3q^{7}+q^{9}+2q^{11}-3q^{13}+\cdots\) | |
Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(600))\) into lower level spaces
\( S_{2}^{\mathrm{old}}(\Gamma_0(600)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 2}\)