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
\( S_{2}^{\mathrm{old}}(\Gamma_0(600)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 2}\)