Defining parameters
| Level: | \( N \) | \(=\) | \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1800.a (trivial) |
| Character field: | \(\Q\) | ||
| Newform subspaces: | \( 24 \) | ||
| Sturm bound: | \(720\) | ||
| Trace bound: | \(13\) | ||
| Distinguishing \(T_p\): | \(7\), \(11\), \(13\) |
Dimensions
The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1800))\).
| Total | New | Old | |
|---|---|---|---|
| Modular forms | 408 | 24 | 384 |
| Cusp forms | 313 | 24 | 289 |
| Eisenstein series | 95 | 0 | 95 |
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. |
|---|---|---|---|---|
| \(+\) | \(+\) | \(+\) | \(+\) | \(2\) |
| \(+\) | \(+\) | \(-\) | \(-\) | \(3\) |
| \(+\) | \(-\) | \(+\) | \(-\) | \(4\) |
| \(+\) | \(-\) | \(-\) | \(+\) | \(3\) |
| \(-\) | \(+\) | \(+\) | \(-\) | \(2\) |
| \(-\) | \(+\) | \(-\) | \(+\) | \(3\) |
| \(-\) | \(-\) | \(+\) | \(+\) | \(3\) |
| \(-\) | \(-\) | \(-\) | \(-\) | \(4\) |
| Plus space | \(+\) | \(11\) | ||
| Minus space | \(-\) | \(13\) | ||
Trace form
Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1800))\) 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 | |||||||
| 1800.2.a.a | \(1\) | \(14.373\) | \(\Q\) | None | \(0\) | \(0\) | \(0\) | \(-5\) | \(-\) | \(-\) | \(-\) | \(q-5q^{7}+6q^{11}-3q^{13}+2q^{17}+q^{19}+\cdots\) | |
| 1800.2.a.b | \(1\) | \(14.373\) | \(\Q\) | None | \(0\) | \(0\) | \(0\) | \(-4\) | \(+\) | \(+\) | \(-\) | \(q-4q^{7}-4q^{11}+4q^{13}+6q^{17}-4q^{19}+\cdots\) | |
| 1800.2.a.c | \(1\) | \(14.373\) | \(\Q\) | None | \(0\) | \(0\) | \(0\) | \(-4\) | \(+\) | \(-\) | \(+\) | \(q-4q^{7}+6q^{13}-2q^{17}+4q^{19}-8q^{23}+\cdots\) | |
| 1800.2.a.d | \(1\) | \(14.373\) | \(\Q\) | None | \(0\) | \(0\) | \(0\) | \(-4\) | \(-\) | \(+\) | \(-\) | \(q-4q^{7}+4q^{11}+4q^{13}-6q^{17}-4q^{19}+\cdots\) | |
| 1800.2.a.e | \(1\) | \(14.373\) | \(\Q\) | None | \(0\) | \(0\) | \(0\) | \(-3\) | \(-\) | \(-\) | \(+\) | \(q-3q^{7}-2q^{11}+3q^{13}+6q^{17}-7q^{19}+\cdots\) | |
| 1800.2.a.f | \(1\) | \(14.373\) | \(\Q\) | None | \(0\) | \(0\) | \(0\) | \(-2\) | \(-\) | \(+\) | \(+\) | \(q-2q^{7}-2q^{11}-4q^{13}-2q^{17}+4q^{19}+\cdots\) | |
| 1800.2.a.g | \(1\) | \(14.373\) | \(\Q\) | None | \(0\) | \(0\) | \(0\) | \(-2\) | \(+\) | \(-\) | \(-\) | \(q-2q^{7}-2q^{11}+2q^{13}-6q^{17}+8q^{19}+\cdots\) | |
| 1800.2.a.h | \(1\) | \(14.373\) | \(\Q\) | None | \(0\) | \(0\) | \(0\) | \(-2\) | \(+\) | \(-\) | \(+\) | \(q-2q^{7}-q^{11}-4q^{13}+5q^{17}+q^{19}+\cdots\) | |
| 1800.2.a.i | \(1\) | \(14.373\) | \(\Q\) | None | \(0\) | \(0\) | \(0\) | \(-2\) | \(+\) | \(+\) | \(+\) | \(q-2q^{7}+2q^{11}-4q^{13}+2q^{17}+4q^{19}+\cdots\) | |
| 1800.2.a.j | \(1\) | \(14.373\) | \(\Q\) | None | \(0\) | \(0\) | \(0\) | \(-2\) | \(+\) | \(-\) | \(-\) | \(q-2q^{7}+4q^{11}-4q^{13}-4q^{19}-2q^{23}+\cdots\) | |
| 1800.2.a.k | \(1\) | \(14.373\) | \(\Q\) | None | \(0\) | \(0\) | \(0\) | \(-1\) | \(-\) | \(+\) | \(-\) | \(q-q^{7}-4q^{11}+q^{13}+4q^{17}+q^{19}+\cdots\) | |
| 1800.2.a.l | \(1\) | \(14.373\) | \(\Q\) | None | \(0\) | \(0\) | \(0\) | \(-1\) | \(+\) | \(+\) | \(-\) | \(q-q^{7}+4q^{11}+q^{13}-4q^{17}+q^{19}+\cdots\) | |
| 1800.2.a.m | \(1\) | \(14.373\) | \(\Q\) | None | \(0\) | \(0\) | \(0\) | \(0\) | \(-\) | \(-\) | \(+\) | \(q-4q^{11}+2q^{13}+2q^{17}-4q^{19}+\cdots\) | |
| 1800.2.a.n | \(1\) | \(14.373\) | \(\Q\) | None | \(0\) | \(0\) | \(0\) | \(0\) | \(-\) | \(-\) | \(+\) | \(q+4q^{11}-6q^{13}-6q^{17}-4q^{19}+\cdots\) | |
| 1800.2.a.o | \(1\) | \(14.373\) | \(\Q\) | None | \(0\) | \(0\) | \(0\) | \(1\) | \(+\) | \(+\) | \(+\) | \(q+q^{7}-4q^{11}-q^{13}-4q^{17}+q^{19}+\cdots\) | |
| 1800.2.a.p | \(1\) | \(14.373\) | \(\Q\) | None | \(0\) | \(0\) | \(0\) | \(1\) | \(-\) | \(+\) | \(+\) | \(q+q^{7}+4q^{11}-q^{13}+4q^{17}+q^{19}+\cdots\) | |
| 1800.2.a.q | \(1\) | \(14.373\) | \(\Q\) | None | \(0\) | \(0\) | \(0\) | \(2\) | \(-\) | \(-\) | \(-\) | \(q+2q^{7}-2q^{11}-2q^{13}+6q^{17}+8q^{19}+\cdots\) | |
| 1800.2.a.r | \(1\) | \(14.373\) | \(\Q\) | None | \(0\) | \(0\) | \(0\) | \(2\) | \(-\) | \(-\) | \(-\) | \(q+2q^{7}-q^{11}+4q^{13}-5q^{17}+q^{19}+\cdots\) | |
| 1800.2.a.s | \(1\) | \(14.373\) | \(\Q\) | None | \(0\) | \(0\) | \(0\) | \(2\) | \(-\) | \(-\) | \(-\) | \(q+2q^{7}+4q^{11}+4q^{13}-4q^{19}+2q^{23}+\cdots\) | |
| 1800.2.a.t | \(1\) | \(14.373\) | \(\Q\) | None | \(0\) | \(0\) | \(0\) | \(3\) | \(+\) | \(-\) | \(-\) | \(q+3q^{7}-2q^{11}-3q^{13}-6q^{17}-7q^{19}+\cdots\) | |
| 1800.2.a.u | \(1\) | \(14.373\) | \(\Q\) | None | \(0\) | \(0\) | \(0\) | \(4\) | \(-\) | \(+\) | \(-\) | \(q+4q^{7}-4q^{11}-4q^{13}-6q^{17}-4q^{19}+\cdots\) | |
| 1800.2.a.v | \(1\) | \(14.373\) | \(\Q\) | None | \(0\) | \(0\) | \(0\) | \(4\) | \(+\) | \(-\) | \(+\) | \(q+4q^{7}-4q^{11}+2q^{13}+2q^{17}+4q^{19}+\cdots\) | |
| 1800.2.a.w | \(1\) | \(14.373\) | \(\Q\) | None | \(0\) | \(0\) | \(0\) | \(4\) | \(+\) | \(+\) | \(-\) | \(q+4q^{7}+4q^{11}-4q^{13}+6q^{17}-4q^{19}+\cdots\) | |
| 1800.2.a.x | \(1\) | \(14.373\) | \(\Q\) | None | \(0\) | \(0\) | \(0\) | \(5\) | \(+\) | \(-\) | \(+\) | \(q+5q^{7}+6q^{11}+3q^{13}-2q^{17}+q^{19}+\cdots\) | |
Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1800))\) into lower level spaces
\( S_{2}^{\mathrm{old}}(\Gamma_0(1800)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(360))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(450))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(600))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(900))\)\(^{\oplus 2}\)