Defining parameters
Level: | \( N \) | \(=\) | \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 684.f (of order \(2\) and degree \(1\)) |
Character conductor: | \(\operatorname{cond}(\chi)\) | \(=\) | \( 76 \) |
Character field: | \(\Q\) | ||
Newform subspaces: | \( 5 \) | ||
Sturm bound: | \(240\) | ||
Trace bound: | \(2\) | ||
Distinguishing \(T_p\): | \(5\), \(31\) |
Dimensions
The following table gives the dimensions of various subspaces of \(M_{2}(684, [\chi])\).
Total | New | Old | |
---|---|---|---|
Modular forms | 128 | 52 | 76 |
Cusp forms | 112 | 48 | 64 |
Eisenstein series | 16 | 4 | 12 |
Trace form
Decomposition of \(S_{2}^{\mathrm{new}}(684, [\chi])\) into newform subspaces
Label | Dim | $A$ | Field | CM | Traces | $q$-expansion | |||
---|---|---|---|---|---|---|---|---|---|
$a_{2}$ | $a_{3}$ | $a_{5}$ | $a_{7}$ | ||||||
684.2.f.a | $4$ | $5.462$ | \(\Q(\sqrt{-2}, \sqrt{19})\) | \(\Q(\sqrt{-57}) \) | \(0\) | \(0\) | \(0\) | \(0\) | \(q-\beta _{1}q^{2}-2q^{4}+2\beta _{1}q^{8}-\beta _{2}q^{11}+\cdots\) |
684.2.f.b | $8$ | $5.462$ | 8.0.\(\cdots\).1 | None | \(0\) | \(0\) | \(4\) | \(0\) | \(q+\beta _{1}q^{2}+(-1+\beta _{2})q^{4}+(1-\beta _{2}+\beta _{6}+\cdots)q^{5}+\cdots\) |
684.2.f.c | $10$ | $5.462$ | 10.0.\(\cdots\).1 | None | \(-2\) | \(0\) | \(0\) | \(0\) | \(q-\beta _{1}q^{2}+\beta _{2}q^{4}-\beta _{4}q^{5}-\beta _{5}q^{7}+\cdots\) |
684.2.f.d | $10$ | $5.462$ | 10.0.\(\cdots\).1 | None | \(2\) | \(0\) | \(0\) | \(0\) | \(q+\beta _{1}q^{2}+\beta _{2}q^{4}-\beta _{4}q^{5}-\beta _{5}q^{7}+\cdots\) |
684.2.f.e | $16$ | $5.462$ | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) | None | \(0\) | \(0\) | \(0\) | \(0\) | \(q-\beta _{3}q^{2}+(1+\beta _{6})q^{4}-\beta _{11}q^{5}+(\beta _{7}+\cdots)q^{7}+\cdots\) |
Decomposition of \(S_{2}^{\mathrm{old}}(684, [\chi])\) into lower level spaces
\( S_{2}^{\mathrm{old}}(684, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(76, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(228, [\chi])\)\(^{\oplus 2}\)