Newspace parameters
| Level: | \( N \) | = | \( 3 \) |
| Weight: | \( k \) | = | \( 22 \) |
| Character orbit: | \([\chi]\) | = | 3.a (trivial) |
Newform invariants
| Self dual: | Yes |
| Analytic conductor: | \(8.38432032861\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{649}) \) |
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2\cdot 3^{2}\cdot 7 \) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 63\sqrt{649}\). We also show the integral \(q\)-expansion of the trace form.
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−1271.96 | 59049.0 | −479282. | −2.12776e7 | −7.51077e7 | 6.32076e8 | 3.27711e9 | 3.48678e9 | 2.70641e10 | ||||||||||||||||||||||||
| 1.2 | 1937.96 | 59049.0 | 1.65852e6 | 2.22745e7 | 1.14434e8 | 4.78205e7 | −8.50053e8 | 3.48678e9 | 4.31669e10 | |||||||||||||||||||||||||
Inner twists
This newform does not admit any (nontrivial) inner twists.
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
Hecke kernels
This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut -\mathstrut 666 T_{2} \) \(\mathstrut -\mathstrut 2464992 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(3))\).