Newspace parameters
| Level: | \( N \) | = | \( 39 = 3 \cdot 13 \) |
| Weight: | \( k \) | = | \( 1 \) |
| Character orbit: | \([\chi]\) | = | 39.d (of order \(2\) and degree \(1\)) |
Newform invariants
| Self dual: | Yes |
| Analytic conductor: | \(0.0194635354927\) |
| Analytic rank: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Projective image | \(D_{2}\) |
| Projective field | Galois closure of \(\Q(\sqrt{-3}, \sqrt{13})\) |
| Artin image size | \(8\) |
| Artin image | $D_4$ |
| Artin field | Galois closure of 4.0.117.1 |
$q$-expansion
Character Values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/39\mathbb{Z}\right)^\times\).
| \(n\) | \(14\) | \(28\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 38.1 |
|
0 | −1.00000 | −1.00000 | 0 | 0 | 0 | 0 | 1.00000 | 0 | |||||||||||||||||||||
Inner twists
| Char. orbit | Parity | Mult. | Self Twist | Proved |
|---|---|---|---|---|
| 1.a | Even | 1 | trivial | yes |
| 3.b | Odd | 1 | CM by \(\Q(\sqrt{-3}) \) | yes |
| 13.b | Even | 1 | RM by \(\Q(\sqrt{13}) \) | yes |
| 39.d | Odd | 1 | CM by \(\Q(\sqrt{-39}) \) | yes |
Hecke kernels
There are no other newforms in \(S_{1}^{\mathrm{new}}(39, [\chi])\).