Base field: \(\Q(\sqrt{-13}) \)
Generator \(a\), with minimal polynomial \(x^2 + 13\); class number \(2\).
Form
| Weight: | 2 | |
| Level: | 144.1 = \( \left(12\right) \) | |
| Level norm: | 144 | |
| Dimension: | 1 | |
| CM: | no | |
| Base change: | no, but is a twist of the base change of a form over \(\mathbb{Q}\) | |
| Newspace: | 2.0.52.1-144.1 (dimension 14) | |
| Sign of functional equation: | $-1$ | |
| Analytic rank: | odd |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| \( 2 \) | 2.1 = \( \left(2, a + 1\right) \) | \( 1 \) |
| \( 9 \) | 9.1 = \( \left(3\right) \) | \( -1 \) |
Hecke eigenvalues
The Hecke eigenvalue field is $\Q$. The eigenvalue of the Hecke operator $T_{\mathfrak{p}}$ is $a_{\mathfrak{p}}$. The database contains 100 eigenvalues, of which 20 are currently shown below. We only show the eigenvalues $a_{\mathfrak{p}}$ for primes $\mathfrak{p}$ which do not divide the level.
| $N(\mathfrak{p})$ | $\mathfrak{p}$ | $a_{\mathfrak{p}}$ |
|---|---|---|
| \( 7 \) | 7.1 = \( \left(7, a + 1\right) \) | \( -2 \) |
| \( 7 \) | 7.2 = \( \left(7, a + 6\right) \) | \( 2 \) |
| \( 11 \) | 11.1 = \( \left(11, a + 3\right) \) | \( 2 \) |
| \( 11 \) | 11.2 = \( \left(11, a + 8\right) \) | \( -2 \) |
| \( 13 \) | 13.1 = \( \left(a\right) \) | \( -6 \) |
| \( 17 \) | 17.1 = \( \left(a + 2\right) \) | \( -2 \) |
| \( 17 \) | 17.2 = \( \left(a - 2\right) \) | \( -2 \) |
| \( 19 \) | 19.1 = \( \left(19, a + 5\right) \) | \( -6 \) |
| \( 19 \) | 19.2 = \( \left(19, a + 14\right) \) | \( 6 \) |
| \( 25 \) | 25.1 = \( \left(5\right) \) | \( 10 \) |
| \( 29 \) | 29.1 = \( \left(a + 4\right) \) | \( 2 \) |
| \( 29 \) | 29.2 = \( \left(a - 4\right) \) | \( 2 \) |
| \( 31 \) | 31.1 = \( \left(31, a + 7\right) \) | \( 2 \) |
| \( 31 \) | 31.2 = \( \left(31, a + 24\right) \) | \( -2 \) |
| \( 47 \) | 47.1 = \( \left(47, a + 9\right) \) | \( 10 \) |
| \( 47 \) | 47.2 = \( \left(47, a + 38\right) \) | \( -10 \) |
| \( 53 \) | 53.1 = \( \left(-2 a + 1\right) \) | \( -10 \) |
| \( 53 \) | 53.2 = \( \left(-2 a - 1\right) \) | \( -10 \) |
| \( 59 \) | 59.1 = \( \left(59, a + 20\right) \) | \( -6 \) |
| \( 59 \) | 59.2 = \( \left(59, a + 39\right) \) | \( 6 \) |