Base field: \(\Q(\sqrt{-163}) \)
Generator \(a\), with minimal polynomial \(x^2 - x + 41\); class number \(1\).
Form
| Weight: | 2 | |
| Level: | 441.1 = \( \left(21\right) \) | |
| Level norm: | 441 | |
| Dimension: | 1 | |
| CM: | no | |
| Base change: | yes | 21.2.a.a |
| Newspace: | 2.0.163.1-441.1 (dimension 1) | |
| Sign of functional equation: | $-1$ | |
| Analytic rank: | odd |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| \( 9 \) | 9.1 = \( \left(3\right) \) | \( -1 \) |
| \( 49 \) | 49.1 = \( \left(7\right) \) | \( -1 \) |
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.
| $N(\mathfrak{p})$ | $\mathfrak{p}$ | $a_{\mathfrak{p}}$ |
|---|---|---|
| \( 4 \) | 4.1 = \( \left(2\right) \) | \( -3 \) |
| \( 25 \) | 25.1 = \( \left(5\right) \) | \( -6 \) |
| \( 41 \) | 41.1 = \( \left(-a\right) \) | \( 2 \) |
| \( 41 \) | 41.2 = \( \left(a - 1\right) \) | \( 2 \) |
| \( 43 \) | 43.1 = \( \left(a + 1\right) \) | \( -4 \) |
| \( 43 \) | 43.2 = \( \left(a - 2\right) \) | \( -4 \) |
| \( 47 \) | 47.1 = \( \left(a + 2\right) \) | \( 0 \) |
| \( 47 \) | 47.2 = \( \left(a - 3\right) \) | \( 0 \) |
| \( 53 \) | 53.1 = \( \left(a + 3\right) \) | \( 6 \) |
| \( 53 \) | 53.2 = \( \left(a - 4\right) \) | \( 6 \) |
| \( 61 \) | 61.1 = \( \left(a + 4\right) \) | \( -2 \) |
| \( 61 \) | 61.2 = \( \left(a - 5\right) \) | \( -2 \) |
| \( 71 \) | 71.1 = \( \left(a + 5\right) \) | \( 0 \) |
| \( 71 \) | 71.2 = \( \left(a - 6\right) \) | \( 0 \) |
| \( 83 \) | 83.1 = \( \left(a + 6\right) \) | \( -12 \) |
| \( 83 \) | 83.2 = \( \left(a - 7\right) \) | \( -12 \) |
| \( 97 \) | 97.1 = \( \left(a + 7\right) \) | \( 18 \) |
| \( 97 \) | 97.2 = \( \left(a - 8\right) \) | \( 18 \) |
| \( 113 \) | 113.1 = \( \left(a + 8\right) \) | \( -14 \) |
| \( 113 \) | 113.2 = \( \left(a - 9\right) \) | \( -14 \) |