Properties

Label 2.0.163.1-652.1-c
Base field \(\Q(\sqrt{-163}) \)
Weight $2$
Level norm $652$
Level \( \left(-4 a + 2\right) \)
Dimension $1$
CM no
Base change yes
Sign $-1$
Analytic rank odd

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Base field: \(\Q(\sqrt{-163}) \)

Generator \(a\), with minimal polynomial \(x^2 - x + 41\); class number \(1\).

Form

Weight: 2
Level: 652.1 = \( \left(-4 a + 2\right) \)
Level norm: 652
Dimension: 1
CM: no
Base change: yes , 326.2.a.a
Newspace:2.0.163.1-652.1 (dimension 14)
Sign of functional equation: $-1$
Analytic rank: odd

Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
\( 4 \) 4.1 = (\( 2 \)) \( -1 \)
\( 163 \) 163.1 = (\( -2 a + 1 \)) \( -1 \)

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.

$N(\mathfrak{p})$ $\mathfrak{p}$ $a_{\mathfrak{p}}$
\( 9 \) 9.1 = (\( 3 \)) \( -2 \)
\( 25 \) 25.1 = (\( 5 \)) \( -1 \)
\( 41 \) 41.1 = (\( -a \)) \( 9 \)
\( 41 \) 41.2 = (\( a - 1 \)) \( 9 \)
\( 43 \) 43.1 = (\( a + 1 \)) \( -1 \)
\( 43 \) 43.2 = (\( a - 2 \)) \( -1 \)
\( 47 \) 47.1 = (\( a + 2 \)) \( -12 \)
\( 47 \) 47.2 = (\( a - 3 \)) \( -12 \)
\( 49 \) 49.1 = (\( 7 \)) \( -13 \)
\( 53 \) 53.1 = (\( a + 3 \)) \( 0 \)
\( 53 \) 53.2 = (\( a - 4 \)) \( 0 \)
\( 61 \) 61.1 = (\( a + 4 \)) \( 8 \)
\( 61 \) 61.2 = (\( a - 5 \)) \( 8 \)
\( 71 \) 71.1 = (\( a + 5 \)) \( -12 \)
\( 71 \) 71.2 = (\( a - 6 \)) \( -12 \)
\( 83 \) 83.1 = (\( a + 6 \)) \( -3 \)
\( 83 \) 83.2 = (\( a - 7 \)) \( -3 \)
\( 97 \) 97.1 = (\( a + 7 \)) \( -1 \)
\( 97 \) 97.2 = (\( a - 8 \)) \( -1 \)
\( 113 \) 113.1 = (\( a + 8 \)) \( 18 \)
\( 113 \) 113.2 = (\( a - 9 \)) \( 18 \)
Display number of eigenvalues