Properties

Label 2.0.8.1-36864.2-a
Base field \(\Q(\sqrt{-2}) \)
Weight $2$
Level norm $36864$
Level \( \left(192\right) \)
Dimension $1$
CM no
Base change yes
Sign $+1$
Analytic rank \(\ge2\), even

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Base field: \(\Q(\sqrt{-2}) \)

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

Form

Weight: 2
Level: 36864.2 = \( \left(192\right) \)
Level norm: 36864
Dimension: 1
CM: no
Base change: yes, of a form over \(\mathbb{Q}\) with coefficients in \(\mathbb{Q}(\sqrt{2})\)
Newspace:2.0.8.1-36864.2 (dimension 40)
Sign of functional equation: $+1$
Analytic rank: \(\ge2\), even

Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
\( 2 \) 2.1 = \( \left(a\right) \) \( -1 \)
\( 3 \) 3.1 = \( \left(-a - 1\right) \) \( 1 \)
\( 3 \) 3.2 = \( \left(a - 1\right) \) \( 1 \)

Hecke eigenvalues

The Hecke eigenvalue field is $\Q$. The eigenvalue of the Hecke operator $T_{\mathfrak{p}}$ is $a_{\mathfrak{p}}$. The database contains 200 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.

$N(\mathfrak{p})$ $\mathfrak{p}$ $a_{\mathfrak{p}}$
\( 11 \) 11.1 = \( \left(a + 3\right) \) \( -2 \)
\( 11 \) 11.2 = \( \left(a - 3\right) \) \( -2 \)
\( 17 \) 17.1 = \( \left(-2 a + 3\right) \) \( 2 \)
\( 17 \) 17.2 = \( \left(2 a + 3\right) \) \( 2 \)
\( 19 \) 19.1 = \( \left(-3 a + 1\right) \) \( -4 \)
\( 19 \) 19.2 = \( \left(3 a + 1\right) \) \( -4 \)
\( 25 \) 25.1 = \( \left(5\right) \) \( -8 \)
\( 41 \) 41.1 = \( \left(-4 a - 3\right) \) \( 6 \)
\( 41 \) 41.2 = \( \left(4 a - 3\right) \) \( 6 \)
\( 43 \) 43.1 = \( \left(-3 a - 5\right) \) \( -8 \)
\( 43 \) 43.2 = \( \left(3 a - 5\right) \) \( -8 \)
\( 49 \) 49.1 = \( \left(7\right) \) \( -12 \)
\( 59 \) 59.1 = \( \left(-5 a + 3\right) \) \( -12 \)
\( 59 \) 59.2 = \( \left(-5 a - 3\right) \) \( -12 \)
\( 67 \) 67.1 = \( \left(-3 a + 7\right) \) \( -8 \)
\( 67 \) 67.2 = \( \left(3 a + 7\right) \) \( -8 \)
\( 73 \) 73.1 = \( \left(-6 a + 1\right) \) \( -8 \)
\( 73 \) 73.2 = \( \left(6 a + 1\right) \) \( -8 \)
\( 83 \) 83.1 = \( \left(a + 9\right) \) \( -6 \)
\( 83 \) 83.2 = \( \left(a - 9\right) \) \( -6 \)
Display number of eigenvalues