# Properties

 Label 2.0.8.1-21609.2-b Base field $$\Q(\sqrt{-2})$$ Weight $2$ Level norm $21609$ Level $$\left(147\right)$$ Dimension $1$ CM no Base change yes Sign $-1$ Analytic rank odd

# Related objects

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

Generator $$a$$, with minimal polynomial $$x^2 + 2$$; class number $$1$$.

## Form

 Weight: 2 Level: 21609.2 = $$\left(147\right)$$ Level norm: 21609 Dimension: 1 CM: no Base change: yes 9408.2.a.k , 147.2.a.b Newspace: 2.0.8.1-21609.2 (dimension 55) Sign of functional equation: $-1$ Analytic rank: odd

## Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
$$3$$ 3.1 = $$\left(-a - 1\right)$$ $$1$$
$$3$$ 3.2 = $$\left(a - 1\right)$$ $$1$$
$$49$$ 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 500 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.

$N(\mathfrak{p})$ $\mathfrak{p}$ $a_{\mathfrak{p}}$
$$2$$ 2.1 = $$\left(a\right)$$ $$2$$
$$11$$ 11.1 = $$\left(a + 3\right)$$ $$-2$$
$$11$$ 11.2 = $$\left(a - 3\right)$$ $$-2$$
$$17$$ 17.1 = $$\left(-2 a + 3\right)$$ $$0$$
$$17$$ 17.2 = $$\left(2 a + 3\right)$$ $$0$$
$$19$$ 19.1 = $$\left(-3 a + 1\right)$$ $$-1$$
$$19$$ 19.2 = $$\left(3 a + 1\right)$$ $$-1$$
$$25$$ 25.1 = $$\left(5\right)$$ $$-6$$
$$41$$ 41.1 = $$\left(-4 a - 3\right)$$ $$10$$
$$41$$ 41.2 = $$\left(4 a - 3\right)$$ $$10$$
$$43$$ 43.1 = $$\left(-3 a - 5\right)$$ $$5$$
$$43$$ 43.2 = $$\left(3 a - 5\right)$$ $$5$$
$$59$$ 59.1 = $$\left(-5 a + 3\right)$$ $$12$$
$$59$$ 59.2 = $$\left(-5 a - 3\right)$$ $$12$$
$$67$$ 67.1 = $$\left(-3 a + 7\right)$$ $$-5$$
$$67$$ 67.2 = $$\left(3 a + 7\right)$$ $$-5$$
$$73$$ 73.1 = $$\left(-6 a + 1\right)$$ $$3$$
$$73$$ 73.2 = $$\left(6 a + 1\right)$$ $$3$$
$$83$$ 83.1 = $$\left(a + 9\right)$$ $$-6$$
$$83$$ 83.2 = $$\left(a - 9\right)$$ $$-6$$
 Display number of eigenvalues