# Properties

 Label 2.0.163.1-441.1-a Base field $$\Q(\sqrt{-163})$$ Weight $2$ Level norm $441$ Level $$\left(21\right)$$ Dimension $1$ CM no Base change yes Sign $-1$ Analytic rank odd

# Related objects

## 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 = ($$3$$) $$-1$$
$$49$$ 49.1 = ($$7$$) $$-1$$

## 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 = ($$2$$) $$-3$$
$$25$$ 25.1 = ($$5$$) $$-6$$
$$41$$ 41.1 = ($$-a$$) $$2$$
$$41$$ 41.2 = ($$a - 1$$) $$2$$
$$43$$ 43.1 = ($$a + 1$$) $$-4$$
$$43$$ 43.2 = ($$a - 2$$) $$-4$$
$$47$$ 47.1 = ($$a + 2$$) $$0$$
$$47$$ 47.2 = ($$a - 3$$) $$0$$
$$53$$ 53.1 = ($$a + 3$$) $$6$$
$$53$$ 53.2 = ($$a - 4$$) $$6$$
$$61$$ 61.1 = ($$a + 4$$) $$-2$$
$$61$$ 61.2 = ($$a - 5$$) $$-2$$
$$71$$ 71.1 = ($$a + 5$$) $$0$$
$$71$$ 71.2 = ($$a - 6$$) $$0$$
$$83$$ 83.1 = ($$a + 6$$) $$-12$$
$$83$$ 83.2 = ($$a - 7$$) $$-12$$
$$97$$ 97.1 = ($$a + 7$$) $$18$$
$$97$$ 97.2 = ($$a - 8$$) $$18$$
$$113$$ 113.1 = ($$a + 8$$) $$-14$$
$$113$$ 113.2 = ($$a - 9$$) $$-14$$
 Display number of eigenvalues