# Properties

 Label 2.0.3.1-21609.3-d Base field $$\Q(\sqrt{-3})$$ Weight $2$ Level norm $21609$ Level $$\left(147\right)$$ Dimension $1$ CM $-7$ Base change yes Sign $+1$ Analytic rank $$0$$

# Related objects

## Base field: $$\Q(\sqrt{-3})$$

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

## Form

 Weight: 2 Level: 21609.3 = $$\left(147\right)$$ Level norm: 21609 Dimension: 1 CM: $-7$ Base change: yes, of a form over $$\mathbb{Q}$$ with coefficients in $$\mathbb{Q}(\sqrt{7})$$ Newspace: 2.0.3.1-21609.3 (dimension 8) Sign of functional equation: $+1$ Analytic rank: $$0$$ L-ratio: 2

## Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
$$3$$ 3.1 = ($$-2 a + 1$$) $$-1$$
$$7$$ 7.1 = ($$-3 a + 1$$) $$-1$$
$$7$$ 7.2 = ($$3 a - 2$$) $$-1$$

## 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.

$N(\mathfrak{p})$ $\mathfrak{p}$ $a_{\mathfrak{p}}$
$$4$$ 4.1 = ($$2$$) $$3$$
$$13$$ 13.1 = ($$-4 a + 1$$) $$0$$
$$13$$ 13.2 = ($$4 a - 3$$) $$0$$
$$19$$ 19.1 = ($$-5 a + 3$$) $$0$$
$$19$$ 19.2 = ($$-5 a + 2$$) $$0$$
$$25$$ 25.1 = ($$5$$) $$-10$$
$$31$$ 31.1 = ($$-6 a + 1$$) $$0$$
$$31$$ 31.2 = ($$6 a - 5$$) $$0$$
$$37$$ 37.1 = ($$-7 a + 4$$) $$6$$
$$37$$ 37.2 = ($$-7 a + 3$$) $$6$$
$$43$$ 43.1 = ($$-7 a + 1$$) $$12$$
$$43$$ 43.2 = ($$7 a - 6$$) $$12$$
$$61$$ 61.1 = ($$-9 a + 5$$) $$0$$
$$61$$ 61.2 = ($$-9 a + 4$$) $$0$$
$$67$$ 67.1 = ($$9 a - 7$$) $$4$$
$$67$$ 67.2 = ($$9 a - 2$$) $$4$$
$$73$$ 73.1 = ($$-9 a + 1$$) $$0$$
$$73$$ 73.2 = ($$9 a - 8$$) $$0$$
$$79$$ 79.1 = ($$10 a - 7$$) $$8$$
$$79$$ 79.2 = ($$10 a - 3$$) $$8$$
 Display number of eigenvalues