# Properties

 Label 819.1.cg.a Level $819$ Weight $1$ Character orbit 819.cg Analytic conductor $0.409$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$819 = 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 819.cg (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.408734245346$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.12960667629.4

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -\zeta_{6}^{2} q^{4} + \zeta_{6} q^{7} +O(q^{10})$$ $$q -\zeta_{6}^{2} q^{4} + \zeta_{6} q^{7} -\zeta_{6}^{2} q^{13} -\zeta_{6} q^{16} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{19} -\zeta_{6} q^{25} + q^{28} -\zeta_{6} q^{37} -\zeta_{6} q^{43} + \zeta_{6}^{2} q^{49} -\zeta_{6} q^{52} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{61} - q^{64} + 2 q^{67} + ( -1 + \zeta_{6}^{2} ) q^{73} + ( 1 + \zeta_{6} ) q^{76} + 2 \zeta_{6}^{2} q^{79} + q^{91} + ( -1 + \zeta_{6}^{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{4} + q^{7} + O(q^{10})$$ $$2 q + q^{4} + q^{7} + q^{13} - q^{16} - q^{25} + 2 q^{28} - q^{37} - q^{43} - q^{49} - q^{52} - 2 q^{64} + 4 q^{67} - 3 q^{73} + 3 q^{76} - 2 q^{79} + 2 q^{91} - 3 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/819\mathbb{Z}\right)^\times$$.

 $$n$$ $$92$$ $$379$$ $$703$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$-\zeta_{6}^{2}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
334.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0.500000 0.866025i 0 0 0.500000 + 0.866025i 0 0 0
640.1 0 0 0.500000 + 0.866025i 0 0 0.500000 0.866025i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
91.m odd 6 1 inner
273.bf even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.1.cg.a 2
3.b odd 2 1 CM 819.1.cg.a 2
7.d odd 6 1 819.1.ef.a yes 2
13.c even 3 1 819.1.ef.a yes 2
21.g even 6 1 819.1.ef.a yes 2
39.i odd 6 1 819.1.ef.a yes 2
91.m odd 6 1 inner 819.1.cg.a 2
273.bf even 6 1 inner 819.1.cg.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.1.cg.a 2 1.a even 1 1 trivial
819.1.cg.a 2 3.b odd 2 1 CM
819.1.cg.a 2 91.m odd 6 1 inner
819.1.cg.a 2 273.bf even 6 1 inner
819.1.ef.a yes 2 7.d odd 6 1
819.1.ef.a yes 2 13.c even 3 1
819.1.ef.a yes 2 21.g even 6 1
819.1.ef.a yes 2 39.i odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(819, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 - T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$1 - T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$3 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$1 + T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$1 + T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$3 + T^{2}$$
$67$ $$( -2 + T )^{2}$$
$71$ $$T^{2}$$
$73$ $$3 + 3 T + T^{2}$$
$79$ $$4 + 2 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$3 + 3 T + T^{2}$$