# Properties

 Label 819.2.bm.c Level $819$ Weight $2$ Character orbit 819.bm Analytic conductor $6.540$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$6.53974792554$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{4} + ( -3 + 2 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + 2 q^{4} + ( -3 + 2 \zeta_{6} ) q^{7} + ( -3 + 4 \zeta_{6} ) q^{13} + 4 q^{16} + ( 5 + 5 \zeta_{6} ) q^{19} + ( -5 + 5 \zeta_{6} ) q^{25} + ( -6 + 4 \zeta_{6} ) q^{28} + ( 5 + 5 \zeta_{6} ) q^{31} + ( 3 - 6 \zeta_{6} ) q^{37} -8 \zeta_{6} q^{43} + ( 5 - 8 \zeta_{6} ) q^{49} + ( -6 + 8 \zeta_{6} ) q^{52} + ( -1 + \zeta_{6} ) q^{61} + 8 q^{64} + ( -14 + 7 \zeta_{6} ) q^{67} + ( 8 + 8 \zeta_{6} ) q^{73} + ( 10 + 10 \zeta_{6} ) q^{76} -13 \zeta_{6} q^{79} + ( 1 - 10 \zeta_{6} ) q^{91} + ( -6 + 3 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{4} - 4 q^{7} + O(q^{10})$$ $$2 q + 4 q^{4} - 4 q^{7} - 2 q^{13} + 8 q^{16} + 15 q^{19} - 5 q^{25} - 8 q^{28} + 15 q^{31} - 8 q^{43} + 2 q^{49} - 4 q^{52} - q^{61} + 16 q^{64} - 21 q^{67} + 24 q^{73} + 30 q^{76} - 13 q^{79} - 8 q^{91} - 9 q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
478.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 2.00000 0 0 −2.00000 1.73205i 0 0 0
550.1 0 0 2.00000 0 0 −2.00000 + 1.73205i 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.k even 6 1 inner
273.bp odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.bm.c 2
3.b odd 2 1 CM 819.2.bm.c 2
7.c even 3 1 819.2.do.b yes 2
13.e even 6 1 819.2.do.b yes 2
21.h odd 6 1 819.2.do.b yes 2
39.h odd 6 1 819.2.do.b yes 2
91.k even 6 1 inner 819.2.bm.c 2
273.bp odd 6 1 inner 819.2.bm.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.bm.c 2 1.a even 1 1 trivial
819.2.bm.c 2 3.b odd 2 1 CM
819.2.bm.c 2 91.k even 6 1 inner
819.2.bm.c 2 273.bp odd 6 1 inner
819.2.do.b yes 2 7.c even 3 1
819.2.do.b yes 2 13.e even 6 1
819.2.do.b yes 2 21.h odd 6 1
819.2.do.b yes 2 39.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(819, [\chi])$$:

 $$T_{2}$$ $$T_{5}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$7 + 4 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$13 + 2 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$75 - 15 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$75 - 15 T + T^{2}$$
$37$ $$27 + T^{2}$$
$41$ $$T^{2}$$
$43$ $$64 + 8 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$1 + T + T^{2}$$
$67$ $$147 + 21 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$192 - 24 T + T^{2}$$
$79$ $$169 + 13 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$27 + 9 T + T^{2}$$