# Properties

 Label 819.2.bm.a Level $819$ Weight $2$ Character orbit 819.bm Analytic conductor $6.540$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + 2 \zeta_{6} ) q^{2} - q^{4} + ( -2 + \zeta_{6} ) q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} + ( -1 + 2 \zeta_{6} ) q^{8} +O(q^{10})$$ $$q + ( -1 + 2 \zeta_{6} ) q^{2} - q^{4} + ( -2 + \zeta_{6} ) q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} + ( -1 + 2 \zeta_{6} ) q^{8} -3 \zeta_{6} q^{10} + ( 6 - 3 \zeta_{6} ) q^{11} + ( -3 + 4 \zeta_{6} ) q^{13} + ( -1 - 4 \zeta_{6} ) q^{14} -5 q^{16} -6 q^{17} + ( -1 - \zeta_{6} ) q^{19} + ( 2 - \zeta_{6} ) q^{20} + 9 \zeta_{6} q^{22} + ( -2 + 2 \zeta_{6} ) q^{25} + ( -5 - 2 \zeta_{6} ) q^{26} + ( 3 - 2 \zeta_{6} ) q^{28} + ( 3 - 3 \zeta_{6} ) q^{29} + ( -1 - \zeta_{6} ) q^{31} + ( 3 - 6 \zeta_{6} ) q^{32} + ( 6 - 12 \zeta_{6} ) q^{34} + ( 4 - 5 \zeta_{6} ) q^{35} + ( 3 - 3 \zeta_{6} ) q^{38} -3 \zeta_{6} q^{40} + ( 3 + 3 \zeta_{6} ) q^{41} -11 \zeta_{6} q^{43} + ( -6 + 3 \zeta_{6} ) q^{44} + ( -10 + 5 \zeta_{6} ) q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} + ( -2 - 2 \zeta_{6} ) q^{50} + ( 3 - 4 \zeta_{6} ) q^{52} + ( -9 + 9 \zeta_{6} ) q^{53} + ( -9 + 9 \zeta_{6} ) q^{55} + ( -1 - 4 \zeta_{6} ) q^{56} + ( 3 + 3 \zeta_{6} ) q^{58} + ( 2 - 4 \zeta_{6} ) q^{59} + ( -7 + 7 \zeta_{6} ) q^{61} + ( 3 - 3 \zeta_{6} ) q^{62} - q^{64} + ( 2 - 7 \zeta_{6} ) q^{65} + ( 10 - 5 \zeta_{6} ) q^{67} + 6 q^{68} + ( 6 + 3 \zeta_{6} ) q^{70} + ( -2 + \zeta_{6} ) q^{71} + ( 5 + 5 \zeta_{6} ) q^{73} + ( 1 + \zeta_{6} ) q^{76} + ( -12 + 15 \zeta_{6} ) q^{77} + 5 \zeta_{6} q^{79} + ( 10 - 5 \zeta_{6} ) q^{80} + ( -9 + 9 \zeta_{6} ) q^{82} + ( -2 + 4 \zeta_{6} ) q^{83} + ( 12 - 6 \zeta_{6} ) q^{85} + ( 22 - 11 \zeta_{6} ) q^{86} + 9 \zeta_{6} q^{88} + ( -4 + 8 \zeta_{6} ) q^{89} + ( 1 - 10 \zeta_{6} ) q^{91} -15 \zeta_{6} q^{94} + 3 q^{95} + ( -6 + 3 \zeta_{6} ) q^{97} + ( 11 + 2 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 3 q^{5} - 4 q^{7} + O(q^{10})$$ $$2 q - 2 q^{4} - 3 q^{5} - 4 q^{7} - 3 q^{10} + 9 q^{11} - 2 q^{13} - 6 q^{14} - 10 q^{16} - 12 q^{17} - 3 q^{19} + 3 q^{20} + 9 q^{22} - 2 q^{25} - 12 q^{26} + 4 q^{28} + 3 q^{29} - 3 q^{31} + 3 q^{35} + 3 q^{38} - 3 q^{40} + 9 q^{41} - 11 q^{43} - 9 q^{44} - 15 q^{47} + 2 q^{49} - 6 q^{50} + 2 q^{52} - 9 q^{53} - 9 q^{55} - 6 q^{56} + 9 q^{58} - 7 q^{61} + 3 q^{62} - 2 q^{64} - 3 q^{65} + 15 q^{67} + 12 q^{68} + 15 q^{70} - 3 q^{71} + 15 q^{73} + 3 q^{76} - 9 q^{77} + 5 q^{79} + 15 q^{80} - 9 q^{82} + 18 q^{85} + 33 q^{86} + 9 q^{88} - 8 q^{91} - 15 q^{94} + 6 q^{95} - 9 q^{97} + 24 q^{98} + O(q^{100})$$

## 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
1.73205i 0 −1.00000 −1.50000 0.866025i 0 −2.00000 1.73205i 1.73205i 0 −1.50000 + 2.59808i
550.1 1.73205i 0 −1.00000 −1.50000 + 0.866025i 0 −2.00000 + 1.73205i 1.73205i 0 −1.50000 2.59808i
 $$n$$: 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
91.k even 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.a 2
3.b odd 2 1 91.2.k.a 2
7.c even 3 1 819.2.do.c 2
13.e even 6 1 819.2.do.c 2
21.c even 2 1 637.2.k.b 2
21.g even 6 1 637.2.q.b 2
21.g even 6 1 637.2.u.a 2
21.h odd 6 1 91.2.u.a yes 2
21.h odd 6 1 637.2.q.c 2
39.h odd 6 1 91.2.u.a yes 2
39.k even 12 2 1183.2.e.e 4
91.k even 6 1 inner 819.2.bm.a 2
273.u even 6 1 637.2.u.a 2
273.x odd 6 1 637.2.q.c 2
273.y even 6 1 637.2.q.b 2
273.bp odd 6 1 91.2.k.a 2
273.br even 6 1 637.2.k.b 2
273.bs odd 12 2 8281.2.a.w 2
273.bv even 12 2 8281.2.a.s 2
273.bw even 12 2 1183.2.e.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.k.a 2 3.b odd 2 1
91.2.k.a 2 273.bp odd 6 1
91.2.u.a yes 2 21.h odd 6 1
91.2.u.a yes 2 39.h odd 6 1
637.2.k.b 2 21.c even 2 1
637.2.k.b 2 273.br even 6 1
637.2.q.b 2 21.g even 6 1
637.2.q.b 2 273.y even 6 1
637.2.q.c 2 21.h odd 6 1
637.2.q.c 2 273.x odd 6 1
637.2.u.a 2 21.g even 6 1
637.2.u.a 2 273.u even 6 1
819.2.bm.a 2 1.a even 1 1 trivial
819.2.bm.a 2 91.k even 6 1 inner
819.2.do.c 2 7.c even 3 1
819.2.do.c 2 13.e even 6 1
1183.2.e.e 4 39.k even 12 2
1183.2.e.e 4 273.bw even 12 2
8281.2.a.s 2 273.bv even 12 2
8281.2.a.w 2 273.bs odd 12 2

## Hecke kernels

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

 $$T_{2}^{2} + 3$$ $$T_{5}^{2} + 3 T_{5} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$3 + 3 T + T^{2}$$
$7$ $$7 + 4 T + T^{2}$$
$11$ $$27 - 9 T + T^{2}$$
$13$ $$13 + 2 T + T^{2}$$
$17$ $$( 6 + T )^{2}$$
$19$ $$3 + 3 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$9 - 3 T + T^{2}$$
$31$ $$3 + 3 T + T^{2}$$
$37$ $$T^{2}$$
$41$ $$27 - 9 T + T^{2}$$
$43$ $$121 + 11 T + T^{2}$$
$47$ $$75 + 15 T + T^{2}$$
$53$ $$81 + 9 T + T^{2}$$
$59$ $$12 + T^{2}$$
$61$ $$49 + 7 T + T^{2}$$
$67$ $$75 - 15 T + T^{2}$$
$71$ $$3 + 3 T + T^{2}$$
$73$ $$75 - 15 T + T^{2}$$
$79$ $$25 - 5 T + T^{2}$$
$83$ $$12 + T^{2}$$
$89$ $$48 + T^{2}$$
$97$ $$27 + 9 T + T^{2}$$