# Properties

 Label 819.2.s.a Level $819$ Weight $2$ Character orbit 819.s 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.s (of order $$3$$, 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{4} + 3 \zeta_{6} q^{5} + ( 3 - 2 \zeta_{6} ) q^{7} + 3 q^{8} +O(q^{10})$$ $$q - q^{2} - q^{4} + 3 \zeta_{6} q^{5} + ( 3 - 2 \zeta_{6} ) q^{7} + 3 q^{8} -3 \zeta_{6} q^{10} -3 \zeta_{6} q^{11} + ( 1 - 4 \zeta_{6} ) q^{13} + ( -3 + 2 \zeta_{6} ) q^{14} - q^{16} + 2 q^{17} + ( 1 - \zeta_{6} ) q^{19} -3 \zeta_{6} q^{20} + 3 \zeta_{6} q^{22} + ( -4 + 4 \zeta_{6} ) q^{25} + ( -1 + 4 \zeta_{6} ) q^{26} + ( -3 + 2 \zeta_{6} ) q^{28} + ( 7 - 7 \zeta_{6} ) q^{29} + ( -3 + 3 \zeta_{6} ) q^{31} -5 q^{32} -2 q^{34} + ( 6 + 3 \zeta_{6} ) q^{35} + 2 q^{37} + ( -1 + \zeta_{6} ) q^{38} + 9 \zeta_{6} q^{40} + ( 3 - 3 \zeta_{6} ) q^{41} + 7 \zeta_{6} q^{43} + 3 \zeta_{6} q^{44} + \zeta_{6} q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} + ( 4 - 4 \zeta_{6} ) q^{50} + ( -1 + 4 \zeta_{6} ) q^{52} + ( 3 - 3 \zeta_{6} ) q^{53} + ( 9 - 9 \zeta_{6} ) q^{55} + ( 9 - 6 \zeta_{6} ) q^{56} + ( -7 + 7 \zeta_{6} ) q^{58} + 4 q^{59} + ( 13 - 13 \zeta_{6} ) q^{61} + ( 3 - 3 \zeta_{6} ) q^{62} + 7 q^{64} + ( 12 - 9 \zeta_{6} ) q^{65} + 3 \zeta_{6} q^{67} -2 q^{68} + ( -6 - 3 \zeta_{6} ) q^{70} + 13 \zeta_{6} q^{71} + ( 13 - 13 \zeta_{6} ) q^{73} -2 q^{74} + ( -1 + \zeta_{6} ) q^{76} + ( -6 - 3 \zeta_{6} ) q^{77} + 3 \zeta_{6} q^{79} -3 \zeta_{6} q^{80} + ( -3 + 3 \zeta_{6} ) q^{82} + 6 \zeta_{6} q^{85} -7 \zeta_{6} q^{86} -9 \zeta_{6} q^{88} -6 q^{89} + ( -5 - 6 \zeta_{6} ) q^{91} -\zeta_{6} q^{94} + 3 q^{95} + 5 \zeta_{6} q^{97} + ( -5 + 8 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 2q^{4} + 3q^{5} + 4q^{7} + 6q^{8} + O(q^{10})$$ $$2q - 2q^{2} - 2q^{4} + 3q^{5} + 4q^{7} + 6q^{8} - 3q^{10} - 3q^{11} - 2q^{13} - 4q^{14} - 2q^{16} + 4q^{17} + q^{19} - 3q^{20} + 3q^{22} - 4q^{25} + 2q^{26} - 4q^{28} + 7q^{29} - 3q^{31} - 10q^{32} - 4q^{34} + 15q^{35} + 4q^{37} - q^{38} + 9q^{40} + 3q^{41} + 7q^{43} + 3q^{44} + q^{47} + 2q^{49} + 4q^{50} + 2q^{52} + 3q^{53} + 9q^{55} + 12q^{56} - 7q^{58} + 8q^{59} + 13q^{61} + 3q^{62} + 14q^{64} + 15q^{65} + 3q^{67} - 4q^{68} - 15q^{70} + 13q^{71} + 13q^{73} - 4q^{74} - q^{76} - 15q^{77} + 3q^{79} - 3q^{80} - 3q^{82} + 6q^{85} - 7q^{86} - 9q^{88} - 12q^{89} - 16q^{91} - q^{94} + 6q^{95} + 5q^{97} - 2q^{98} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.00000 0 −1.00000 1.50000 2.59808i 0 2.00000 + 1.73205i 3.00000 0 −1.50000 + 2.59808i
802.1 −1.00000 0 −1.00000 1.50000 + 2.59808i 0 2.00000 1.73205i 3.00000 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.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.s.a 2
3.b odd 2 1 91.2.h.a yes 2
7.c even 3 1 819.2.n.c 2
13.c even 3 1 819.2.n.c 2
21.c even 2 1 637.2.h.a 2
21.g even 6 1 637.2.f.a 2
21.g even 6 1 637.2.g.a 2
21.h odd 6 1 91.2.g.a 2
21.h odd 6 1 637.2.f.b 2
39.h odd 6 1 1183.2.e.c 2
39.i odd 6 1 91.2.g.a 2
39.i odd 6 1 1183.2.e.a 2
91.h even 3 1 inner 819.2.s.a 2
273.r even 6 1 637.2.h.a 2
273.r even 6 1 8281.2.a.j 1
273.s odd 6 1 91.2.h.a yes 2
273.s odd 6 1 8281.2.a.i 1
273.x odd 6 1 1183.2.e.c 2
273.bf even 6 1 637.2.f.a 2
273.bm odd 6 1 637.2.f.b 2
273.bm odd 6 1 1183.2.e.a 2
273.bn even 6 1 637.2.g.a 2
273.bp odd 6 1 8281.2.a.c 1
273.br even 6 1 8281.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.g.a 2 21.h odd 6 1
91.2.g.a 2 39.i odd 6 1
91.2.h.a yes 2 3.b odd 2 1
91.2.h.a yes 2 273.s odd 6 1
637.2.f.a 2 21.g even 6 1
637.2.f.a 2 273.bf even 6 1
637.2.f.b 2 21.h odd 6 1
637.2.f.b 2 273.bm odd 6 1
637.2.g.a 2 21.g even 6 1
637.2.g.a 2 273.bn even 6 1
637.2.h.a 2 21.c even 2 1
637.2.h.a 2 273.r even 6 1
819.2.n.c 2 7.c even 3 1
819.2.n.c 2 13.c even 3 1
819.2.s.a 2 1.a even 1 1 trivial
819.2.s.a 2 91.h even 3 1 inner
1183.2.e.a 2 39.i odd 6 1
1183.2.e.a 2 273.bm odd 6 1
1183.2.e.c 2 39.h odd 6 1
1183.2.e.c 2 273.x odd 6 1
8281.2.a.c 1 273.bp odd 6 1
8281.2.a.g 1 273.br even 6 1
8281.2.a.i 1 273.s odd 6 1
8281.2.a.j 1 273.r even 6 1

## Hecke kernels

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

 $$T_{2} + 1$$ $$T_{11}^{2} + 3 T_{11} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$9 - 3 T + T^{2}$$
$7$ $$7 - 4 T + T^{2}$$
$11$ $$9 + 3 T + T^{2}$$
$13$ $$13 + 2 T + T^{2}$$
$17$ $$( -2 + T )^{2}$$
$19$ $$1 - T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$49 - 7 T + T^{2}$$
$31$ $$9 + 3 T + T^{2}$$
$37$ $$( -2 + T )^{2}$$
$41$ $$9 - 3 T + T^{2}$$
$43$ $$49 - 7 T + T^{2}$$
$47$ $$1 - T + T^{2}$$
$53$ $$9 - 3 T + T^{2}$$
$59$ $$( -4 + T )^{2}$$
$61$ $$169 - 13 T + T^{2}$$
$67$ $$9 - 3 T + T^{2}$$
$71$ $$169 - 13 T + T^{2}$$
$73$ $$169 - 13 T + T^{2}$$
$79$ $$9 - 3 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$25 - 5 T + T^{2}$$