# Properties

 Label 819.2.o.c Level $819$ Weight $2$ Character orbit 819.o Analytic conductor $6.540$ Analytic rank $0$ Dimension $4$ 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.o (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.53974792554$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{5})$$ Defining polynomial: $$x^{4} - x^{3} + 2 x^{2} + x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{1} + \beta_{3} ) q^{2} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{4} + ( -1 + \beta_{2} ) q^{5} + \beta_{3} q^{7} + ( -1 + 4 \beta_{2} ) q^{8} +O(q^{10})$$ $$q + ( 1 + \beta_{1} + \beta_{3} ) q^{2} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{4} + ( -1 + \beta_{2} ) q^{5} + \beta_{3} q^{7} + ( -1 + 4 \beta_{2} ) q^{8} + ( -2 - 3 \beta_{1} - 2 \beta_{3} ) q^{10} + ( -3 + 3 \beta_{1} - 3 \beta_{3} ) q^{11} + ( -4 - 3 \beta_{3} ) q^{13} + ( -1 + \beta_{2} ) q^{14} + ( -5 - 3 \beta_{1} - 5 \beta_{3} ) q^{16} + ( 4 \beta_{1} + 4 \beta_{2} - 5 \beta_{3} ) q^{17} + ( -3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{19} + ( -6 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{20} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{22} + ( 2 - 4 \beta_{1} + 2 \beta_{3} ) q^{23} + ( -3 - 3 \beta_{2} ) q^{25} + ( -1 - 4 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{26} -3 \beta_{1} q^{28} + ( -1 + 5 \beta_{1} - \beta_{3} ) q^{29} + ( 5 + 6 \beta_{2} ) q^{31} + ( -3 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} ) q^{32} + ( 1 + 3 \beta_{2} ) q^{34} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{35} + ( -4 - 4 \beta_{3} ) q^{37} -3 \beta_{2} q^{38} + ( 5 - 9 \beta_{2} ) q^{40} + ( 4 - 2 \beta_{1} + 4 \beta_{3} ) q^{41} + ( 9 \beta_{1} + 9 \beta_{2} - 2 \beta_{3} ) q^{43} -9 q^{44} + ( -6 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} ) q^{46} + ( -1 - 2 \beta_{2} ) q^{47} + ( -1 - \beta_{3} ) q^{49} + 3 \beta_{1} q^{50} + ( -3 \beta_{1} - 12 \beta_{2} ) q^{52} + ( -7 - 2 \beta_{2} ) q^{53} -3 \beta_{1} q^{55} + ( -4 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{56} + ( 9 \beta_{1} + 9 \beta_{2} + 4 \beta_{3} ) q^{58} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{59} -6 \beta_{3} q^{61} + ( -1 - 7 \beta_{1} - \beta_{3} ) q^{62} + ( -1 - 6 \beta_{2} ) q^{64} + ( 4 + 3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{65} + ( 3 + 6 \beta_{1} + 3 \beta_{3} ) q^{67} + ( -12 + 3 \beta_{1} - 12 \beta_{3} ) q^{68} + ( 2 - 3 \beta_{2} ) q^{70} + ( 10 \beta_{1} + 10 \beta_{2} - 2 \beta_{3} ) q^{71} -2 q^{73} + ( -4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{74} + ( 9 + 9 \beta_{3} ) q^{76} + ( 3 + 3 \beta_{2} ) q^{77} + 4 q^{79} + ( 8 + 11 \beta_{1} + 8 \beta_{3} ) q^{80} + 2 \beta_{3} q^{82} + ( 3 + 6 \beta_{2} ) q^{83} + ( -3 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{85} + ( -7 + 16 \beta_{2} ) q^{86} + ( -9 - 3 \beta_{1} - 9 \beta_{3} ) q^{88} + ( 13 - 5 \beta_{1} + 13 \beta_{3} ) q^{89} + ( 3 - \beta_{3} ) q^{91} + ( 12 - 6 \beta_{2} ) q^{92} + ( 1 + 3 \beta_{1} + \beta_{3} ) q^{94} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{95} + ( 3 \beta_{1} + 3 \beta_{2} + 14 \beta_{3} ) q^{97} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 3q^{2} - 3q^{4} - 6q^{5} - 2q^{7} - 12q^{8} + O(q^{10})$$ $$4q + 3q^{2} - 3q^{4} - 6q^{5} - 2q^{7} - 12q^{8} - 7q^{10} - 3q^{11} - 10q^{13} - 6q^{14} - 13q^{16} + 6q^{17} - 3q^{19} + 12q^{20} - 3q^{22} - 6q^{25} + 6q^{26} - 3q^{28} + 3q^{29} + 8q^{31} + 15q^{32} - 2q^{34} + 3q^{35} - 8q^{37} + 6q^{38} + 38q^{40} + 6q^{41} - 5q^{43} - 36q^{44} + 10q^{46} - 2q^{49} + 3q^{50} + 21q^{52} - 24q^{53} - 3q^{55} + 6q^{56} - 17q^{58} + 12q^{61} - 9q^{62} + 8q^{64} + 15q^{65} + 12q^{67} - 21q^{68} + 14q^{70} - 6q^{71} - 8q^{73} + 12q^{74} + 18q^{76} + 6q^{77} + 16q^{79} + 27q^{80} - 4q^{82} + q^{85} - 60q^{86} - 21q^{88} + 21q^{89} + 14q^{91} + 60q^{92} + 5q^{94} - 3q^{95} - 31q^{97} + 3q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 2 x^{2} + x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 1$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu^{2} - 2 \nu - 1$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{2} - 1$$

## Character values

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

 $$n$$ $$92$$ $$379$$ $$703$$ $$\chi(n)$$ $$1$$ $$\beta_{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
568.1
 −0.309017 + 0.535233i 0.809017 − 1.40126i −0.309017 − 0.535233i 0.809017 + 1.40126i
0.190983 0.330792i 0 0.927051 + 1.60570i −0.381966 0 −0.500000 0.866025i 1.47214 0 −0.0729490 + 0.126351i
568.2 1.30902 2.26728i 0 −2.42705 4.20378i −2.61803 0 −0.500000 0.866025i −7.47214 0 −3.42705 + 5.93583i
757.1 0.190983 + 0.330792i 0 0.927051 1.60570i −0.381966 0 −0.500000 + 0.866025i 1.47214 0 −0.0729490 0.126351i
757.2 1.30902 + 2.26728i 0 −2.42705 + 4.20378i −2.61803 0 −0.500000 + 0.866025i −7.47214 0 −3.42705 5.93583i
 $$n$$: 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
13.c 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.o.c 4
3.b odd 2 1 91.2.f.a 4
12.b even 2 1 1456.2.s.h 4
13.c even 3 1 inner 819.2.o.c 4
21.c even 2 1 637.2.f.c 4
21.g even 6 1 637.2.g.c 4
21.g even 6 1 637.2.h.f 4
21.h odd 6 1 637.2.g.b 4
21.h odd 6 1 637.2.h.g 4
39.h odd 6 1 1183.2.a.c 2
39.i odd 6 1 91.2.f.a 4
39.i odd 6 1 1183.2.a.g 2
39.k even 12 2 1183.2.c.c 4
156.p even 6 1 1456.2.s.h 4
273.r even 6 1 637.2.g.c 4
273.s odd 6 1 637.2.g.b 4
273.u even 6 1 8281.2.a.n 2
273.bf even 6 1 637.2.h.f 4
273.bm odd 6 1 637.2.h.g 4
273.bn even 6 1 637.2.f.c 4
273.bn even 6 1 8281.2.a.bb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.a 4 3.b odd 2 1
91.2.f.a 4 39.i odd 6 1
637.2.f.c 4 21.c even 2 1
637.2.f.c 4 273.bn even 6 1
637.2.g.b 4 21.h odd 6 1
637.2.g.b 4 273.s odd 6 1
637.2.g.c 4 21.g even 6 1
637.2.g.c 4 273.r even 6 1
637.2.h.f 4 21.g even 6 1
637.2.h.f 4 273.bf even 6 1
637.2.h.g 4 21.h odd 6 1
637.2.h.g 4 273.bm odd 6 1
819.2.o.c 4 1.a even 1 1 trivial
819.2.o.c 4 13.c even 3 1 inner
1183.2.a.c 2 39.h odd 6 1
1183.2.a.g 2 39.i odd 6 1
1183.2.c.c 4 39.k even 12 2
1456.2.s.h 4 12.b even 2 1
1456.2.s.h 4 156.p even 6 1
8281.2.a.n 2 273.u even 6 1
8281.2.a.bb 2 273.bn even 6 1

## Hecke kernels

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

 $$T_{2}^{4} - 3 T_{2}^{3} + 8 T_{2}^{2} - 3 T_{2} + 1$$ $$T_{11}^{4} + 3 T_{11}^{3} + 18 T_{11}^{2} - 27 T_{11} + 81$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 3 T + 8 T^{2} - 3 T^{3} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 1 + 3 T + T^{2} )^{2}$$
$7$ $$( 1 + T + T^{2} )^{2}$$
$11$ $$81 - 27 T + 18 T^{2} + 3 T^{3} + T^{4}$$
$13$ $$( 13 + 5 T + T^{2} )^{2}$$
$17$ $$121 + 66 T + 47 T^{2} - 6 T^{3} + T^{4}$$
$19$ $$81 - 27 T + 18 T^{2} + 3 T^{3} + T^{4}$$
$23$ $$400 + 20 T^{2} + T^{4}$$
$29$ $$841 + 87 T + 38 T^{2} - 3 T^{3} + T^{4}$$
$31$ $$( -41 - 4 T + T^{2} )^{2}$$
$37$ $$( 16 + 4 T + T^{2} )^{2}$$
$41$ $$16 - 24 T + 32 T^{2} - 6 T^{3} + T^{4}$$
$43$ $$9025 - 475 T + 120 T^{2} + 5 T^{3} + T^{4}$$
$47$ $$( -5 + T^{2} )^{2}$$
$53$ $$( 31 + 12 T + T^{2} )^{2}$$
$59$ $$25 + 5 T^{2} + T^{4}$$
$61$ $$( 36 - 6 T + T^{2} )^{2}$$
$67$ $$81 + 108 T + 153 T^{2} - 12 T^{3} + T^{4}$$
$71$ $$13456 - 696 T + 152 T^{2} + 6 T^{3} + T^{4}$$
$73$ $$( 2 + T )^{4}$$
$79$ $$( -4 + T )^{4}$$
$83$ $$( -45 + T^{2} )^{2}$$
$89$ $$6241 - 1659 T + 362 T^{2} - 21 T^{3} + T^{4}$$
$97$ $$52441 + 7099 T + 732 T^{2} + 31 T^{3} + T^{4}$$