# Properties

 Label 819.2.j.c Level $819$ Weight $2$ Character orbit 819.j Analytic conductor $6.540$ Analytic rank $1$ 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.j (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.53974792554$$ Analytic rank: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{1} - \beta_{3} ) q^{2} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{4} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{5} + ( -1 + 2 \beta_{3} ) q^{7} + ( 1 - 4 \beta_{2} ) q^{8} +O(q^{10})$$ $$q + ( -1 - \beta_{1} - \beta_{3} ) q^{2} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{4} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{5} + ( -1 + 2 \beta_{3} ) q^{7} + ( 1 - 4 \beta_{2} ) q^{8} + ( -3 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{10} + 3 \beta_{3} q^{11} - q^{13} + ( 3 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{14} + ( -5 - 3 \beta_{1} - 5 \beta_{3} ) q^{16} + ( -4 \beta_{1} - 4 \beta_{2} + 5 \beta_{3} ) q^{17} + ( -3 - 3 \beta_{3} ) q^{19} + ( -6 + 3 \beta_{2} ) q^{20} + ( 3 - 3 \beta_{2} ) q^{22} + ( -5 - 2 \beta_{1} - 5 \beta_{3} ) q^{23} + ( 1 + \beta_{1} + \beta_{3} ) q^{26} + ( -9 \beta_{1} - 3 \beta_{2} ) q^{28} + ( 2 + 4 \beta_{2} ) q^{29} + 5 \beta_{3} q^{31} + ( 3 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} ) q^{32} + ( 1 + 3 \beta_{2} ) q^{34} + ( 3 - 2 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{35} + ( 5 - 6 \beta_{1} + 5 \beta_{3} ) q^{37} + ( 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{38} + ( 7 + 6 \beta_{1} + 7 \beta_{3} ) q^{40} + ( -2 - 4 \beta_{2} ) q^{41} -8 q^{43} -9 \beta_{1} q^{44} + ( 9 \beta_{1} + 9 \beta_{2} + 7 \beta_{3} ) q^{46} + ( -1 - 4 \beta_{1} - \beta_{3} ) q^{47} + ( -3 - 8 \beta_{3} ) q^{49} + ( -3 \beta_{1} - 3 \beta_{2} ) q^{52} + ( 4 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{53} + ( 3 + 6 \beta_{2} ) q^{55} + ( -1 + 8 \beta_{1} + 12 \beta_{2} + 2 \beta_{3} ) q^{56} + ( 2 + 6 \beta_{1} + 2 \beta_{3} ) q^{58} + ( 4 \beta_{1} + 4 \beta_{2} - 5 \beta_{3} ) q^{59} + ( -3 - 3 \beta_{3} ) q^{61} + ( 5 - 5 \beta_{2} ) q^{62} + ( -1 - 6 \beta_{2} ) q^{64} + ( 1 - 2 \beta_{1} + \beta_{3} ) q^{65} -3 \beta_{3} q^{67} + ( 12 - 3 \beta_{1} + 12 \beta_{3} ) q^{68} + ( 2 + 9 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{70} + ( -4 - 8 \beta_{2} ) q^{71} + ( -6 \beta_{1} - 6 \beta_{2} + 7 \beta_{3} ) q^{73} + ( 7 \beta_{1} + 7 \beta_{2} + \beta_{3} ) q^{74} -9 \beta_{2} q^{76} + ( -6 - 9 \beta_{3} ) q^{77} + ( -7 + 6 \beta_{1} - 7 \beta_{3} ) q^{79} + ( -13 \beta_{1} - 13 \beta_{2} - \beta_{3} ) q^{80} + ( -2 - 6 \beta_{1} - 2 \beta_{3} ) q^{82} + ( 13 + 6 \beta_{2} ) q^{85} + ( 8 + 8 \beta_{1} + 8 \beta_{3} ) q^{86} + ( 12 \beta_{1} + 12 \beta_{2} + 3 \beta_{3} ) q^{88} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{89} + ( 1 - 2 \beta_{3} ) q^{91} + ( 6 - 21 \beta_{2} ) q^{92} + ( 9 \beta_{1} + 9 \beta_{2} + 5 \beta_{3} ) q^{94} + ( -6 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{95} + ( -10 - 12 \beta_{2} ) q^{97} + ( -5 + 3 \beta_{1} + 8 \beta_{2} + 3 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 3 q^{2} - 3 q^{4} - 8 q^{7} + 12 q^{8} + O(q^{10})$$ $$4 q - 3 q^{2} - 3 q^{4} - 8 q^{7} + 12 q^{8} + 5 q^{10} - 6 q^{11} - 4 q^{13} + 15 q^{14} - 13 q^{16} - 6 q^{17} - 6 q^{19} - 30 q^{20} + 18 q^{22} - 12 q^{23} + 3 q^{26} - 3 q^{28} - 10 q^{31} - 15 q^{32} - 2 q^{34} + 4 q^{37} - 9 q^{38} + 20 q^{40} - 32 q^{43} - 9 q^{44} - 23 q^{46} - 6 q^{47} + 4 q^{49} + 3 q^{52} - 6 q^{53} - 24 q^{56} + 10 q^{58} + 6 q^{59} - 6 q^{61} + 30 q^{62} + 8 q^{64} + 6 q^{67} + 21 q^{68} + 5 q^{70} - 8 q^{73} - 9 q^{74} + 18 q^{76} - 6 q^{77} - 8 q^{79} + 15 q^{80} - 10 q^{82} + 40 q^{85} + 24 q^{86} - 18 q^{88} + 8 q^{91} + 66 q^{92} - 19 q^{94} - 16 q^{97} - 39 q^{98} + O(q^{100})$$

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 $$\left(\mathbb{Z}/819\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
235.1
 0.809017 + 1.40126i −0.309017 − 0.535233i 0.809017 − 1.40126i −0.309017 + 0.535233i
−1.30902 2.26728i 0 −2.42705 + 4.20378i 1.11803 + 1.93649i 0 −2.00000 + 1.73205i 7.47214 0 2.92705 5.06980i
235.2 −0.190983 0.330792i 0 0.927051 1.60570i −1.11803 1.93649i 0 −2.00000 + 1.73205i −1.47214 0 −0.427051 + 0.739674i
352.1 −1.30902 + 2.26728i 0 −2.42705 4.20378i 1.11803 1.93649i 0 −2.00000 1.73205i 7.47214 0 2.92705 + 5.06980i
352.2 −0.190983 + 0.330792i 0 0.927051 + 1.60570i −1.11803 + 1.93649i 0 −2.00000 1.73205i −1.47214 0 −0.427051 0.739674i
 $$n$$: 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
7.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.j.c 4
3.b odd 2 1 91.2.e.b 4
7.c even 3 1 inner 819.2.j.c 4
7.c even 3 1 5733.2.a.v 2
7.d odd 6 1 5733.2.a.w 2
12.b even 2 1 1456.2.r.j 4
21.c even 2 1 637.2.e.h 4
21.g even 6 1 637.2.a.e 2
21.g even 6 1 637.2.e.h 4
21.h odd 6 1 91.2.e.b 4
21.h odd 6 1 637.2.a.f 2
39.d odd 2 1 1183.2.e.d 4
84.n even 6 1 1456.2.r.j 4
273.w odd 6 1 1183.2.e.d 4
273.w odd 6 1 8281.2.a.z 2
273.ba even 6 1 8281.2.a.ba 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.b 4 3.b odd 2 1
91.2.e.b 4 21.h odd 6 1
637.2.a.e 2 21.g even 6 1
637.2.a.f 2 21.h odd 6 1
637.2.e.h 4 21.c even 2 1
637.2.e.h 4 21.g even 6 1
819.2.j.c 4 1.a even 1 1 trivial
819.2.j.c 4 7.c even 3 1 inner
1183.2.e.d 4 39.d odd 2 1
1183.2.e.d 4 273.w odd 6 1
1456.2.r.j 4 12.b even 2 1
1456.2.r.j 4 84.n even 6 1
5733.2.a.v 2 7.c even 3 1
5733.2.a.w 2 7.d odd 6 1
8281.2.a.z 2 273.w odd 6 1
8281.2.a.ba 2 273.ba even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 3 T_{2}^{3} + 8 T_{2}^{2} + 3 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(819, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T + 8 T^{2} + 3 T^{3} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$25 + 5 T^{2} + T^{4}$$
$7$ $$( 7 + 4 T + T^{2} )^{2}$$
$11$ $$( 9 + 3 T + T^{2} )^{2}$$
$13$ $$( 1 + T )^{4}$$
$17$ $$121 - 66 T + 47 T^{2} + 6 T^{3} + T^{4}$$
$19$ $$( 9 + 3 T + T^{2} )^{2}$$
$23$ $$961 + 372 T + 113 T^{2} + 12 T^{3} + T^{4}$$
$29$ $$( -20 + T^{2} )^{2}$$
$31$ $$( 25 + 5 T + T^{2} )^{2}$$
$37$ $$1681 + 164 T + 57 T^{2} - 4 T^{3} + T^{4}$$
$41$ $$( -20 + T^{2} )^{2}$$
$43$ $$( 8 + T )^{4}$$
$47$ $$121 - 66 T + 47 T^{2} + 6 T^{3} + T^{4}$$
$53$ $$121 - 66 T + 47 T^{2} + 6 T^{3} + T^{4}$$
$59$ $$121 + 66 T + 47 T^{2} - 6 T^{3} + T^{4}$$
$61$ $$( 9 + 3 T + T^{2} )^{2}$$
$67$ $$( 9 - 3 T + T^{2} )^{2}$$
$71$ $$( -80 + T^{2} )^{2}$$
$73$ $$841 - 232 T + 93 T^{2} + 8 T^{3} + T^{4}$$
$79$ $$841 - 232 T + 93 T^{2} + 8 T^{3} + T^{4}$$
$83$ $$T^{4}$$
$89$ $$25 + 5 T^{2} + T^{4}$$
$97$ $$( -164 + 8 T + T^{2} )^{2}$$