# Properties

 Label 819.2.do.d Level $819$ Weight $2$ Character orbit 819.do 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.do (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.53974792554$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-7})$$ Defining polynomial: $$x^{4} - x^{3} - x^{2} - 2 x + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 273) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{3} ) q^{2} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{4} + ( 1 + \beta_{2} ) q^{5} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{7} + ( -1 + 2 \beta_{2} ) q^{8} +O(q^{10})$$ $$q + ( -1 + \beta_{3} ) q^{2} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{4} + ( 1 + \beta_{2} ) q^{5} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{7} + ( -1 + 2 \beta_{2} ) q^{8} + ( -2 + \beta_{1} + \beta_{3} ) q^{10} + ( 2 - 4 \beta_{2} ) q^{11} + ( -3 + 4 \beta_{2} ) q^{13} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{14} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{16} + ( 1 - \beta_{2} ) q^{17} + ( -2 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{19} + ( 1 + \beta_{2} - 3 \beta_{3} ) q^{20} + ( 2 - 4 \beta_{1} + 2 \beta_{3} ) q^{22} + ( -2 + 4 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{23} -2 \beta_{2} q^{25} + ( -1 + 4 \beta_{1} - 3 \beta_{3} ) q^{26} + ( -4 + \beta_{1} - 3 \beta_{2} ) q^{28} + ( -7 + 7 \beta_{2} ) q^{29} + ( -4 + \beta_{2} + 2 \beta_{3} ) q^{31} + ( -4 + \beta_{1} - 3 \beta_{2} ) q^{32} + ( -\beta_{1} + \beta_{3} ) q^{34} + ( -1 + 2 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{35} + ( -2 - \beta_{2} + 4 \beta_{3} ) q^{37} + ( 2 - 4 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{38} + ( -3 + 3 \beta_{2} ) q^{40} + ( 1 - 4 \beta_{1} - 3 \beta_{2} ) q^{41} + ( -2 + 4 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{43} + ( -4 + 6 \beta_{1} + 2 \beta_{2} ) q^{44} + ( 9 - 7 \beta_{1} + 2 \beta_{2} ) q^{46} + ( -1 - 2 \beta_{1} - 3 \beta_{2} ) q^{47} -7 q^{49} + ( 2 - 2 \beta_{1} ) q^{50} + ( 4 - 7 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{52} + ( -4 + 8 \beta_{1} + 9 \beta_{2} - 4 \beta_{3} ) q^{53} + ( 6 - 6 \beta_{2} ) q^{55} + ( 1 - 2 \beta_{1} - 2 \beta_{3} ) q^{56} + ( 7 \beta_{1} - 7 \beta_{3} ) q^{58} + ( 3 + 2 \beta_{1} + 5 \beta_{2} ) q^{59} + ( 2 + 4 \beta_{1} + 4 \beta_{3} ) q^{61} + ( 5 + 3 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{62} + ( 9 - 2 \beta_{1} - 2 \beta_{3} ) q^{64} + ( -7 + 5 \beta_{2} ) q^{65} + ( -2 + 8 \beta_{1} + 4 \beta_{2} - 8 \beta_{3} ) q^{67} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{68} + ( -2 - 3 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{70} + ( -4 + 5 \beta_{2} - 6 \beta_{3} ) q^{71} + ( 10 - 5 \beta_{2} ) q^{73} + ( 7 + 3 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} ) q^{74} + ( -8 + 2 \beta_{1} - 6 \beta_{2} ) q^{76} + ( -2 + 4 \beta_{1} + 4 \beta_{3} ) q^{77} + ( 7 - 2 \beta_{1} - 9 \beta_{2} + 4 \beta_{3} ) q^{79} + ( 2 - 3 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} ) q^{80} + ( -6 + \beta_{1} + \beta_{3} ) q^{82} + ( -2 + 4 \beta_{2} ) q^{83} + ( 2 - \beta_{2} ) q^{85} + ( 5 - 3 \beta_{1} + 2 \beta_{2} ) q^{86} + 6 q^{88} + ( 18 - 9 \beta_{2} ) q^{89} + ( 3 - 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{91} + ( -15 + 5 \beta_{1} + 5 \beta_{3} ) q^{92} + ( -\beta_{1} - \beta_{3} ) q^{94} + ( -2 + 4 \beta_{1} + 6 \beta_{2} - 8 \beta_{3} ) q^{95} + ( 6 - \beta_{2} - 4 \beta_{3} ) q^{97} + ( 7 - 7 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 3q^{2} + q^{4} + 6q^{5} + O(q^{10})$$ $$4q - 3q^{2} + q^{4} + 6q^{5} - 6q^{10} - 4q^{13} + 7q^{14} - q^{16} + 2q^{17} + 3q^{20} + 6q^{22} - 8q^{23} - 4q^{25} - 3q^{26} - 21q^{28} - 14q^{29} - 12q^{31} - 21q^{32} - 6q^{37} + 14q^{38} - 6q^{40} - 6q^{41} - 6q^{44} + 33q^{46} - 12q^{47} - 28q^{49} + 6q^{50} + 5q^{52} + 6q^{53} + 12q^{55} + 24q^{59} + 16q^{61} + 13q^{62} + 32q^{64} - 18q^{65} - q^{68} - 12q^{71} + 30q^{73} + 17q^{74} - 42q^{76} + 12q^{79} - 22q^{82} + 6q^{85} + 21q^{86} + 24q^{88} + 54q^{89} - 50q^{92} - 2q^{94} + 18q^{97} + 21q^{98} + O(q^{100})$$

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

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

## Character values

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

 $$n$$ $$92$$ $$379$$ $$703$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$ $$-1 + \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 −0.895644 + 1.09445i 1.39564 − 0.228425i −0.895644 − 1.09445i 1.39564 + 0.228425i
−1.89564 1.09445i 0 1.39564 + 2.41733i 1.50000 0.866025i 0 2.64575i 1.73205i 0 −3.79129
361.2 0.395644 + 0.228425i 0 −0.895644 1.55130i 1.50000 0.866025i 0 2.64575i 1.73205i 0 0.791288
667.1 −1.89564 + 1.09445i 0 1.39564 2.41733i 1.50000 + 0.866025i 0 2.64575i 1.73205i 0 −3.79129
667.2 0.395644 0.228425i 0 −0.895644 + 1.55130i 1.50000 + 0.866025i 0 2.64575i 1.73205i 0 0.791288
 $$n$$: 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.u 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.do.d 4
3.b odd 2 1 273.2.bl.b yes 4
7.c even 3 1 819.2.bm.d 4
13.e even 6 1 819.2.bm.d 4
21.h odd 6 1 273.2.t.b 4
39.h odd 6 1 273.2.t.b 4
91.u even 6 1 inner 819.2.do.d 4
273.x odd 6 1 273.2.bl.b yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.t.b 4 21.h odd 6 1
273.2.t.b 4 39.h odd 6 1
273.2.bl.b yes 4 3.b odd 2 1
273.2.bl.b yes 4 273.x odd 6 1
819.2.bm.d 4 7.c even 3 1
819.2.bm.d 4 13.e even 6 1
819.2.do.d 4 1.a even 1 1 trivial
819.2.do.d 4 91.u even 6 1 inner

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 3 T + 2 T^{2} + 3 T^{3} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 3 - 3 T + T^{2} )^{2}$$
$7$ $$( 7 + T^{2} )^{2}$$
$11$ $$( 12 + T^{2} )^{2}$$
$13$ $$( 13 + 2 T + T^{2} )^{2}$$
$17$ $$( 1 - T + T^{2} )^{2}$$
$19$ $$( 28 + T^{2} )^{2}$$
$23$ $$25 - 40 T + 69 T^{2} + 8 T^{3} + T^{4}$$
$29$ $$( 49 + 7 T + T^{2} )^{2}$$
$31$ $$25 + 60 T + 53 T^{2} + 12 T^{3} + T^{4}$$
$37$ $$625 - 150 T - 13 T^{2} + 6 T^{3} + T^{4}$$
$41$ $$625 - 150 T - 13 T^{2} + 6 T^{3} + T^{4}$$
$43$ $$441 + 21 T^{2} + T^{4}$$
$47$ $$25 + 60 T + 53 T^{2} + 12 T^{3} + T^{4}$$
$53$ $$5625 + 450 T + 111 T^{2} - 6 T^{3} + T^{4}$$
$59$ $$1681 - 984 T + 233 T^{2} - 24 T^{3} + T^{4}$$
$61$ $$( -68 - 8 T + T^{2} )^{2}$$
$67$ $$10000 + 248 T^{2} + T^{4}$$
$71$ $$2601 - 612 T - 3 T^{2} + 12 T^{3} + T^{4}$$
$73$ $$( 75 - 15 T + T^{2} )^{2}$$
$79$ $$225 - 180 T + 129 T^{2} - 12 T^{3} + T^{4}$$
$83$ $$( 12 + T^{2} )^{2}$$
$89$ $$( 243 - 27 T + T^{2} )^{2}$$
$97$ $$1 + 18 T + 107 T^{2} - 18 T^{3} + T^{4}$$