# Properties

 Label 882.4.g.a Level $882$ Weight $4$ Character orbit 882.g Analytic conductor $52.040$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 882.g (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$52.0396846251$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 126) 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.

 $$f(q)$$ $$=$$ $$q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} -22 \zeta_{6} q^{5} + 8 q^{8} +O(q^{10})$$ $$q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} -22 \zeta_{6} q^{5} + 8 q^{8} + ( -44 + 44 \zeta_{6} ) q^{10} + ( 26 - 26 \zeta_{6} ) q^{11} + 54 q^{13} -16 \zeta_{6} q^{16} + ( -74 + 74 \zeta_{6} ) q^{17} + 116 \zeta_{6} q^{19} + 88 q^{20} -52 q^{22} -58 \zeta_{6} q^{23} + ( -359 + 359 \zeta_{6} ) q^{25} -108 \zeta_{6} q^{26} -208 q^{29} + ( -252 + 252 \zeta_{6} ) q^{31} + ( -32 + 32 \zeta_{6} ) q^{32} + 148 q^{34} -50 \zeta_{6} q^{37} + ( 232 - 232 \zeta_{6} ) q^{38} -176 \zeta_{6} q^{40} + 126 q^{41} + 164 q^{43} + 104 \zeta_{6} q^{44} + ( -116 + 116 \zeta_{6} ) q^{46} + 444 \zeta_{6} q^{47} + 718 q^{50} + ( -216 + 216 \zeta_{6} ) q^{52} + ( 12 - 12 \zeta_{6} ) q^{53} -572 q^{55} + 416 \zeta_{6} q^{58} + ( -124 + 124 \zeta_{6} ) q^{59} -162 \zeta_{6} q^{61} + 504 q^{62} + 64 q^{64} -1188 \zeta_{6} q^{65} + ( 860 - 860 \zeta_{6} ) q^{67} -296 \zeta_{6} q^{68} + 238 q^{71} + ( -146 + 146 \zeta_{6} ) q^{73} + ( -100 + 100 \zeta_{6} ) q^{74} -464 q^{76} + 984 \zeta_{6} q^{79} + ( -352 + 352 \zeta_{6} ) q^{80} -252 \zeta_{6} q^{82} + 656 q^{83} + 1628 q^{85} -328 \zeta_{6} q^{86} + ( 208 - 208 \zeta_{6} ) q^{88} + 954 \zeta_{6} q^{89} + 232 q^{92} + ( 888 - 888 \zeta_{6} ) q^{94} + ( 2552 - 2552 \zeta_{6} ) q^{95} -526 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 4q^{4} - 22q^{5} + 16q^{8} + O(q^{10})$$ $$2q - 2q^{2} - 4q^{4} - 22q^{5} + 16q^{8} - 44q^{10} + 26q^{11} + 108q^{13} - 16q^{16} - 74q^{17} + 116q^{19} + 176q^{20} - 104q^{22} - 58q^{23} - 359q^{25} - 108q^{26} - 416q^{29} - 252q^{31} - 32q^{32} + 296q^{34} - 50q^{37} + 232q^{38} - 176q^{40} + 252q^{41} + 328q^{43} + 104q^{44} - 116q^{46} + 444q^{47} + 1436q^{50} - 216q^{52} + 12q^{53} - 1144q^{55} + 416q^{58} - 124q^{59} - 162q^{61} + 1008q^{62} + 128q^{64} - 1188q^{65} + 860q^{67} - 296q^{68} + 476q^{71} - 146q^{73} - 100q^{74} - 928q^{76} + 984q^{79} - 352q^{80} - 252q^{82} + 1312q^{83} + 3256q^{85} - 328q^{86} + 208q^{88} + 954q^{89} + 464q^{92} + 888q^{94} + 2552q^{95} - 1052q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$785$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.00000 1.73205i 0 −2.00000 + 3.46410i −11.0000 19.0526i 0 0 8.00000 0 −22.0000 + 38.1051i
667.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i −11.0000 + 19.0526i 0 0 8.00000 0 −22.0000 38.1051i
 $$n$$: 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 882.4.g.a 2
3.b odd 2 1 882.4.g.x 2
7.b odd 2 1 882.4.g.m 2
7.c even 3 1 882.4.a.s 1
7.c even 3 1 inner 882.4.g.a 2
7.d odd 6 1 126.4.a.f yes 1
7.d odd 6 1 882.4.g.m 2
21.c even 2 1 882.4.g.n 2
21.g even 6 1 126.4.a.e 1
21.g even 6 1 882.4.g.n 2
21.h odd 6 1 882.4.a.a 1
21.h odd 6 1 882.4.g.x 2
28.f even 6 1 1008.4.a.a 1
84.j odd 6 1 1008.4.a.w 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.a.e 1 21.g even 6 1
126.4.a.f yes 1 7.d odd 6 1
882.4.a.a 1 21.h odd 6 1
882.4.a.s 1 7.c even 3 1
882.4.g.a 2 1.a even 1 1 trivial
882.4.g.a 2 7.c even 3 1 inner
882.4.g.m 2 7.b odd 2 1
882.4.g.m 2 7.d odd 6 1
882.4.g.n 2 21.c even 2 1
882.4.g.n 2 21.g even 6 1
882.4.g.x 2 3.b odd 2 1
882.4.g.x 2 21.h odd 6 1
1008.4.a.a 1 28.f even 6 1
1008.4.a.w 1 84.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{2} + 22 T_{5} + 484$$ $$T_{11}^{2} - 26 T_{11} + 676$$ $$T_{13} - 54$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$484 + 22 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$676 - 26 T + T^{2}$$
$13$ $$( -54 + T )^{2}$$
$17$ $$5476 + 74 T + T^{2}$$
$19$ $$13456 - 116 T + T^{2}$$
$23$ $$3364 + 58 T + T^{2}$$
$29$ $$( 208 + T )^{2}$$
$31$ $$63504 + 252 T + T^{2}$$
$37$ $$2500 + 50 T + T^{2}$$
$41$ $$( -126 + T )^{2}$$
$43$ $$( -164 + T )^{2}$$
$47$ $$197136 - 444 T + T^{2}$$
$53$ $$144 - 12 T + T^{2}$$
$59$ $$15376 + 124 T + T^{2}$$
$61$ $$26244 + 162 T + T^{2}$$
$67$ $$739600 - 860 T + T^{2}$$
$71$ $$( -238 + T )^{2}$$
$73$ $$21316 + 146 T + T^{2}$$
$79$ $$968256 - 984 T + T^{2}$$
$83$ $$( -656 + T )^{2}$$
$89$ $$910116 - 954 T + T^{2}$$
$97$ $$( 526 + T )^{2}$$