# Properties

 Label 882.4.g.j 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 294) 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.

 $$f(q)$$ $$=$$ $$q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + 8 \zeta_{6} q^{5} + 8 q^{8} +O(q^{10})$$ $$q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + 8 \zeta_{6} q^{5} + 8 q^{8} + ( 16 - 16 \zeta_{6} ) q^{10} + ( 40 - 40 \zeta_{6} ) q^{11} + 4 q^{13} -16 \zeta_{6} q^{16} + ( -84 + 84 \zeta_{6} ) q^{17} -148 \zeta_{6} q^{19} -32 q^{20} -80 q^{22} + 84 \zeta_{6} q^{23} + ( 61 - 61 \zeta_{6} ) q^{25} -8 \zeta_{6} q^{26} -58 q^{29} + ( 136 - 136 \zeta_{6} ) q^{31} + ( -32 + 32 \zeta_{6} ) q^{32} + 168 q^{34} + 222 \zeta_{6} q^{37} + ( -296 + 296 \zeta_{6} ) q^{38} + 64 \zeta_{6} q^{40} -420 q^{41} -164 q^{43} + 160 \zeta_{6} q^{44} + ( 168 - 168 \zeta_{6} ) q^{46} + 488 \zeta_{6} q^{47} -122 q^{50} + ( -16 + 16 \zeta_{6} ) q^{52} + ( 478 - 478 \zeta_{6} ) q^{53} + 320 q^{55} + 116 \zeta_{6} q^{58} + ( 548 - 548 \zeta_{6} ) q^{59} -692 \zeta_{6} q^{61} -272 q^{62} + 64 q^{64} + 32 \zeta_{6} q^{65} + ( 908 - 908 \zeta_{6} ) q^{67} -336 \zeta_{6} q^{68} + 524 q^{71} + ( -440 + 440 \zeta_{6} ) q^{73} + ( 444 - 444 \zeta_{6} ) q^{74} + 592 q^{76} -1216 \zeta_{6} q^{79} + ( 128 - 128 \zeta_{6} ) q^{80} + 840 \zeta_{6} q^{82} + 684 q^{83} -672 q^{85} + 328 \zeta_{6} q^{86} + ( 320 - 320 \zeta_{6} ) q^{88} + 604 \zeta_{6} q^{89} -336 q^{92} + ( 976 - 976 \zeta_{6} ) q^{94} + ( 1184 - 1184 \zeta_{6} ) q^{95} -832 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 4q^{4} + 8q^{5} + 16q^{8} + O(q^{10})$$ $$2q - 2q^{2} - 4q^{4} + 8q^{5} + 16q^{8} + 16q^{10} + 40q^{11} + 8q^{13} - 16q^{16} - 84q^{17} - 148q^{19} - 64q^{20} - 160q^{22} + 84q^{23} + 61q^{25} - 8q^{26} - 116q^{29} + 136q^{31} - 32q^{32} + 336q^{34} + 222q^{37} - 296q^{38} + 64q^{40} - 840q^{41} - 328q^{43} + 160q^{44} + 168q^{46} + 488q^{47} - 244q^{50} - 16q^{52} + 478q^{53} + 640q^{55} + 116q^{58} + 548q^{59} - 692q^{61} - 544q^{62} + 128q^{64} + 32q^{65} + 908q^{67} - 336q^{68} + 1048q^{71} - 440q^{73} + 444q^{74} + 1184q^{76} - 1216q^{79} + 128q^{80} + 840q^{82} + 1368q^{83} - 1344q^{85} + 328q^{86} + 320q^{88} + 604q^{89} - 672q^{92} + 976q^{94} + 1184q^{95} - 1664q^{97} + O(q^{100})$$

## 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 4.00000 + 6.92820i 0 0 8.00000 0 8.00000 13.8564i
667.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i 4.00000 6.92820i 0 0 8.00000 0 8.00000 + 13.8564i
 $$n$$: 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.j 2
3.b odd 2 1 294.4.e.f 2
7.b odd 2 1 882.4.g.c 2
7.c even 3 1 882.4.a.j 1
7.c even 3 1 inner 882.4.g.j 2
7.d odd 6 1 882.4.a.q 1
7.d odd 6 1 882.4.g.c 2
21.c even 2 1 294.4.e.j 2
21.g even 6 1 294.4.a.b 1
21.g even 6 1 294.4.e.j 2
21.h odd 6 1 294.4.a.f yes 1
21.h odd 6 1 294.4.e.f 2
84.j odd 6 1 2352.4.a.z 1
84.n even 6 1 2352.4.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.4.a.b 1 21.g even 6 1
294.4.a.f yes 1 21.h odd 6 1
294.4.e.f 2 3.b odd 2 1
294.4.e.f 2 21.h odd 6 1
294.4.e.j 2 21.c even 2 1
294.4.e.j 2 21.g even 6 1
882.4.a.j 1 7.c even 3 1
882.4.a.q 1 7.d odd 6 1
882.4.g.c 2 7.b odd 2 1
882.4.g.c 2 7.d odd 6 1
882.4.g.j 2 1.a even 1 1 trivial
882.4.g.j 2 7.c even 3 1 inner
2352.4.a.m 1 84.n even 6 1
2352.4.a.z 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 $$S_{4}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{2} - 8 T_{5} + 64$$ $$T_{11}^{2} - 40 T_{11} + 1600$$ $$T_{13} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$64 - 8 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$1600 - 40 T + T^{2}$$
$13$ $$( -4 + T )^{2}$$
$17$ $$7056 + 84 T + T^{2}$$
$19$ $$21904 + 148 T + T^{2}$$
$23$ $$7056 - 84 T + T^{2}$$
$29$ $$( 58 + T )^{2}$$
$31$ $$18496 - 136 T + T^{2}$$
$37$ $$49284 - 222 T + T^{2}$$
$41$ $$( 420 + T )^{2}$$
$43$ $$( 164 + T )^{2}$$
$47$ $$238144 - 488 T + T^{2}$$
$53$ $$228484 - 478 T + T^{2}$$
$59$ $$300304 - 548 T + T^{2}$$
$61$ $$478864 + 692 T + T^{2}$$
$67$ $$824464 - 908 T + T^{2}$$
$71$ $$( -524 + T )^{2}$$
$73$ $$193600 + 440 T + T^{2}$$
$79$ $$1478656 + 1216 T + T^{2}$$
$83$ $$( -684 + T )^{2}$$
$89$ $$364816 - 604 T + T^{2}$$
$97$ $$( 832 + T )^{2}$$