# Properties

 Label 882.4.g.w 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 42) 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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + 18 \zeta_{6} q^{5} -8 q^{8} +O(q^{10})$$ $$q + 2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + 18 \zeta_{6} q^{5} -8 q^{8} + ( -36 + 36 \zeta_{6} ) q^{10} + ( -72 + 72 \zeta_{6} ) q^{11} -34 q^{13} -16 \zeta_{6} q^{16} + ( 6 - 6 \zeta_{6} ) q^{17} -92 \zeta_{6} q^{19} -72 q^{20} -144 q^{22} -180 \zeta_{6} q^{23} + ( -199 + 199 \zeta_{6} ) q^{25} -68 \zeta_{6} q^{26} + 114 q^{29} + ( -56 + 56 \zeta_{6} ) q^{31} + ( 32 - 32 \zeta_{6} ) q^{32} + 12 q^{34} + 34 \zeta_{6} q^{37} + ( 184 - 184 \zeta_{6} ) q^{38} -144 \zeta_{6} q^{40} -6 q^{41} + 164 q^{43} -288 \zeta_{6} q^{44} + ( 360 - 360 \zeta_{6} ) q^{46} + 168 \zeta_{6} q^{47} -398 q^{50} + ( 136 - 136 \zeta_{6} ) q^{52} + ( 654 - 654 \zeta_{6} ) q^{53} -1296 q^{55} + 228 \zeta_{6} q^{58} + ( -492 + 492 \zeta_{6} ) q^{59} + 250 \zeta_{6} q^{61} -112 q^{62} + 64 q^{64} -612 \zeta_{6} q^{65} + ( 124 - 124 \zeta_{6} ) q^{67} + 24 \zeta_{6} q^{68} -36 q^{71} + ( -1010 + 1010 \zeta_{6} ) q^{73} + ( -68 + 68 \zeta_{6} ) q^{74} + 368 q^{76} -56 \zeta_{6} q^{79} + ( 288 - 288 \zeta_{6} ) q^{80} -12 \zeta_{6} q^{82} -228 q^{83} + 108 q^{85} + 328 \zeta_{6} q^{86} + ( 576 - 576 \zeta_{6} ) q^{88} + 390 \zeta_{6} q^{89} + 720 q^{92} + ( -336 + 336 \zeta_{6} ) q^{94} + ( 1656 - 1656 \zeta_{6} ) q^{95} -70 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 4q^{4} + 18q^{5} - 16q^{8} + O(q^{10})$$ $$2q + 2q^{2} - 4q^{4} + 18q^{5} - 16q^{8} - 36q^{10} - 72q^{11} - 68q^{13} - 16q^{16} + 6q^{17} - 92q^{19} - 144q^{20} - 288q^{22} - 180q^{23} - 199q^{25} - 68q^{26} + 228q^{29} - 56q^{31} + 32q^{32} + 24q^{34} + 34q^{37} + 184q^{38} - 144q^{40} - 12q^{41} + 328q^{43} - 288q^{44} + 360q^{46} + 168q^{47} - 796q^{50} + 136q^{52} + 654q^{53} - 2592q^{55} + 228q^{58} - 492q^{59} + 250q^{61} - 224q^{62} + 128q^{64} - 612q^{65} + 124q^{67} + 24q^{68} - 72q^{71} - 1010q^{73} - 68q^{74} + 736q^{76} - 56q^{79} + 288q^{80} - 12q^{82} - 456q^{83} + 216q^{85} + 328q^{86} + 576q^{88} + 390q^{89} + 1440q^{92} - 336q^{94} + 1656q^{95} - 140q^{97} + O(q^{100})$$

## 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 9.00000 + 15.5885i 0 0 −8.00000 0 −18.0000 + 31.1769i
667.1 1.00000 1.73205i 0 −2.00000 3.46410i 9.00000 15.5885i 0 0 −8.00000 0 −18.0000 31.1769i
 $$n$$: 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.w 2
3.b odd 2 1 294.4.e.c 2
7.b odd 2 1 882.4.g.o 2
7.c even 3 1 126.4.a.a 1
7.c even 3 1 inner 882.4.g.w 2
7.d odd 6 1 882.4.a.g 1
7.d odd 6 1 882.4.g.o 2
21.c even 2 1 294.4.e.b 2
21.g even 6 1 294.4.a.i 1
21.g even 6 1 294.4.e.b 2
21.h odd 6 1 42.4.a.a 1
21.h odd 6 1 294.4.e.c 2
28.g odd 6 1 1008.4.a.b 1
84.j odd 6 1 2352.4.a.a 1
84.n even 6 1 336.4.a.l 1
105.o odd 6 1 1050.4.a.g 1
105.x even 12 2 1050.4.g.a 2
168.s odd 6 1 1344.4.a.o 1
168.v even 6 1 1344.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.a.a 1 21.h odd 6 1
126.4.a.a 1 7.c even 3 1
294.4.a.i 1 21.g even 6 1
294.4.e.b 2 21.c even 2 1
294.4.e.b 2 21.g even 6 1
294.4.e.c 2 3.b odd 2 1
294.4.e.c 2 21.h odd 6 1
336.4.a.l 1 84.n even 6 1
882.4.a.g 1 7.d odd 6 1
882.4.g.o 2 7.b odd 2 1
882.4.g.o 2 7.d odd 6 1
882.4.g.w 2 1.a even 1 1 trivial
882.4.g.w 2 7.c even 3 1 inner
1008.4.a.b 1 28.g odd 6 1
1050.4.a.g 1 105.o odd 6 1
1050.4.g.a 2 105.x even 12 2
1344.4.a.a 1 168.v even 6 1
1344.4.a.o 1 168.s odd 6 1
2352.4.a.a 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 $$S_{4}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{2} - 18 T_{5} + 324$$ $$T_{11}^{2} + 72 T_{11} + 5184$$ $$T_{13} + 34$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 2 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$324 - 18 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$5184 + 72 T + T^{2}$$
$13$ $$( 34 + T )^{2}$$
$17$ $$36 - 6 T + T^{2}$$
$19$ $$8464 + 92 T + T^{2}$$
$23$ $$32400 + 180 T + T^{2}$$
$29$ $$( -114 + T )^{2}$$
$31$ $$3136 + 56 T + T^{2}$$
$37$ $$1156 - 34 T + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$( -164 + T )^{2}$$
$47$ $$28224 - 168 T + T^{2}$$
$53$ $$427716 - 654 T + T^{2}$$
$59$ $$242064 + 492 T + T^{2}$$
$61$ $$62500 - 250 T + T^{2}$$
$67$ $$15376 - 124 T + T^{2}$$
$71$ $$( 36 + T )^{2}$$
$73$ $$1020100 + 1010 T + T^{2}$$
$79$ $$3136 + 56 T + T^{2}$$
$83$ $$( 228 + T )^{2}$$
$89$ $$152100 - 390 T + T^{2}$$
$97$ $$( 70 + T )^{2}$$