# Properties

 Label 882.4.g.g 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: $$\Z[a_1, \ldots, a_{19}]$$ 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.

 $$f(q)$$ $$=$$ $$q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + 6 \zeta_{6} q^{5} + 8 q^{8} +O(q^{10})$$ $$q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + 6 \zeta_{6} q^{5} + 8 q^{8} + ( 12 - 12 \zeta_{6} ) q^{10} + ( -30 + 30 \zeta_{6} ) q^{11} -53 q^{13} -16 \zeta_{6} q^{16} + ( 84 - 84 \zeta_{6} ) q^{17} -97 \zeta_{6} q^{19} -24 q^{20} + 60 q^{22} + 84 \zeta_{6} q^{23} + ( 89 - 89 \zeta_{6} ) q^{25} + 106 \zeta_{6} q^{26} + 180 q^{29} + ( 179 - 179 \zeta_{6} ) q^{31} + ( -32 + 32 \zeta_{6} ) q^{32} -168 q^{34} + 145 \zeta_{6} q^{37} + ( -194 + 194 \zeta_{6} ) q^{38} + 48 \zeta_{6} q^{40} + 126 q^{41} -325 q^{43} -120 \zeta_{6} q^{44} + ( 168 - 168 \zeta_{6} ) q^{46} + 366 \zeta_{6} q^{47} -178 q^{50} + ( 212 - 212 \zeta_{6} ) q^{52} + ( -768 + 768 \zeta_{6} ) q^{53} -180 q^{55} -360 \zeta_{6} q^{58} + ( 264 - 264 \zeta_{6} ) q^{59} + 818 \zeta_{6} q^{61} -358 q^{62} + 64 q^{64} -318 \zeta_{6} q^{65} + ( 523 - 523 \zeta_{6} ) q^{67} + 336 \zeta_{6} q^{68} + 342 q^{71} + ( -43 + 43 \zeta_{6} ) q^{73} + ( 290 - 290 \zeta_{6} ) q^{74} + 388 q^{76} + 1171 \zeta_{6} q^{79} + ( 96 - 96 \zeta_{6} ) q^{80} -252 \zeta_{6} q^{82} -810 q^{83} + 504 q^{85} + 650 \zeta_{6} q^{86} + ( -240 + 240 \zeta_{6} ) q^{88} + 600 \zeta_{6} q^{89} -336 q^{92} + ( 732 - 732 \zeta_{6} ) q^{94} + ( 582 - 582 \zeta_{6} ) q^{95} -386 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 4q^{4} + 6q^{5} + 16q^{8} + O(q^{10})$$ $$2q - 2q^{2} - 4q^{4} + 6q^{5} + 16q^{8} + 12q^{10} - 30q^{11} - 106q^{13} - 16q^{16} + 84q^{17} - 97q^{19} - 48q^{20} + 120q^{22} + 84q^{23} + 89q^{25} + 106q^{26} + 360q^{29} + 179q^{31} - 32q^{32} - 336q^{34} + 145q^{37} - 194q^{38} + 48q^{40} + 252q^{41} - 650q^{43} - 120q^{44} + 168q^{46} + 366q^{47} - 356q^{50} + 212q^{52} - 768q^{53} - 360q^{55} - 360q^{58} + 264q^{59} + 818q^{61} - 716q^{62} + 128q^{64} - 318q^{65} + 523q^{67} + 336q^{68} + 684q^{71} - 43q^{73} + 290q^{74} + 776q^{76} + 1171q^{79} + 96q^{80} - 252q^{82} - 1620q^{83} + 1008q^{85} + 650q^{86} - 240q^{88} + 600q^{89} - 672q^{92} + 732q^{94} + 582q^{95} - 772q^{97} + O(q^{100})$$

## 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 3.00000 + 5.19615i 0 0 8.00000 0 6.00000 10.3923i
667.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i 3.00000 5.19615i 0 0 8.00000 0 6.00000 + 10.3923i
 $$n$$: 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.g 2
3.b odd 2 1 294.4.e.i 2
7.b odd 2 1 126.4.g.b 2
7.c even 3 1 882.4.a.l 1
7.c even 3 1 inner 882.4.g.g 2
7.d odd 6 1 126.4.g.b 2
7.d odd 6 1 882.4.a.o 1
21.c even 2 1 42.4.e.a 2
21.g even 6 1 42.4.e.a 2
21.g even 6 1 294.4.a.d 1
21.h odd 6 1 294.4.a.c 1
21.h odd 6 1 294.4.e.i 2
84.h odd 2 1 336.4.q.f 2
84.j odd 6 1 336.4.q.f 2
84.j odd 6 1 2352.4.a.f 1
84.n even 6 1 2352.4.a.bf 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.a 2 21.c even 2 1
42.4.e.a 2 21.g even 6 1
126.4.g.b 2 7.b odd 2 1
126.4.g.b 2 7.d odd 6 1
294.4.a.c 1 21.h odd 6 1
294.4.a.d 1 21.g even 6 1
294.4.e.i 2 3.b odd 2 1
294.4.e.i 2 21.h odd 6 1
336.4.q.f 2 84.h odd 2 1
336.4.q.f 2 84.j odd 6 1
882.4.a.l 1 7.c even 3 1
882.4.a.o 1 7.d odd 6 1
882.4.g.g 2 1.a even 1 1 trivial
882.4.g.g 2 7.c even 3 1 inner
2352.4.a.f 1 84.j odd 6 1
2352.4.a.bf 1 84.n even 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 6 T_{5} + 36$$ $$T_{11}^{2} + 30 T_{11} + 900$$ $$T_{13} + 53$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$36 - 6 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$900 + 30 T + T^{2}$$
$13$ $$( 53 + T )^{2}$$
$17$ $$7056 - 84 T + T^{2}$$
$19$ $$9409 + 97 T + T^{2}$$
$23$ $$7056 - 84 T + T^{2}$$
$29$ $$( -180 + T )^{2}$$
$31$ $$32041 - 179 T + T^{2}$$
$37$ $$21025 - 145 T + T^{2}$$
$41$ $$( -126 + T )^{2}$$
$43$ $$( 325 + T )^{2}$$
$47$ $$133956 - 366 T + T^{2}$$
$53$ $$589824 + 768 T + T^{2}$$
$59$ $$69696 - 264 T + T^{2}$$
$61$ $$669124 - 818 T + T^{2}$$
$67$ $$273529 - 523 T + T^{2}$$
$71$ $$( -342 + T )^{2}$$
$73$ $$1849 + 43 T + T^{2}$$
$79$ $$1371241 - 1171 T + T^{2}$$
$83$ $$( 810 + T )^{2}$$
$89$ $$360000 - 600 T + T^{2}$$
$97$ $$( 386 + T )^{2}$$