# Properties

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

# Learn more

## 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.

 $$f(q)$$ $$=$$ $$q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} -6 \zeta_{6} q^{5} + 8 q^{8} +O(q^{10})$$ $$q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} -6 \zeta_{6} q^{5} + 8 q^{8} + ( -12 + 12 \zeta_{6} ) q^{10} + ( -30 + 30 \zeta_{6} ) q^{11} + 2 q^{13} -16 \zeta_{6} q^{16} + ( -66 + 66 \zeta_{6} ) q^{17} + 52 \zeta_{6} q^{19} + 24 q^{20} + 60 q^{22} -114 \zeta_{6} q^{23} + ( 89 - 89 \zeta_{6} ) q^{25} -4 \zeta_{6} q^{26} + 72 q^{29} + ( 196 - 196 \zeta_{6} ) q^{31} + ( -32 + 32 \zeta_{6} ) q^{32} + 132 q^{34} + 286 \zeta_{6} q^{37} + ( 104 - 104 \zeta_{6} ) q^{38} -48 \zeta_{6} q^{40} -378 q^{41} + 164 q^{43} -120 \zeta_{6} q^{44} + ( -228 + 228 \zeta_{6} ) q^{46} + 228 \zeta_{6} q^{47} -178 q^{50} + ( -8 + 8 \zeta_{6} ) q^{52} + ( 348 - 348 \zeta_{6} ) q^{53} + 180 q^{55} -144 \zeta_{6} q^{58} + ( 348 - 348 \zeta_{6} ) q^{59} + 106 \zeta_{6} q^{61} -392 q^{62} + 64 q^{64} -12 \zeta_{6} q^{65} + ( -596 + 596 \zeta_{6} ) q^{67} -264 \zeta_{6} q^{68} + 630 q^{71} + ( 1042 - 1042 \zeta_{6} ) q^{73} + ( 572 - 572 \zeta_{6} ) q^{74} -208 q^{76} + 88 \zeta_{6} q^{79} + ( -96 + 96 \zeta_{6} ) q^{80} + 756 \zeta_{6} q^{82} -1440 q^{83} + 396 q^{85} -328 \zeta_{6} q^{86} + ( -240 + 240 \zeta_{6} ) q^{88} -1374 \zeta_{6} q^{89} + 456 q^{92} + ( 456 - 456 \zeta_{6} ) q^{94} + ( 312 - 312 \zeta_{6} ) q^{95} -34 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 4q^{4} - 6q^{5} + 16q^{8} + O(q^{10})$$ $$2q - 2q^{2} - 4q^{4} - 6q^{5} + 16q^{8} - 12q^{10} - 30q^{11} + 4q^{13} - 16q^{16} - 66q^{17} + 52q^{19} + 48q^{20} + 120q^{22} - 114q^{23} + 89q^{25} - 4q^{26} + 144q^{29} + 196q^{31} - 32q^{32} + 264q^{34} + 286q^{37} + 104q^{38} - 48q^{40} - 756q^{41} + 328q^{43} - 120q^{44} - 228q^{46} + 228q^{47} - 356q^{50} - 8q^{52} + 348q^{53} + 360q^{55} - 144q^{58} + 348q^{59} + 106q^{61} - 784q^{62} + 128q^{64} - 12q^{65} - 596q^{67} - 264q^{68} + 1260q^{71} + 1042q^{73} + 572q^{74} - 416q^{76} + 88q^{79} - 96q^{80} + 756q^{82} - 2880q^{83} + 792q^{85} - 328q^{86} - 240q^{88} - 1374q^{89} + 912q^{92} + 456q^{94} + 312q^{95} - 68q^{97} + O(q^{100})$$

## 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.e 2
3.b odd 2 1 882.4.g.t 2
7.b odd 2 1 882.4.g.h 2
7.c even 3 1 126.4.a.g yes 1
7.c even 3 1 inner 882.4.g.e 2
7.d odd 6 1 882.4.a.m 1
7.d odd 6 1 882.4.g.h 2
21.c even 2 1 882.4.g.q 2
21.g even 6 1 882.4.a.e 1
21.g even 6 1 882.4.g.q 2
21.h odd 6 1 126.4.a.b 1
21.h odd 6 1 882.4.g.t 2
28.g odd 6 1 1008.4.a.n 1
84.n even 6 1 1008.4.a.g 1

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$36 + 6 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$900 + 30 T + T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$4356 + 66 T + T^{2}$$
$19$ $$2704 - 52 T + T^{2}$$
$23$ $$12996 + 114 T + T^{2}$$
$29$ $$( -72 + T )^{2}$$
$31$ $$38416 - 196 T + T^{2}$$
$37$ $$81796 - 286 T + T^{2}$$
$41$ $$( 378 + T )^{2}$$
$43$ $$( -164 + T )^{2}$$
$47$ $$51984 - 228 T + T^{2}$$
$53$ $$121104 - 348 T + T^{2}$$
$59$ $$121104 - 348 T + T^{2}$$
$61$ $$11236 - 106 T + T^{2}$$
$67$ $$355216 + 596 T + T^{2}$$
$71$ $$( -630 + T )^{2}$$
$73$ $$1085764 - 1042 T + T^{2}$$
$79$ $$7744 - 88 T + T^{2}$$
$83$ $$( 1440 + T )^{2}$$
$89$ $$1887876 + 1374 T + T^{2}$$
$97$ $$( 34 + T )^{2}$$
show more
show less