# Properties

 Label 882.4.g.c 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: $$x^{2} - x + 1$$ 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.

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

## 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.c 2
3.b odd 2 1 294.4.e.j 2
7.b odd 2 1 882.4.g.j 2
7.c even 3 1 882.4.a.q 1
7.c even 3 1 inner 882.4.g.c 2
7.d odd 6 1 882.4.a.j 1
7.d odd 6 1 882.4.g.j 2
21.c even 2 1 294.4.e.f 2
21.g even 6 1 294.4.a.f yes 1
21.g even 6 1 294.4.e.f 2
21.h odd 6 1 294.4.a.b 1
21.h odd 6 1 294.4.e.j 2
84.j odd 6 1 2352.4.a.m 1
84.n even 6 1 2352.4.a.z 1

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

 $$T_{5}^{2} + 8T_{5} + 64$$ T5^2 + 8*T5 + 64 $$T_{11}^{2} - 40T_{11} + 1600$$ T11^2 - 40*T11 + 1600 $$T_{13} + 4$$ T13 + 4

## Hecke characteristic polynomials

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