# Properties

 Label 882.4.g.v 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 14) 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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} + 12 \zeta_{6} q^{5} - 8 q^{8}+O(q^{10})$$ q + 2*z * q^2 + (4*z - 4) * q^4 + 12*z * q^5 - 8 * q^8 $$q + 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} + 12 \zeta_{6} q^{5} - 8 q^{8} + (24 \zeta_{6} - 24) q^{10} + ( - 48 \zeta_{6} + 48) q^{11} - 56 q^{13} - 16 \zeta_{6} q^{16} + ( - 114 \zeta_{6} + 114) q^{17} + 2 \zeta_{6} q^{19} - 48 q^{20} + 96 q^{22} - 120 \zeta_{6} q^{23} + (19 \zeta_{6} - 19) q^{25} - 112 \zeta_{6} q^{26} + 54 q^{29} + ( - 236 \zeta_{6} + 236) q^{31} + ( - 32 \zeta_{6} + 32) q^{32} + 228 q^{34} - 146 \zeta_{6} q^{37} + (4 \zeta_{6} - 4) q^{38} - 96 \zeta_{6} q^{40} + 126 q^{41} - 376 q^{43} + 192 \zeta_{6} q^{44} + ( - 240 \zeta_{6} + 240) q^{46} + 12 \zeta_{6} q^{47} - 38 q^{50} + ( - 224 \zeta_{6} + 224) q^{52} + ( - 174 \zeta_{6} + 174) q^{53} + 576 q^{55} + 108 \zeta_{6} q^{58} + (138 \zeta_{6} - 138) q^{59} + 380 \zeta_{6} q^{61} + 472 q^{62} + 64 q^{64} - 672 \zeta_{6} q^{65} + ( - 484 \zeta_{6} + 484) q^{67} + 456 \zeta_{6} q^{68} - 576 q^{71} + (1150 \zeta_{6} - 1150) q^{73} + ( - 292 \zeta_{6} + 292) q^{74} - 8 q^{76} - 776 \zeta_{6} q^{79} + ( - 192 \zeta_{6} + 192) q^{80} + 252 \zeta_{6} q^{82} + 378 q^{83} + 1368 q^{85} - 752 \zeta_{6} q^{86} + (384 \zeta_{6} - 384) q^{88} + 390 \zeta_{6} q^{89} + 480 q^{92} + (24 \zeta_{6} - 24) q^{94} + (24 \zeta_{6} - 24) q^{95} + 1330 q^{97} +O(q^{100})$$ q + 2*z * q^2 + (4*z - 4) * q^4 + 12*z * q^5 - 8 * q^8 + (24*z - 24) * q^10 + (-48*z + 48) * q^11 - 56 * q^13 - 16*z * q^16 + (-114*z + 114) * q^17 + 2*z * q^19 - 48 * q^20 + 96 * q^22 - 120*z * q^23 + (19*z - 19) * q^25 - 112*z * q^26 + 54 * q^29 + (-236*z + 236) * q^31 + (-32*z + 32) * q^32 + 228 * q^34 - 146*z * q^37 + (4*z - 4) * q^38 - 96*z * q^40 + 126 * q^41 - 376 * q^43 + 192*z * q^44 + (-240*z + 240) * q^46 + 12*z * q^47 - 38 * q^50 + (-224*z + 224) * q^52 + (-174*z + 174) * q^53 + 576 * q^55 + 108*z * q^58 + (138*z - 138) * q^59 + 380*z * q^61 + 472 * q^62 + 64 * q^64 - 672*z * q^65 + (-484*z + 484) * q^67 + 456*z * q^68 - 576 * q^71 + (1150*z - 1150) * q^73 + (-292*z + 292) * q^74 - 8 * q^76 - 776*z * q^79 + (-192*z + 192) * q^80 + 252*z * q^82 + 378 * q^83 + 1368 * q^85 - 752*z * q^86 + (384*z - 384) * q^88 + 390*z * q^89 + 480 * q^92 + (24*z - 24) * q^94 + (24*z - 24) * q^95 + 1330 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 4 q^{4} + 12 q^{5} - 16 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 - 4 * q^4 + 12 * q^5 - 16 * q^8 $$2 q + 2 q^{2} - 4 q^{4} + 12 q^{5} - 16 q^{8} - 24 q^{10} + 48 q^{11} - 112 q^{13} - 16 q^{16} + 114 q^{17} + 2 q^{19} - 96 q^{20} + 192 q^{22} - 120 q^{23} - 19 q^{25} - 112 q^{26} + 108 q^{29} + 236 q^{31} + 32 q^{32} + 456 q^{34} - 146 q^{37} - 4 q^{38} - 96 q^{40} + 252 q^{41} - 752 q^{43} + 192 q^{44} + 240 q^{46} + 12 q^{47} - 76 q^{50} + 224 q^{52} + 174 q^{53} + 1152 q^{55} + 108 q^{58} - 138 q^{59} + 380 q^{61} + 944 q^{62} + 128 q^{64} - 672 q^{65} + 484 q^{67} + 456 q^{68} - 1152 q^{71} - 1150 q^{73} + 292 q^{74} - 16 q^{76} - 776 q^{79} + 192 q^{80} + 252 q^{82} + 756 q^{83} + 2736 q^{85} - 752 q^{86} - 384 q^{88} + 390 q^{89} + 960 q^{92} - 24 q^{94} - 24 q^{95} + 2660 q^{97}+O(q^{100})$$ 2 * q + 2 * q^2 - 4 * q^4 + 12 * q^5 - 16 * q^8 - 24 * q^10 + 48 * q^11 - 112 * q^13 - 16 * q^16 + 114 * q^17 + 2 * q^19 - 96 * q^20 + 192 * q^22 - 120 * q^23 - 19 * q^25 - 112 * q^26 + 108 * q^29 + 236 * q^31 + 32 * q^32 + 456 * q^34 - 146 * q^37 - 4 * q^38 - 96 * q^40 + 252 * q^41 - 752 * q^43 + 192 * q^44 + 240 * q^46 + 12 * q^47 - 76 * q^50 + 224 * q^52 + 174 * q^53 + 1152 * q^55 + 108 * q^58 - 138 * q^59 + 380 * q^61 + 944 * q^62 + 128 * q^64 - 672 * q^65 + 484 * q^67 + 456 * q^68 - 1152 * q^71 - 1150 * q^73 + 292 * q^74 - 16 * q^76 - 776 * q^79 + 192 * q^80 + 252 * q^82 + 756 * q^83 + 2736 * q^85 - 752 * q^86 - 384 * q^88 + 390 * q^89 + 960 * q^92 - 24 * q^94 - 24 * q^95 + 2660 * 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 6.00000 + 10.3923i 0 0 −8.00000 0 −12.0000 + 20.7846i
667.1 1.00000 1.73205i 0 −2.00000 3.46410i 6.00000 10.3923i 0 0 −8.00000 0 −12.0000 20.7846i
 $$n$$: 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.v 2
3.b odd 2 1 98.4.c.b 2
7.b odd 2 1 882.4.g.p 2
7.c even 3 1 882.4.a.b 1
7.c even 3 1 inner 882.4.g.v 2
7.d odd 6 1 126.4.a.d 1
7.d odd 6 1 882.4.g.p 2
21.c even 2 1 98.4.c.c 2
21.g even 6 1 14.4.a.b 1
21.g even 6 1 98.4.c.c 2
21.h odd 6 1 98.4.a.e 1
21.h odd 6 1 98.4.c.b 2
28.f even 6 1 1008.4.a.r 1
84.j odd 6 1 112.4.a.e 1
84.n even 6 1 784.4.a.h 1
105.o odd 6 1 2450.4.a.i 1
105.p even 6 1 350.4.a.f 1
105.w odd 12 2 350.4.c.g 2
168.ba even 6 1 448.4.a.k 1
168.be odd 6 1 448.4.a.g 1
231.k odd 6 1 1694.4.a.b 1
273.ba even 6 1 2366.4.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.b 1 21.g even 6 1
98.4.a.e 1 21.h odd 6 1
98.4.c.b 2 3.b odd 2 1
98.4.c.b 2 21.h odd 6 1
98.4.c.c 2 21.c even 2 1
98.4.c.c 2 21.g even 6 1
112.4.a.e 1 84.j odd 6 1
126.4.a.d 1 7.d odd 6 1
350.4.a.f 1 105.p even 6 1
350.4.c.g 2 105.w odd 12 2
448.4.a.g 1 168.be odd 6 1
448.4.a.k 1 168.ba even 6 1
784.4.a.h 1 84.n even 6 1
882.4.a.b 1 7.c even 3 1
882.4.g.p 2 7.b odd 2 1
882.4.g.p 2 7.d odd 6 1
882.4.g.v 2 1.a even 1 1 trivial
882.4.g.v 2 7.c even 3 1 inner
1008.4.a.r 1 28.f even 6 1
1694.4.a.b 1 231.k odd 6 1
2366.4.a.c 1 273.ba even 6 1
2450.4.a.i 1 105.o odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 12T_{5} + 144$$ T5^2 - 12*T5 + 144 $$T_{11}^{2} - 48T_{11} + 2304$$ T11^2 - 48*T11 + 2304 $$T_{13} + 56$$ T13 + 56

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 12T + 144$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 48T + 2304$$
$13$ $$(T + 56)^{2}$$
$17$ $$T^{2} - 114T + 12996$$
$19$ $$T^{2} - 2T + 4$$
$23$ $$T^{2} + 120T + 14400$$
$29$ $$(T - 54)^{2}$$
$31$ $$T^{2} - 236T + 55696$$
$37$ $$T^{2} + 146T + 21316$$
$41$ $$(T - 126)^{2}$$
$43$ $$(T + 376)^{2}$$
$47$ $$T^{2} - 12T + 144$$
$53$ $$T^{2} - 174T + 30276$$
$59$ $$T^{2} + 138T + 19044$$
$61$ $$T^{2} - 380T + 144400$$
$67$ $$T^{2} - 484T + 234256$$
$71$ $$(T + 576)^{2}$$
$73$ $$T^{2} + 1150 T + 1322500$$
$79$ $$T^{2} + 776T + 602176$$
$83$ $$(T - 378)^{2}$$
$89$ $$T^{2} - 390T + 152100$$
$97$ $$(T - 1330)^{2}$$