# Properties

 Label 882.4.g.s Level $882$ Weight $4$ Character orbit 882.g Analytic conductor $52.040$ Analytic rank $1$ 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: $$1$$ 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 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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} + 2 \zeta_{6} q^{5} - 8 q^{8}+O(q^{10})$$ q + 2*z * q^2 + (4*z - 4) * q^4 + 2*z * q^5 - 8 * q^8 $$q + 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} + 2 \zeta_{6} q^{5} - 8 q^{8} + (4 \zeta_{6} - 4) q^{10} + (8 \zeta_{6} - 8) q^{11} - 42 q^{13} - 16 \zeta_{6} q^{16} + (2 \zeta_{6} - 2) q^{17} + 124 \zeta_{6} q^{19} - 8 q^{20} - 16 q^{22} + 76 \zeta_{6} q^{23} + ( - 121 \zeta_{6} + 121) q^{25} - 84 \zeta_{6} q^{26} - 254 q^{29} + ( - 72 \zeta_{6} + 72) q^{31} + ( - 32 \zeta_{6} + 32) q^{32} - 4 q^{34} - 398 \zeta_{6} q^{37} + (248 \zeta_{6} - 248) q^{38} - 16 \zeta_{6} q^{40} - 462 q^{41} + 212 q^{43} - 32 \zeta_{6} q^{44} + (152 \zeta_{6} - 152) q^{46} - 264 \zeta_{6} q^{47} + 242 q^{50} + ( - 168 \zeta_{6} + 168) q^{52} + (162 \zeta_{6} - 162) q^{53} - 16 q^{55} - 508 \zeta_{6} q^{58} + (772 \zeta_{6} - 772) q^{59} - 30 \zeta_{6} q^{61} + 144 q^{62} + 64 q^{64} - 84 \zeta_{6} q^{65} + ( - 764 \zeta_{6} + 764) q^{67} - 8 \zeta_{6} q^{68} + 236 q^{71} + (418 \zeta_{6} - 418) q^{73} + ( - 796 \zeta_{6} + 796) q^{74} - 496 q^{76} - 552 \zeta_{6} q^{79} + ( - 32 \zeta_{6} + 32) q^{80} - 924 \zeta_{6} q^{82} - 1036 q^{83} - 4 q^{85} + 424 \zeta_{6} q^{86} + ( - 64 \zeta_{6} + 64) q^{88} + 30 \zeta_{6} q^{89} - 304 q^{92} + ( - 528 \zeta_{6} + 528) q^{94} + (248 \zeta_{6} - 248) q^{95} - 1190 q^{97} +O(q^{100})$$ q + 2*z * q^2 + (4*z - 4) * q^4 + 2*z * q^5 - 8 * q^8 + (4*z - 4) * q^10 + (8*z - 8) * q^11 - 42 * q^13 - 16*z * q^16 + (2*z - 2) * q^17 + 124*z * q^19 - 8 * q^20 - 16 * q^22 + 76*z * q^23 + (-121*z + 121) * q^25 - 84*z * q^26 - 254 * q^29 + (-72*z + 72) * q^31 + (-32*z + 32) * q^32 - 4 * q^34 - 398*z * q^37 + (248*z - 248) * q^38 - 16*z * q^40 - 462 * q^41 + 212 * q^43 - 32*z * q^44 + (152*z - 152) * q^46 - 264*z * q^47 + 242 * q^50 + (-168*z + 168) * q^52 + (162*z - 162) * q^53 - 16 * q^55 - 508*z * q^58 + (772*z - 772) * q^59 - 30*z * q^61 + 144 * q^62 + 64 * q^64 - 84*z * q^65 + (-764*z + 764) * q^67 - 8*z * q^68 + 236 * q^71 + (418*z - 418) * q^73 + (-796*z + 796) * q^74 - 496 * q^76 - 552*z * q^79 + (-32*z + 32) * q^80 - 924*z * q^82 - 1036 * q^83 - 4 * q^85 + 424*z * q^86 + (-64*z + 64) * q^88 + 30*z * q^89 - 304 * q^92 + (-528*z + 528) * q^94 + (248*z - 248) * q^95 - 1190 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 4 q^{4} + 2 q^{5} - 16 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 - 4 * q^4 + 2 * q^5 - 16 * q^8 $$2 q + 2 q^{2} - 4 q^{4} + 2 q^{5} - 16 q^{8} - 4 q^{10} - 8 q^{11} - 84 q^{13} - 16 q^{16} - 2 q^{17} + 124 q^{19} - 16 q^{20} - 32 q^{22} + 76 q^{23} + 121 q^{25} - 84 q^{26} - 508 q^{29} + 72 q^{31} + 32 q^{32} - 8 q^{34} - 398 q^{37} - 248 q^{38} - 16 q^{40} - 924 q^{41} + 424 q^{43} - 32 q^{44} - 152 q^{46} - 264 q^{47} + 484 q^{50} + 168 q^{52} - 162 q^{53} - 32 q^{55} - 508 q^{58} - 772 q^{59} - 30 q^{61} + 288 q^{62} + 128 q^{64} - 84 q^{65} + 764 q^{67} - 8 q^{68} + 472 q^{71} - 418 q^{73} + 796 q^{74} - 992 q^{76} - 552 q^{79} + 32 q^{80} - 924 q^{82} - 2072 q^{83} - 8 q^{85} + 424 q^{86} + 64 q^{88} + 30 q^{89} - 608 q^{92} + 528 q^{94} - 248 q^{95} - 2380 q^{97}+O(q^{100})$$ 2 * q + 2 * q^2 - 4 * q^4 + 2 * q^5 - 16 * q^8 - 4 * q^10 - 8 * q^11 - 84 * q^13 - 16 * q^16 - 2 * q^17 + 124 * q^19 - 16 * q^20 - 32 * q^22 + 76 * q^23 + 121 * q^25 - 84 * q^26 - 508 * q^29 + 72 * q^31 + 32 * q^32 - 8 * q^34 - 398 * q^37 - 248 * q^38 - 16 * q^40 - 924 * q^41 + 424 * q^43 - 32 * q^44 - 152 * q^46 - 264 * q^47 + 484 * q^50 + 168 * q^52 - 162 * q^53 - 32 * q^55 - 508 * q^58 - 772 * q^59 - 30 * q^61 + 288 * q^62 + 128 * q^64 - 84 * q^65 + 764 * q^67 - 8 * q^68 + 472 * q^71 - 418 * q^73 + 796 * q^74 - 992 * q^76 - 552 * q^79 + 32 * q^80 - 924 * q^82 - 2072 * q^83 - 8 * q^85 + 424 * q^86 + 64 * q^88 + 30 * q^89 - 608 * q^92 + 528 * q^94 - 248 * q^95 - 2380 * 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 1.00000 + 1.73205i 0 0 −8.00000 0 −2.00000 + 3.46410i
667.1 1.00000 1.73205i 0 −2.00000 3.46410i 1.00000 1.73205i 0 0 −8.00000 0 −2.00000 3.46410i
 $$n$$: 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.s 2
3.b odd 2 1 294.4.e.a 2
7.b odd 2 1 882.4.g.r 2
7.c even 3 1 126.4.a.c 1
7.c even 3 1 inner 882.4.g.s 2
7.d odd 6 1 882.4.a.d 1
7.d odd 6 1 882.4.g.r 2
21.c even 2 1 294.4.e.d 2
21.g even 6 1 294.4.a.h 1
21.g even 6 1 294.4.e.d 2
21.h odd 6 1 42.4.a.b 1
21.h odd 6 1 294.4.e.a 2
28.g odd 6 1 1008.4.a.j 1
84.j odd 6 1 2352.4.a.ba 1
84.n even 6 1 336.4.a.d 1
105.o odd 6 1 1050.4.a.d 1
105.x even 12 2 1050.4.g.n 2
168.s odd 6 1 1344.4.a.f 1
168.v even 6 1 1344.4.a.t 1

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

 $$T_{5}^{2} - 2T_{5} + 4$$ T5^2 - 2*T5 + 4 $$T_{11}^{2} + 8T_{11} + 64$$ T11^2 + 8*T11 + 64 $$T_{13} + 42$$ T13 + 42

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 2T + 4$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 8T + 64$$
$13$ $$(T + 42)^{2}$$
$17$ $$T^{2} + 2T + 4$$
$19$ $$T^{2} - 124T + 15376$$
$23$ $$T^{2} - 76T + 5776$$
$29$ $$(T + 254)^{2}$$
$31$ $$T^{2} - 72T + 5184$$
$37$ $$T^{2} + 398T + 158404$$
$41$ $$(T + 462)^{2}$$
$43$ $$(T - 212)^{2}$$
$47$ $$T^{2} + 264T + 69696$$
$53$ $$T^{2} + 162T + 26244$$
$59$ $$T^{2} + 772T + 595984$$
$61$ $$T^{2} + 30T + 900$$
$67$ $$T^{2} - 764T + 583696$$
$71$ $$(T - 236)^{2}$$
$73$ $$T^{2} + 418T + 174724$$
$79$ $$T^{2} + 552T + 304704$$
$83$ $$(T + 1036)^{2}$$
$89$ $$T^{2} - 30T + 900$$
$97$ $$(T + 1190)^{2}$$