# Properties

 Label 882.4.g.n 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 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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} - 22 \zeta_{6} q^{5} - 8 q^{8} +O(q^{10})$$ q + 2*z * q^2 + (4*z - 4) * q^4 - 22*z * q^5 - 8 * q^8 $$q + 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} - 22 \zeta_{6} q^{5} - 8 q^{8} + ( - 44 \zeta_{6} + 44) q^{10} + (26 \zeta_{6} - 26) q^{11} - 54 q^{13} - 16 \zeta_{6} q^{16} + (74 \zeta_{6} - 74) q^{17} - 116 \zeta_{6} q^{19} + 88 q^{20} - 52 q^{22} + 58 \zeta_{6} q^{23} + (359 \zeta_{6} - 359) q^{25} - 108 \zeta_{6} q^{26} + 208 q^{29} + ( - 252 \zeta_{6} + 252) q^{31} + ( - 32 \zeta_{6} + 32) q^{32} - 148 q^{34} - 50 \zeta_{6} q^{37} + ( - 232 \zeta_{6} + 232) q^{38} + 176 \zeta_{6} q^{40} + 126 q^{41} + 164 q^{43} - 104 \zeta_{6} q^{44} + (116 \zeta_{6} - 116) q^{46} + 444 \zeta_{6} q^{47} - 718 q^{50} + ( - 216 \zeta_{6} + 216) q^{52} + (12 \zeta_{6} - 12) q^{53} + 572 q^{55} + 416 \zeta_{6} q^{58} + (124 \zeta_{6} - 124) q^{59} + 162 \zeta_{6} q^{61} + 504 q^{62} + 64 q^{64} + 1188 \zeta_{6} q^{65} + ( - 860 \zeta_{6} + 860) q^{67} - 296 \zeta_{6} q^{68} - 238 q^{71} + ( - 146 \zeta_{6} + 146) q^{73} + ( - 100 \zeta_{6} + 100) q^{74} + 464 q^{76} + 984 \zeta_{6} q^{79} + (352 \zeta_{6} - 352) q^{80} + 252 \zeta_{6} q^{82} + 656 q^{83} + 1628 q^{85} + 328 \zeta_{6} q^{86} + ( - 208 \zeta_{6} + 208) q^{88} + 954 \zeta_{6} q^{89} - 232 q^{92} + (888 \zeta_{6} - 888) q^{94} + (2552 \zeta_{6} - 2552) q^{95} + 526 q^{97} +O(q^{100})$$ q + 2*z * q^2 + (4*z - 4) * q^4 - 22*z * q^5 - 8 * q^8 + (-44*z + 44) * q^10 + (26*z - 26) * q^11 - 54 * q^13 - 16*z * q^16 + (74*z - 74) * q^17 - 116*z * q^19 + 88 * q^20 - 52 * q^22 + 58*z * q^23 + (359*z - 359) * q^25 - 108*z * q^26 + 208 * q^29 + (-252*z + 252) * q^31 + (-32*z + 32) * q^32 - 148 * q^34 - 50*z * q^37 + (-232*z + 232) * q^38 + 176*z * q^40 + 126 * q^41 + 164 * q^43 - 104*z * q^44 + (116*z - 116) * q^46 + 444*z * q^47 - 718 * q^50 + (-216*z + 216) * q^52 + (12*z - 12) * q^53 + 572 * q^55 + 416*z * q^58 + (124*z - 124) * q^59 + 162*z * q^61 + 504 * q^62 + 64 * q^64 + 1188*z * q^65 + (-860*z + 860) * q^67 - 296*z * q^68 - 238 * q^71 + (-146*z + 146) * q^73 + (-100*z + 100) * q^74 + 464 * q^76 + 984*z * q^79 + (352*z - 352) * q^80 + 252*z * q^82 + 656 * q^83 + 1628 * q^85 + 328*z * q^86 + (-208*z + 208) * q^88 + 954*z * q^89 - 232 * q^92 + (888*z - 888) * q^94 + (2552*z - 2552) * q^95 + 526 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 4 q^{4} - 22 q^{5} - 16 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 - 4 * q^4 - 22 * q^5 - 16 * q^8 $$2 q + 2 q^{2} - 4 q^{4} - 22 q^{5} - 16 q^{8} + 44 q^{10} - 26 q^{11} - 108 q^{13} - 16 q^{16} - 74 q^{17} - 116 q^{19} + 176 q^{20} - 104 q^{22} + 58 q^{23} - 359 q^{25} - 108 q^{26} + 416 q^{29} + 252 q^{31} + 32 q^{32} - 296 q^{34} - 50 q^{37} + 232 q^{38} + 176 q^{40} + 252 q^{41} + 328 q^{43} - 104 q^{44} - 116 q^{46} + 444 q^{47} - 1436 q^{50} + 216 q^{52} - 12 q^{53} + 1144 q^{55} + 416 q^{58} - 124 q^{59} + 162 q^{61} + 1008 q^{62} + 128 q^{64} + 1188 q^{65} + 860 q^{67} - 296 q^{68} - 476 q^{71} + 146 q^{73} + 100 q^{74} + 928 q^{76} + 984 q^{79} - 352 q^{80} + 252 q^{82} + 1312 q^{83} + 3256 q^{85} + 328 q^{86} + 208 q^{88} + 954 q^{89} - 464 q^{92} - 888 q^{94} - 2552 q^{95} + 1052 q^{97}+O(q^{100})$$ 2 * q + 2 * q^2 - 4 * q^4 - 22 * q^5 - 16 * q^8 + 44 * q^10 - 26 * q^11 - 108 * q^13 - 16 * q^16 - 74 * q^17 - 116 * q^19 + 176 * q^20 - 104 * q^22 + 58 * q^23 - 359 * q^25 - 108 * q^26 + 416 * q^29 + 252 * q^31 + 32 * q^32 - 296 * q^34 - 50 * q^37 + 232 * q^38 + 176 * q^40 + 252 * q^41 + 328 * q^43 - 104 * q^44 - 116 * q^46 + 444 * q^47 - 1436 * q^50 + 216 * q^52 - 12 * q^53 + 1144 * q^55 + 416 * q^58 - 124 * q^59 + 162 * q^61 + 1008 * q^62 + 128 * q^64 + 1188 * q^65 + 860 * q^67 - 296 * q^68 - 476 * q^71 + 146 * q^73 + 100 * q^74 + 928 * q^76 + 984 * q^79 - 352 * q^80 + 252 * q^82 + 1312 * q^83 + 3256 * q^85 + 328 * q^86 + 208 * q^88 + 954 * q^89 - 464 * q^92 - 888 * q^94 - 2552 * q^95 + 1052 * 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 −11.0000 19.0526i 0 0 −8.00000 0 22.0000 38.1051i
667.1 1.00000 1.73205i 0 −2.00000 3.46410i −11.0000 + 19.0526i 0 0 −8.00000 0 22.0000 + 38.1051i
 $$n$$: 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.n 2
3.b odd 2 1 882.4.g.m 2
7.b odd 2 1 882.4.g.x 2
7.c even 3 1 126.4.a.e 1
7.c even 3 1 inner 882.4.g.n 2
7.d odd 6 1 882.4.a.a 1
7.d odd 6 1 882.4.g.x 2
21.c even 2 1 882.4.g.a 2
21.g even 6 1 882.4.a.s 1
21.g even 6 1 882.4.g.a 2
21.h odd 6 1 126.4.a.f yes 1
21.h odd 6 1 882.4.g.m 2
28.g odd 6 1 1008.4.a.w 1
84.n even 6 1 1008.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.a.e 1 7.c even 3 1
126.4.a.f yes 1 21.h odd 6 1
882.4.a.a 1 7.d odd 6 1
882.4.a.s 1 21.g even 6 1
882.4.g.a 2 21.c even 2 1
882.4.g.a 2 21.g even 6 1
882.4.g.m 2 3.b odd 2 1
882.4.g.m 2 21.h odd 6 1
882.4.g.n 2 1.a even 1 1 trivial
882.4.g.n 2 7.c even 3 1 inner
882.4.g.x 2 7.b odd 2 1
882.4.g.x 2 7.d odd 6 1
1008.4.a.a 1 84.n even 6 1
1008.4.a.w 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 $$S_{4}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{2} + 22T_{5} + 484$$ T5^2 + 22*T5 + 484 $$T_{11}^{2} + 26T_{11} + 676$$ T11^2 + 26*T11 + 676 $$T_{13} + 54$$ T13 + 54

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 22T + 484$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 26T + 676$$
$13$ $$(T + 54)^{2}$$
$17$ $$T^{2} + 74T + 5476$$
$19$ $$T^{2} + 116T + 13456$$
$23$ $$T^{2} - 58T + 3364$$
$29$ $$(T - 208)^{2}$$
$31$ $$T^{2} - 252T + 63504$$
$37$ $$T^{2} + 50T + 2500$$
$41$ $$(T - 126)^{2}$$
$43$ $$(T - 164)^{2}$$
$47$ $$T^{2} - 444T + 197136$$
$53$ $$T^{2} + 12T + 144$$
$59$ $$T^{2} + 124T + 15376$$
$61$ $$T^{2} - 162T + 26244$$
$67$ $$T^{2} - 860T + 739600$$
$71$ $$(T + 238)^{2}$$
$73$ $$T^{2} - 146T + 21316$$
$79$ $$T^{2} - 984T + 968256$$
$83$ $$(T - 656)^{2}$$
$89$ $$T^{2} - 954T + 910116$$
$97$ $$(T - 526)^{2}$$