# Properties

 Label 882.4.g.k 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.

 $$f(q)$$ $$=$$ $$q - 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} + 14 \zeta_{6} q^{5} + 8 q^{8} +O(q^{10})$$ q - 2*z * q^2 + (4*z - 4) * q^4 + 14*z * q^5 + 8 * q^8 $$q - 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} + 14 \zeta_{6} q^{5} + 8 q^{8} + ( - 28 \zeta_{6} + 28) q^{10} + (28 \zeta_{6} - 28) q^{11} - 18 q^{13} - 16 \zeta_{6} q^{16} + (74 \zeta_{6} - 74) q^{17} + 80 \zeta_{6} q^{19} - 56 q^{20} + 56 q^{22} - 112 \zeta_{6} q^{23} + (71 \zeta_{6} - 71) q^{25} + 36 \zeta_{6} q^{26} - 190 q^{29} + ( - 72 \zeta_{6} + 72) q^{31} + (32 \zeta_{6} - 32) q^{32} + 148 q^{34} + 346 \zeta_{6} q^{37} + ( - 160 \zeta_{6} + 160) q^{38} + 112 \zeta_{6} q^{40} + 162 q^{41} - 412 q^{43} - 112 \zeta_{6} q^{44} + (224 \zeta_{6} - 224) q^{46} - 24 \zeta_{6} q^{47} + 142 q^{50} + ( - 72 \zeta_{6} + 72) q^{52} + ( - 318 \zeta_{6} + 318) q^{53} - 392 q^{55} + 380 \zeta_{6} q^{58} + ( - 200 \zeta_{6} + 200) q^{59} - 198 \zeta_{6} q^{61} - 144 q^{62} + 64 q^{64} - 252 \zeta_{6} q^{65} + ( - 716 \zeta_{6} + 716) q^{67} - 296 \zeta_{6} q^{68} - 392 q^{71} + ( - 538 \zeta_{6} + 538) q^{73} + ( - 692 \zeta_{6} + 692) q^{74} - 320 q^{76} - 240 \zeta_{6} q^{79} + ( - 224 \zeta_{6} + 224) q^{80} - 324 \zeta_{6} q^{82} - 1072 q^{83} - 1036 q^{85} + 824 \zeta_{6} q^{86} + (224 \zeta_{6} - 224) q^{88} - 810 \zeta_{6} q^{89} + 448 q^{92} + (48 \zeta_{6} - 48) q^{94} + (1120 \zeta_{6} - 1120) q^{95} - 1354 q^{97} +O(q^{100})$$ q - 2*z * q^2 + (4*z - 4) * q^4 + 14*z * q^5 + 8 * q^8 + (-28*z + 28) * q^10 + (28*z - 28) * q^11 - 18 * q^13 - 16*z * q^16 + (74*z - 74) * q^17 + 80*z * q^19 - 56 * q^20 + 56 * q^22 - 112*z * q^23 + (71*z - 71) * q^25 + 36*z * q^26 - 190 * q^29 + (-72*z + 72) * q^31 + (32*z - 32) * q^32 + 148 * q^34 + 346*z * q^37 + (-160*z + 160) * q^38 + 112*z * q^40 + 162 * q^41 - 412 * q^43 - 112*z * q^44 + (224*z - 224) * q^46 - 24*z * q^47 + 142 * q^50 + (-72*z + 72) * q^52 + (-318*z + 318) * q^53 - 392 * q^55 + 380*z * q^58 + (-200*z + 200) * q^59 - 198*z * q^61 - 144 * q^62 + 64 * q^64 - 252*z * q^65 + (-716*z + 716) * q^67 - 296*z * q^68 - 392 * q^71 + (-538*z + 538) * q^73 + (-692*z + 692) * q^74 - 320 * q^76 - 240*z * q^79 + (-224*z + 224) * q^80 - 324*z * q^82 - 1072 * q^83 - 1036 * q^85 + 824*z * q^86 + (224*z - 224) * q^88 - 810*z * q^89 + 448 * q^92 + (48*z - 48) * q^94 + (1120*z - 1120) * q^95 - 1354 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 4 q^{4} + 14 q^{5} + 16 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 - 4 * q^4 + 14 * q^5 + 16 * q^8 $$2 q - 2 q^{2} - 4 q^{4} + 14 q^{5} + 16 q^{8} + 28 q^{10} - 28 q^{11} - 36 q^{13} - 16 q^{16} - 74 q^{17} + 80 q^{19} - 112 q^{20} + 112 q^{22} - 112 q^{23} - 71 q^{25} + 36 q^{26} - 380 q^{29} + 72 q^{31} - 32 q^{32} + 296 q^{34} + 346 q^{37} + 160 q^{38} + 112 q^{40} + 324 q^{41} - 824 q^{43} - 112 q^{44} - 224 q^{46} - 24 q^{47} + 284 q^{50} + 72 q^{52} + 318 q^{53} - 784 q^{55} + 380 q^{58} + 200 q^{59} - 198 q^{61} - 288 q^{62} + 128 q^{64} - 252 q^{65} + 716 q^{67} - 296 q^{68} - 784 q^{71} + 538 q^{73} + 692 q^{74} - 640 q^{76} - 240 q^{79} + 224 q^{80} - 324 q^{82} - 2144 q^{83} - 2072 q^{85} + 824 q^{86} - 224 q^{88} - 810 q^{89} + 896 q^{92} - 48 q^{94} - 1120 q^{95} - 2708 q^{97}+O(q^{100})$$ 2 * q - 2 * q^2 - 4 * q^4 + 14 * q^5 + 16 * q^8 + 28 * q^10 - 28 * q^11 - 36 * q^13 - 16 * q^16 - 74 * q^17 + 80 * q^19 - 112 * q^20 + 112 * q^22 - 112 * q^23 - 71 * q^25 + 36 * q^26 - 380 * q^29 + 72 * q^31 - 32 * q^32 + 296 * q^34 + 346 * q^37 + 160 * q^38 + 112 * q^40 + 324 * q^41 - 824 * q^43 - 112 * q^44 - 224 * q^46 - 24 * q^47 + 284 * q^50 + 72 * q^52 + 318 * q^53 - 784 * q^55 + 380 * q^58 + 200 * q^59 - 198 * q^61 - 288 * q^62 + 128 * q^64 - 252 * q^65 + 716 * q^67 - 296 * q^68 - 784 * q^71 + 538 * q^73 + 692 * q^74 - 640 * q^76 - 240 * q^79 + 224 * q^80 - 324 * q^82 - 2144 * q^83 - 2072 * q^85 + 824 * q^86 - 224 * q^88 - 810 * q^89 + 896 * q^92 - 48 * q^94 - 1120 * q^95 - 2708 * 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 7.00000 + 12.1244i 0 0 8.00000 0 14.0000 24.2487i
667.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i 7.00000 12.1244i 0 0 8.00000 0 14.0000 + 24.2487i
 $$n$$: 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.k 2
3.b odd 2 1 98.4.c.f 2
7.b odd 2 1 882.4.g.b 2
7.c even 3 1 882.4.a.i 1
7.c even 3 1 inner 882.4.g.k 2
7.d odd 6 1 126.4.a.h 1
7.d odd 6 1 882.4.g.b 2
21.c even 2 1 98.4.c.d 2
21.g even 6 1 14.4.a.a 1
21.g even 6 1 98.4.c.d 2
21.h odd 6 1 98.4.a.a 1
21.h odd 6 1 98.4.c.f 2
28.f even 6 1 1008.4.a.s 1
84.j odd 6 1 112.4.a.a 1
84.n even 6 1 784.4.a.s 1
105.o odd 6 1 2450.4.a.bo 1
105.p even 6 1 350.4.a.l 1
105.w odd 12 2 350.4.c.b 2
168.ba even 6 1 448.4.a.b 1
168.be odd 6 1 448.4.a.o 1
231.k odd 6 1 1694.4.a.g 1
273.ba even 6 1 2366.4.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.a 1 21.g even 6 1
98.4.a.a 1 21.h odd 6 1
98.4.c.d 2 21.c even 2 1
98.4.c.d 2 21.g even 6 1
98.4.c.f 2 3.b odd 2 1
98.4.c.f 2 21.h odd 6 1
112.4.a.a 1 84.j odd 6 1
126.4.a.h 1 7.d odd 6 1
350.4.a.l 1 105.p even 6 1
350.4.c.b 2 105.w odd 12 2
448.4.a.b 1 168.ba even 6 1
448.4.a.o 1 168.be odd 6 1
784.4.a.s 1 84.n even 6 1
882.4.a.i 1 7.c even 3 1
882.4.g.b 2 7.b odd 2 1
882.4.g.b 2 7.d odd 6 1
882.4.g.k 2 1.a even 1 1 trivial
882.4.g.k 2 7.c even 3 1 inner
1008.4.a.s 1 28.f even 6 1
1694.4.a.g 1 231.k odd 6 1
2366.4.a.h 1 273.ba even 6 1
2450.4.a.bo 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 $$S_{4}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{2} - 14T_{5} + 196$$ T5^2 - 14*T5 + 196 $$T_{11}^{2} + 28T_{11} + 784$$ T11^2 + 28*T11 + 784 $$T_{13} + 18$$ T13 + 18

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 14T + 196$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 28T + 784$$
$13$ $$(T + 18)^{2}$$
$17$ $$T^{2} + 74T + 5476$$
$19$ $$T^{2} - 80T + 6400$$
$23$ $$T^{2} + 112T + 12544$$
$29$ $$(T + 190)^{2}$$
$31$ $$T^{2} - 72T + 5184$$
$37$ $$T^{2} - 346T + 119716$$
$41$ $$(T - 162)^{2}$$
$43$ $$(T + 412)^{2}$$
$47$ $$T^{2} + 24T + 576$$
$53$ $$T^{2} - 318T + 101124$$
$59$ $$T^{2} - 200T + 40000$$
$61$ $$T^{2} + 198T + 39204$$
$67$ $$T^{2} - 716T + 512656$$
$71$ $$(T + 392)^{2}$$
$73$ $$T^{2} - 538T + 289444$$
$79$ $$T^{2} + 240T + 57600$$
$83$ $$(T + 1072)^{2}$$
$89$ $$T^{2} + 810T + 656100$$
$97$ $$(T + 1354)^{2}$$