Properties

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

 $$f(q)$$ $$=$$ $$q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} -14 \zeta_{6} q^{5} + 8 q^{8} +O(q^{10})$$ $$q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} -14 \zeta_{6} q^{5} + 8 q^{8} + ( -28 + 28 \zeta_{6} ) q^{10} + ( -28 + 28 \zeta_{6} ) q^{11} + 18 q^{13} -16 \zeta_{6} q^{16} + ( 74 - 74 \zeta_{6} ) q^{17} -80 \zeta_{6} q^{19} + 56 q^{20} + 56 q^{22} -112 \zeta_{6} q^{23} + ( -71 + 71 \zeta_{6} ) q^{25} -36 \zeta_{6} q^{26} -190 q^{29} + ( -72 + 72 \zeta_{6} ) q^{31} + ( -32 + 32 \zeta_{6} ) q^{32} -148 q^{34} + 346 \zeta_{6} q^{37} + ( -160 + 160 \zeta_{6} ) q^{38} -112 \zeta_{6} q^{40} -162 q^{41} -412 q^{43} -112 \zeta_{6} q^{44} + ( -224 + 224 \zeta_{6} ) q^{46} + 24 \zeta_{6} q^{47} + 142 q^{50} + ( -72 + 72 \zeta_{6} ) q^{52} + ( 318 - 318 \zeta_{6} ) q^{53} + 392 q^{55} + 380 \zeta_{6} q^{58} + ( -200 + 200 \zeta_{6} ) q^{59} + 198 \zeta_{6} q^{61} + 144 q^{62} + 64 q^{64} -252 \zeta_{6} q^{65} + ( 716 - 716 \zeta_{6} ) q^{67} + 296 \zeta_{6} q^{68} -392 q^{71} + ( -538 + 538 \zeta_{6} ) q^{73} + ( 692 - 692 \zeta_{6} ) q^{74} + 320 q^{76} -240 \zeta_{6} q^{79} + ( -224 + 224 \zeta_{6} ) q^{80} + 324 \zeta_{6} q^{82} + 1072 q^{83} -1036 q^{85} + 824 \zeta_{6} q^{86} + ( -224 + 224 \zeta_{6} ) q^{88} + 810 \zeta_{6} q^{89} + 448 q^{92} + ( 48 - 48 \zeta_{6} ) q^{94} + ( -1120 + 1120 \zeta_{6} ) q^{95} + 1354 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 4q^{4} - 14q^{5} + 16q^{8} + O(q^{10})$$ $$2q - 2q^{2} - 4q^{4} - 14q^{5} + 16q^{8} - 28q^{10} - 28q^{11} + 36q^{13} - 16q^{16} + 74q^{17} - 80q^{19} + 112q^{20} + 112q^{22} - 112q^{23} - 71q^{25} - 36q^{26} - 380q^{29} - 72q^{31} - 32q^{32} - 296q^{34} + 346q^{37} - 160q^{38} - 112q^{40} - 324q^{41} - 824q^{43} - 112q^{44} - 224q^{46} + 24q^{47} + 284q^{50} - 72q^{52} + 318q^{53} + 784q^{55} + 380q^{58} - 200q^{59} + 198q^{61} + 288q^{62} + 128q^{64} - 252q^{65} + 716q^{67} + 296q^{68} - 784q^{71} - 538q^{73} + 692q^{74} + 640q^{76} - 240q^{79} - 224q^{80} + 324q^{82} + 2144q^{83} - 2072q^{85} + 824q^{86} - 224q^{88} + 810q^{89} + 896q^{92} + 48q^{94} - 1120q^{95} + 2708q^{97} + O(q^{100})$$

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.b 2
3.b odd 2 1 98.4.c.d 2
7.b odd 2 1 882.4.g.k 2
7.c even 3 1 126.4.a.h 1
7.c even 3 1 inner 882.4.g.b 2
7.d odd 6 1 882.4.a.i 1
7.d odd 6 1 882.4.g.k 2
21.c even 2 1 98.4.c.f 2
21.g even 6 1 98.4.a.a 1
21.g even 6 1 98.4.c.f 2
21.h odd 6 1 14.4.a.a 1
21.h odd 6 1 98.4.c.d 2
28.g odd 6 1 1008.4.a.s 1
84.j odd 6 1 784.4.a.s 1
84.n even 6 1 112.4.a.a 1
105.o odd 6 1 350.4.a.l 1
105.p even 6 1 2450.4.a.bo 1
105.x even 12 2 350.4.c.b 2
168.s odd 6 1 448.4.a.b 1
168.v even 6 1 448.4.a.o 1
231.l even 6 1 1694.4.a.g 1
273.w odd 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.h odd 6 1
98.4.a.a 1 21.g even 6 1
98.4.c.d 2 3.b odd 2 1
98.4.c.d 2 21.h odd 6 1
98.4.c.f 2 21.c even 2 1
98.4.c.f 2 21.g even 6 1
112.4.a.a 1 84.n even 6 1
126.4.a.h 1 7.c even 3 1
350.4.a.l 1 105.o odd 6 1
350.4.c.b 2 105.x even 12 2
448.4.a.b 1 168.s odd 6 1
448.4.a.o 1 168.v even 6 1
784.4.a.s 1 84.j odd 6 1
882.4.a.i 1 7.d odd 6 1
882.4.g.b 2 1.a even 1 1 trivial
882.4.g.b 2 7.c even 3 1 inner
882.4.g.k 2 7.b odd 2 1
882.4.g.k 2 7.d odd 6 1
1008.4.a.s 1 28.g odd 6 1
1694.4.a.g 1 231.l even 6 1
2366.4.a.h 1 273.w odd 6 1
2450.4.a.bo 1 105.p even 6 1

Hecke kernels

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

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

Hecke characteristic polynomials

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