# Properties

 Label 882.2.g.f Level 882 Weight 2 Character orbit 882.g Analytic conductor 7.043 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: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 882.g (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.04280545828$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 294) 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} + 4 \zeta_{6} q^{5} + q^{8} +O(q^{10})$$ $$q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} + 4 \zeta_{6} q^{5} + q^{8} + ( 4 - 4 \zeta_{6} ) q^{10} + ( -4 + 4 \zeta_{6} ) q^{11} + 4 q^{13} -\zeta_{6} q^{16} -4 \zeta_{6} q^{19} -4 q^{20} + 4 q^{22} + ( -11 + 11 \zeta_{6} ) q^{25} -4 \zeta_{6} q^{26} -2 q^{29} + ( -8 + 8 \zeta_{6} ) q^{31} + ( -1 + \zeta_{6} ) q^{32} + 6 \zeta_{6} q^{37} + ( -4 + 4 \zeta_{6} ) q^{38} + 4 \zeta_{6} q^{40} + 4 q^{43} -4 \zeta_{6} q^{44} -8 \zeta_{6} q^{47} + 11 q^{50} + ( -4 + 4 \zeta_{6} ) q^{52} + ( -10 + 10 \zeta_{6} ) q^{53} -16 q^{55} + 2 \zeta_{6} q^{58} + ( 4 - 4 \zeta_{6} ) q^{59} + 4 \zeta_{6} q^{61} + 8 q^{62} + q^{64} + 16 \zeta_{6} q^{65} + ( -4 + 4 \zeta_{6} ) q^{67} -8 q^{71} + ( 16 - 16 \zeta_{6} ) q^{73} + ( 6 - 6 \zeta_{6} ) q^{74} + 4 q^{76} + 8 \zeta_{6} q^{79} + ( 4 - 4 \zeta_{6} ) q^{80} + 12 q^{83} -4 \zeta_{6} q^{86} + ( -4 + 4 \zeta_{6} ) q^{88} + 8 \zeta_{6} q^{89} + ( -8 + 8 \zeta_{6} ) q^{94} + ( 16 - 16 \zeta_{6} ) q^{95} + 8 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} + 4q^{5} + 2q^{8} + O(q^{10})$$ $$2q - q^{2} - q^{4} + 4q^{5} + 2q^{8} + 4q^{10} - 4q^{11} + 8q^{13} - q^{16} - 4q^{19} - 8q^{20} + 8q^{22} - 11q^{25} - 4q^{26} - 4q^{29} - 8q^{31} - q^{32} + 6q^{37} - 4q^{38} + 4q^{40} + 8q^{43} - 4q^{44} - 8q^{47} + 22q^{50} - 4q^{52} - 10q^{53} - 32q^{55} + 2q^{58} + 4q^{59} + 4q^{61} + 16q^{62} + 2q^{64} + 16q^{65} - 4q^{67} - 16q^{71} + 16q^{73} + 6q^{74} + 8q^{76} + 8q^{79} + 4q^{80} + 24q^{83} - 4q^{86} - 4q^{88} + 8q^{89} - 8q^{94} + 16q^{95} + 16q^{97} + O(q^{100})$$

## 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
−0.500000 0.866025i 0 −0.500000 + 0.866025i 2.00000 + 3.46410i 0 0 1.00000 0 2.00000 3.46410i
667.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i 2.00000 3.46410i 0 0 1.00000 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 882.2.g.f 2
3.b odd 2 1 294.2.e.e 2
7.b odd 2 1 882.2.g.a 2
7.c even 3 1 882.2.a.f 1
7.c even 3 1 inner 882.2.g.f 2
7.d odd 6 1 882.2.a.l 1
7.d odd 6 1 882.2.g.a 2
12.b even 2 1 2352.2.q.a 2
21.c even 2 1 294.2.e.d 2
21.g even 6 1 294.2.a.c yes 1
21.g even 6 1 294.2.e.d 2
21.h odd 6 1 294.2.a.b 1
21.h odd 6 1 294.2.e.e 2
28.f even 6 1 7056.2.a.ca 1
28.g odd 6 1 7056.2.a.a 1
84.h odd 2 1 2352.2.q.y 2
84.j odd 6 1 2352.2.a.b 1
84.j odd 6 1 2352.2.q.y 2
84.n even 6 1 2352.2.a.y 1
84.n even 6 1 2352.2.q.a 2
105.o odd 6 1 7350.2.a.cj 1
105.p even 6 1 7350.2.a.br 1
168.s odd 6 1 9408.2.a.br 1
168.v even 6 1 9408.2.a.b 1
168.ba even 6 1 9408.2.a.bo 1
168.be odd 6 1 9408.2.a.de 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.2.a.b 1 21.h odd 6 1
294.2.a.c yes 1 21.g even 6 1
294.2.e.d 2 21.c even 2 1
294.2.e.d 2 21.g even 6 1
294.2.e.e 2 3.b odd 2 1
294.2.e.e 2 21.h odd 6 1
882.2.a.f 1 7.c even 3 1
882.2.a.l 1 7.d odd 6 1
882.2.g.a 2 7.b odd 2 1
882.2.g.a 2 7.d odd 6 1
882.2.g.f 2 1.a even 1 1 trivial
882.2.g.f 2 7.c even 3 1 inner
2352.2.a.b 1 84.j odd 6 1
2352.2.a.y 1 84.n even 6 1
2352.2.q.a 2 12.b even 2 1
2352.2.q.a 2 84.n even 6 1
2352.2.q.y 2 84.h odd 2 1
2352.2.q.y 2 84.j odd 6 1
7056.2.a.a 1 28.g odd 6 1
7056.2.a.ca 1 28.f even 6 1
7350.2.a.br 1 105.p even 6 1
7350.2.a.cj 1 105.o odd 6 1
9408.2.a.b 1 168.v even 6 1
9408.2.a.bo 1 168.ba even 6 1
9408.2.a.br 1 168.s odd 6 1
9408.2.a.de 1 168.be odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{2} - 4 T_{5} + 16$$ $$T_{11}^{2} + 4 T_{11} + 16$$ $$T_{13} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ 1
$5$ $$1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4}$$
$7$ 1
$11$ $$1 + 4 T + 5 T^{2} + 44 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 4 T + 13 T^{2} )^{2}$$
$17$ $$1 - 17 T^{2} + 289 T^{4}$$
$19$ $$1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4}$$
$23$ $$1 - 23 T^{2} + 529 T^{4}$$
$29$ $$( 1 + 2 T + 29 T^{2} )^{2}$$
$31$ $$1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4}$$
$37$ $$1 - 6 T - T^{2} - 222 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 41 T^{2} )^{2}$$
$43$ $$( 1 - 4 T + 43 T^{2} )^{2}$$
$47$ $$1 + 8 T + 17 T^{2} + 376 T^{3} + 2209 T^{4}$$
$53$ $$1 + 10 T + 47 T^{2} + 530 T^{3} + 2809 T^{4}$$
$59$ $$1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4}$$
$61$ $$1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4}$$
$67$ $$1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 8 T + 71 T^{2} )^{2}$$
$73$ $$1 - 16 T + 183 T^{2} - 1168 T^{3} + 5329 T^{4}$$
$79$ $$1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4}$$
$83$ $$( 1 - 12 T + 83 T^{2} )^{2}$$
$89$ $$1 - 8 T - 25 T^{2} - 712 T^{3} + 7921 T^{4}$$
$97$ $$( 1 - 8 T + 97 T^{2} )^{2}$$