# Properties

 Label 882.4.a.s Level $882$ Weight $4$ Character orbit 882.a Self dual yes Analytic conductor $52.040$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 882.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$52.0396846251$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 126) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2q^{2} + 4q^{4} + 22q^{5} + 8q^{8} + O(q^{10})$$ $$q + 2q^{2} + 4q^{4} + 22q^{5} + 8q^{8} + 44q^{10} - 26q^{11} + 54q^{13} + 16q^{16} + 74q^{17} - 116q^{19} + 88q^{20} - 52q^{22} + 58q^{23} + 359q^{25} + 108q^{26} - 208q^{29} + 252q^{31} + 32q^{32} + 148q^{34} + 50q^{37} - 232q^{38} + 176q^{40} + 126q^{41} + 164q^{43} - 104q^{44} + 116q^{46} - 444q^{47} + 718q^{50} + 216q^{52} - 12q^{53} - 572q^{55} - 416q^{58} + 124q^{59} + 162q^{61} + 504q^{62} + 64q^{64} + 1188q^{65} - 860q^{67} + 296q^{68} + 238q^{71} + 146q^{73} + 100q^{74} - 464q^{76} - 984q^{79} + 352q^{80} + 252q^{82} + 656q^{83} + 1628q^{85} + 328q^{86} - 208q^{88} - 954q^{89} + 232q^{92} - 888q^{94} - 2552q^{95} - 526q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
2.00000 0 4.00000 22.0000 0 0 8.00000 0 44.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.4.a.s 1
3.b odd 2 1 882.4.a.a 1
7.b odd 2 1 126.4.a.f yes 1
7.c even 3 2 882.4.g.a 2
7.d odd 6 2 882.4.g.m 2
21.c even 2 1 126.4.a.e 1
21.g even 6 2 882.4.g.n 2
21.h odd 6 2 882.4.g.x 2
28.d even 2 1 1008.4.a.a 1
84.h odd 2 1 1008.4.a.w 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.a.e 1 21.c even 2 1
126.4.a.f yes 1 7.b odd 2 1
882.4.a.a 1 3.b odd 2 1
882.4.a.s 1 1.a even 1 1 trivial
882.4.g.a 2 7.c even 3 2
882.4.g.m 2 7.d odd 6 2
882.4.g.n 2 21.g even 6 2
882.4.g.x 2 21.h odd 6 2
1008.4.a.a 1 28.d even 2 1
1008.4.a.w 1 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(882))$$:

 $$T_{5} - 22$$ $$T_{11} + 26$$ $$T_{13} - 54$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + T$$
$3$ $$T$$
$5$ $$-22 + T$$
$7$ $$T$$
$11$ $$26 + T$$
$13$ $$-54 + T$$
$17$ $$-74 + T$$
$19$ $$116 + T$$
$23$ $$-58 + T$$
$29$ $$208 + T$$
$31$ $$-252 + T$$
$37$ $$-50 + T$$
$41$ $$-126 + T$$
$43$ $$-164 + T$$
$47$ $$444 + T$$
$53$ $$12 + T$$
$59$ $$-124 + T$$
$61$ $$-162 + T$$
$67$ $$860 + T$$
$71$ $$-238 + T$$
$73$ $$-146 + T$$
$79$ $$984 + T$$
$83$ $$-656 + T$$
$89$ $$954 + T$$
$97$ $$526 + T$$