# Properties

 Label 882.4.a.l Level $882$ Weight $4$ Character orbit 882.a Self dual yes Analytic conductor $52.040$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2q^{2} + 4q^{4} - 6q^{5} + 8q^{8} + O(q^{10})$$ $$q + 2q^{2} + 4q^{4} - 6q^{5} + 8q^{8} - 12q^{10} + 30q^{11} - 53q^{13} + 16q^{16} - 84q^{17} + 97q^{19} - 24q^{20} + 60q^{22} - 84q^{23} - 89q^{25} - 106q^{26} + 180q^{29} - 179q^{31} + 32q^{32} - 168q^{34} - 145q^{37} + 194q^{38} - 48q^{40} + 126q^{41} - 325q^{43} + 120q^{44} - 168q^{46} - 366q^{47} - 178q^{50} - 212q^{52} + 768q^{53} - 180q^{55} + 360q^{58} - 264q^{59} - 818q^{61} - 358q^{62} + 64q^{64} + 318q^{65} - 523q^{67} - 336q^{68} + 342q^{71} + 43q^{73} - 290q^{74} + 388q^{76} - 1171q^{79} - 96q^{80} + 252q^{82} - 810q^{83} + 504q^{85} - 650q^{86} + 240q^{88} - 600q^{89} - 336q^{92} - 732q^{94} - 582q^{95} - 386q^{97} + O(q^{100})$$

## 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 −6.00000 0 0 8.00000 0 −12.0000
 $$n$$: 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.l 1
3.b odd 2 1 294.4.a.c 1
7.b odd 2 1 882.4.a.o 1
7.c even 3 2 882.4.g.g 2
7.d odd 6 2 126.4.g.b 2
12.b even 2 1 2352.4.a.bf 1
21.c even 2 1 294.4.a.d 1
21.g even 6 2 42.4.e.a 2
21.h odd 6 2 294.4.e.i 2
84.h odd 2 1 2352.4.a.f 1
84.j odd 6 2 336.4.q.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.a 2 21.g even 6 2
126.4.g.b 2 7.d odd 6 2
294.4.a.c 1 3.b odd 2 1
294.4.a.d 1 21.c even 2 1
294.4.e.i 2 21.h odd 6 2
336.4.q.f 2 84.j odd 6 2
882.4.a.l 1 1.a even 1 1 trivial
882.4.a.o 1 7.b odd 2 1
882.4.g.g 2 7.c even 3 2
2352.4.a.f 1 84.h odd 2 1
2352.4.a.bf 1 12.b even 2 1

## Hecke kernels

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

 $$T_{5} + 6$$ $$T_{11} - 30$$ $$T_{13} + 53$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + T$$
$3$ $$T$$
$5$ $$6 + T$$
$7$ $$T$$
$11$ $$-30 + T$$
$13$ $$53 + T$$
$17$ $$84 + T$$
$19$ $$-97 + T$$
$23$ $$84 + T$$
$29$ $$-180 + T$$
$31$ $$179 + T$$
$37$ $$145 + T$$
$41$ $$-126 + T$$
$43$ $$325 + T$$
$47$ $$366 + T$$
$53$ $$-768 + T$$
$59$ $$264 + T$$
$61$ $$818 + T$$
$67$ $$523 + T$$
$71$ $$-342 + T$$
$73$ $$-43 + T$$
$79$ $$1171 + T$$
$83$ $$810 + T$$
$89$ $$600 + T$$
$97$ $$386 + T$$