# Properties

 Label 882.4.a.be Level $882$ Weight $4$ Character orbit 882.a Self dual yes Analytic conductor $52.040$ Analytic rank $1$ 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.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$52.0396846251$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 4 q^{4} + 5 \beta q^{5} + 8 q^{8} +O(q^{10})$$ $$q + 2 q^{2} + 4 q^{4} + 5 \beta q^{5} + 8 q^{8} + 10 \beta q^{10} -40 q^{11} -45 \beta q^{13} + 16 q^{16} -\beta q^{17} + 8 \beta q^{19} + 20 \beta q^{20} -80 q^{22} -68 q^{23} -75 q^{25} -90 \beta q^{26} -110 q^{29} -84 \beta q^{31} + 32 q^{32} -2 \beta q^{34} -20 q^{37} + 16 \beta q^{38} + 40 \beta q^{40} + 35 \beta q^{41} -340 q^{43} -160 q^{44} -136 q^{46} -64 \beta q^{47} -150 q^{50} -180 \beta q^{52} -628 q^{53} -200 \beta q^{55} -220 q^{58} + 620 \beta q^{59} + 649 \beta q^{61} -168 \beta q^{62} + 64 q^{64} -450 q^{65} + 540 q^{67} -4 \beta q^{68} + 420 q^{71} + 205 \beta q^{73} -40 q^{74} + 32 \beta q^{76} -760 q^{79} + 80 \beta q^{80} + 70 \beta q^{82} -668 \beta q^{83} -10 q^{85} -680 q^{86} -320 q^{88} -815 \beta q^{89} -272 q^{92} -128 \beta q^{94} + 80 q^{95} -355 \beta q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{2} + 8q^{4} + 16q^{8} + O(q^{10})$$ $$2q + 4q^{2} + 8q^{4} + 16q^{8} - 80q^{11} + 32q^{16} - 160q^{22} - 136q^{23} - 150q^{25} - 220q^{29} + 64q^{32} - 40q^{37} - 680q^{43} - 320q^{44} - 272q^{46} - 300q^{50} - 1256q^{53} - 440q^{58} + 128q^{64} - 900q^{65} + 1080q^{67} + 840q^{71} - 80q^{74} - 1520q^{79} - 20q^{85} - 1360q^{86} - 640q^{88} - 544q^{92} + 160q^{95} + O(q^{100})$$

## 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
 −1.41421 1.41421
2.00000 0 4.00000 −7.07107 0 0 8.00000 0 −14.1421
1.2 2.00000 0 4.00000 7.07107 0 0 8.00000 0 14.1421
 $$n$$: 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.4.a.be yes 2
3.b odd 2 1 882.4.a.y 2
7.b odd 2 1 inner 882.4.a.be yes 2
7.c even 3 2 882.4.g.bc 4
7.d odd 6 2 882.4.g.bc 4
21.c even 2 1 882.4.a.y 2
21.g even 6 2 882.4.g.bg 4
21.h odd 6 2 882.4.g.bg 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.4.a.y 2 3.b odd 2 1
882.4.a.y 2 21.c even 2 1
882.4.a.be yes 2 1.a even 1 1 trivial
882.4.a.be yes 2 7.b odd 2 1 inner
882.4.g.bc 4 7.c even 3 2
882.4.g.bc 4 7.d odd 6 2
882.4.g.bg 4 21.g even 6 2
882.4.g.bg 4 21.h odd 6 2

## Hecke kernels

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

 $$T_{5}^{2} - 50$$ $$T_{11} + 40$$ $$T_{13}^{2} - 4050$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$-50 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 40 + T )^{2}$$
$13$ $$-4050 + T^{2}$$
$17$ $$-2 + T^{2}$$
$19$ $$-128 + T^{2}$$
$23$ $$( 68 + T )^{2}$$
$29$ $$( 110 + T )^{2}$$
$31$ $$-14112 + T^{2}$$
$37$ $$( 20 + T )^{2}$$
$41$ $$-2450 + T^{2}$$
$43$ $$( 340 + T )^{2}$$
$47$ $$-8192 + T^{2}$$
$53$ $$( 628 + T )^{2}$$
$59$ $$-768800 + T^{2}$$
$61$ $$-842402 + T^{2}$$
$67$ $$( -540 + T )^{2}$$
$71$ $$( -420 + T )^{2}$$
$73$ $$-84050 + T^{2}$$
$79$ $$( 760 + T )^{2}$$
$83$ $$-892448 + T^{2}$$
$89$ $$-1328450 + T^{2}$$
$97$ $$-252050 + T^{2}$$