# Properties

 Label 882.4.a.w Level $882$ Weight $4$ Character orbit 882.a Self dual yes 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.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$52.0396846251$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{22})$$ Defining polynomial: $$x^{2} - 22$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 98) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{22}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 q^{2} + 4 q^{4} + \beta q^{5} -8 q^{8} +O(q^{10})$$ $$q -2 q^{2} + 4 q^{4} + \beta q^{5} -8 q^{8} -2 \beta q^{10} -20 q^{11} -7 \beta q^{13} + 16 q^{16} + 6 \beta q^{17} -\beta q^{19} + 4 \beta q^{20} + 40 q^{22} -48 q^{23} -37 q^{25} + 14 \beta q^{26} + 166 q^{29} + 22 \beta q^{31} -32 q^{32} -12 \beta q^{34} -78 q^{37} + 2 \beta q^{38} -8 \beta q^{40} + 42 \beta q^{41} + 436 q^{43} -80 q^{44} + 96 q^{46} + 22 \beta q^{47} + 74 q^{50} -28 \beta q^{52} -62 q^{53} -20 \beta q^{55} -332 q^{58} -71 \beta q^{59} -29 \beta q^{61} -44 \beta q^{62} + 64 q^{64} -616 q^{65} + 580 q^{67} + 24 \beta q^{68} + 544 q^{71} + 64 \beta q^{73} + 156 q^{74} -4 \beta q^{76} -680 q^{79} + 16 \beta q^{80} -84 \beta q^{82} + 21 \beta q^{83} + 528 q^{85} -872 q^{86} + 160 q^{88} -160 \beta q^{89} -192 q^{92} -44 \beta q^{94} -88 q^{95} + 70 \beta q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{2} + 8q^{4} - 16q^{8} + O(q^{10})$$ $$2q - 4q^{2} + 8q^{4} - 16q^{8} - 40q^{11} + 32q^{16} + 80q^{22} - 96q^{23} - 74q^{25} + 332q^{29} - 64q^{32} - 156q^{37} + 872q^{43} - 160q^{44} + 192q^{46} + 148q^{50} - 124q^{53} - 664q^{58} + 128q^{64} - 1232q^{65} + 1160q^{67} + 1088q^{71} + 312q^{74} - 1360q^{79} + 1056q^{85} - 1744q^{86} + 320q^{88} - 384q^{92} - 176q^{95} + O(q^{100})$$

## 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
 −4.69042 4.69042
−2.00000 0 4.00000 −9.38083 0 0 −8.00000 0 18.7617
1.2 −2.00000 0 4.00000 9.38083 0 0 −8.00000 0 −18.7617
 $$n$$: 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.w 2
3.b odd 2 1 98.4.a.h 2
7.b odd 2 1 inner 882.4.a.w 2
7.c even 3 2 882.4.g.bi 4
7.d odd 6 2 882.4.g.bi 4
12.b even 2 1 784.4.a.z 2
15.d odd 2 1 2450.4.a.bs 2
21.c even 2 1 98.4.a.h 2
21.g even 6 2 98.4.c.g 4
21.h odd 6 2 98.4.c.g 4
84.h odd 2 1 784.4.a.z 2
105.g even 2 1 2450.4.a.bs 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.4.a.h 2 3.b odd 2 1
98.4.a.h 2 21.c even 2 1
98.4.c.g 4 21.g even 6 2
98.4.c.g 4 21.h odd 6 2
784.4.a.z 2 12.b even 2 1
784.4.a.z 2 84.h odd 2 1
882.4.a.w 2 1.a even 1 1 trivial
882.4.a.w 2 7.b odd 2 1 inner
882.4.g.bi 4 7.c even 3 2
882.4.g.bi 4 7.d odd 6 2
2450.4.a.bs 2 15.d odd 2 1
2450.4.a.bs 2 105.g even 2 1

## Hecke kernels

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

 $$T_{5}^{2} - 88$$ $$T_{11} + 20$$ $$T_{13}^{2} - 4312$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$-88 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 20 + T )^{2}$$
$13$ $$-4312 + T^{2}$$
$17$ $$-3168 + T^{2}$$
$19$ $$-88 + T^{2}$$
$23$ $$( 48 + T )^{2}$$
$29$ $$( -166 + T )^{2}$$
$31$ $$-42592 + T^{2}$$
$37$ $$( 78 + T )^{2}$$
$41$ $$-155232 + T^{2}$$
$43$ $$( -436 + T )^{2}$$
$47$ $$-42592 + T^{2}$$
$53$ $$( 62 + T )^{2}$$
$59$ $$-443608 + T^{2}$$
$61$ $$-74008 + T^{2}$$
$67$ $$( -580 + T )^{2}$$
$71$ $$( -544 + T )^{2}$$
$73$ $$-360448 + T^{2}$$
$79$ $$( 680 + T )^{2}$$
$83$ $$-38808 + T^{2}$$
$89$ $$-2252800 + T^{2}$$
$97$ $$-431200 + T^{2}$$