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

## 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(882))$$:

 $$T_{5}^{2} - 88$$ T5^2 - 88 $$T_{11} + 20$$ T11 + 20 $$T_{13}^{2} - 4312$$ T13^2 - 4312

## Hecke characteristic polynomials

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