# Properties

 Label 882.3.b.d Level $882$ Weight $3$ Character orbit 882.b Analytic conductor $24.033$ 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: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 882.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.0327593166$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 126) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} - 2 q^{4} + \beta q^{5} + 2 \beta q^{8} +O(q^{10})$$ q - b * q^2 - 2 * q^4 + b * q^5 + 2*b * q^8 $$q - \beta q^{2} - 2 q^{4} + \beta q^{5} + 2 \beta q^{8} + 2 q^{10} - 5 \beta q^{11} - 15 q^{13} + 4 q^{16} + 8 \beta q^{17} + 13 q^{19} - 2 \beta q^{20} - 10 q^{22} + 16 \beta q^{23} + 23 q^{25} + 15 \beta q^{26} - 16 \beta q^{29} - 3 q^{31} - 4 \beta q^{32} + 16 q^{34} + 17 q^{37} - 13 \beta q^{38} - 4 q^{40} + 57 \beta q^{41} - 85 q^{43} + 10 \beta q^{44} + 32 q^{46} + 51 \beta q^{47} - 23 \beta q^{50} + 30 q^{52} + 24 \beta q^{53} + 10 q^{55} - 32 q^{58} + 64 \beta q^{59} + 72 q^{61} + 3 \beta q^{62} - 8 q^{64} - 15 \beta q^{65} + 43 q^{67} - 16 \beta q^{68} + 37 \beta q^{71} + 95 q^{73} - 17 \beta q^{74} - 26 q^{76} + 69 q^{79} + 4 \beta q^{80} + 114 q^{82} - 43 \beta q^{83} - 16 q^{85} + 85 \beta q^{86} + 20 q^{88} + 96 \beta q^{89} - 32 \beta q^{92} + 102 q^{94} + 13 \beta q^{95} - 16 q^{97} +O(q^{100})$$ q - b * q^2 - 2 * q^4 + b * q^5 + 2*b * q^8 + 2 * q^10 - 5*b * q^11 - 15 * q^13 + 4 * q^16 + 8*b * q^17 + 13 * q^19 - 2*b * q^20 - 10 * q^22 + 16*b * q^23 + 23 * q^25 + 15*b * q^26 - 16*b * q^29 - 3 * q^31 - 4*b * q^32 + 16 * q^34 + 17 * q^37 - 13*b * q^38 - 4 * q^40 + 57*b * q^41 - 85 * q^43 + 10*b * q^44 + 32 * q^46 + 51*b * q^47 - 23*b * q^50 + 30 * q^52 + 24*b * q^53 + 10 * q^55 - 32 * q^58 + 64*b * q^59 + 72 * q^61 + 3*b * q^62 - 8 * q^64 - 15*b * q^65 + 43 * q^67 - 16*b * q^68 + 37*b * q^71 + 95 * q^73 - 17*b * q^74 - 26 * q^76 + 69 * q^79 + 4*b * q^80 + 114 * q^82 - 43*b * q^83 - 16 * q^85 + 85*b * q^86 + 20 * q^88 + 96*b * q^89 - 32*b * q^92 + 102 * q^94 + 13*b * q^95 - 16 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4}+O(q^{10})$$ 2 * q - 4 * q^4 $$2 q - 4 q^{4} + 4 q^{10} - 30 q^{13} + 8 q^{16} + 26 q^{19} - 20 q^{22} + 46 q^{25} - 6 q^{31} + 32 q^{34} + 34 q^{37} - 8 q^{40} - 170 q^{43} + 64 q^{46} + 60 q^{52} + 20 q^{55} - 64 q^{58} + 144 q^{61} - 16 q^{64} + 86 q^{67} + 190 q^{73} - 52 q^{76} + 138 q^{79} + 228 q^{82} - 32 q^{85} + 40 q^{88} + 204 q^{94} - 32 q^{97}+O(q^{100})$$ 2 * q - 4 * q^4 + 4 * q^10 - 30 * q^13 + 8 * q^16 + 26 * q^19 - 20 * q^22 + 46 * q^25 - 6 * q^31 + 32 * q^34 + 34 * q^37 - 8 * q^40 - 170 * q^43 + 64 * q^46 + 60 * q^52 + 20 * q^55 - 64 * q^58 + 144 * q^61 - 16 * q^64 + 86 * q^67 + 190 * q^73 - 52 * q^76 + 138 * q^79 + 228 * q^82 - 32 * q^85 + 40 * q^88 + 204 * q^94 - 32 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$785$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
197.1
 1.41421i − 1.41421i
1.41421i 0 −2.00000 1.41421i 0 0 2.82843i 0 2.00000
197.2 1.41421i 0 −2.00000 1.41421i 0 0 2.82843i 0 2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.3.b.d 2
3.b odd 2 1 inner 882.3.b.d 2
7.b odd 2 1 882.3.b.c 2
7.c even 3 2 882.3.s.c 4
7.d odd 6 2 126.3.s.b 4
21.c even 2 1 882.3.b.c 2
21.g even 6 2 126.3.s.b 4
21.h odd 6 2 882.3.s.c 4
28.f even 6 2 1008.3.dc.a 4
84.j odd 6 2 1008.3.dc.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.s.b 4 7.d odd 6 2
126.3.s.b 4 21.g even 6 2
882.3.b.c 2 7.b odd 2 1
882.3.b.c 2 21.c even 2 1
882.3.b.d 2 1.a even 1 1 trivial
882.3.b.d 2 3.b odd 2 1 inner
882.3.s.c 4 7.c even 3 2
882.3.s.c 4 21.h odd 6 2
1008.3.dc.a 4 28.f even 6 2
1008.3.dc.a 4 84.j odd 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{2} + 2$$ T5^2 + 2 $$T_{11}^{2} + 50$$ T11^2 + 50 $$T_{13} + 15$$ T13 + 15

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 50$$
$13$ $$(T + 15)^{2}$$
$17$ $$T^{2} + 128$$
$19$ $$(T - 13)^{2}$$
$23$ $$T^{2} + 512$$
$29$ $$T^{2} + 512$$
$31$ $$(T + 3)^{2}$$
$37$ $$(T - 17)^{2}$$
$41$ $$T^{2} + 6498$$
$43$ $$(T + 85)^{2}$$
$47$ $$T^{2} + 5202$$
$53$ $$T^{2} + 1152$$
$59$ $$T^{2} + 8192$$
$61$ $$(T - 72)^{2}$$
$67$ $$(T - 43)^{2}$$
$71$ $$T^{2} + 2738$$
$73$ $$(T - 95)^{2}$$
$79$ $$(T - 69)^{2}$$
$83$ $$T^{2} + 3698$$
$89$ $$T^{2} + 18432$$
$97$ $$(T + 16)^{2}$$