# Properties

 Label 882.3.b.e 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: $$x^{2} + 2$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} -2 q^{4} + 3 \beta q^{5} + 2 \beta q^{8} +O(q^{10})$$ $$q -\beta q^{2} -2 q^{4} + 3 \beta q^{5} + 2 \beta q^{8} + 6 q^{10} + 9 \beta q^{11} + q^{13} + 4 q^{16} -12 \beta q^{17} -23 q^{19} -6 \beta q^{20} + 18 q^{22} + 12 \beta q^{23} + 7 q^{25} -\beta q^{26} -24 \beta q^{29} -47 q^{31} -4 \beta q^{32} -24 q^{34} -55 q^{37} + 23 \beta q^{38} -12 q^{40} -33 \beta q^{41} + 23 q^{43} -18 \beta q^{44} + 24 q^{46} -3 \beta q^{47} -7 \beta q^{50} -2 q^{52} + 36 \beta q^{53} -54 q^{55} -48 q^{58} + 60 \beta q^{59} -104 q^{61} + 47 \beta q^{62} -8 q^{64} + 3 \beta q^{65} -97 q^{67} + 24 \beta q^{68} -69 \beta q^{71} -65 q^{73} + 55 \beta q^{74} + 46 q^{76} + 113 q^{79} + 12 \beta q^{80} -66 q^{82} -21 \beta q^{83} + 72 q^{85} -23 \beta q^{86} -36 q^{88} + 96 \beta q^{89} -24 \beta q^{92} -6 q^{94} -69 \beta q^{95} -104 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{4} + O(q^{10})$$ $$2q - 4q^{4} + 12q^{10} + 2q^{13} + 8q^{16} - 46q^{19} + 36q^{22} + 14q^{25} - 94q^{31} - 48q^{34} - 110q^{37} - 24q^{40} + 46q^{43} + 48q^{46} - 4q^{52} - 108q^{55} - 96q^{58} - 208q^{61} - 16q^{64} - 194q^{67} - 130q^{73} + 92q^{76} + 226q^{79} - 132q^{82} + 144q^{85} - 72q^{88} - 12q^{94} - 208q^{97} + O(q^{100})$$

## 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 4.24264i 0 0 2.82843i 0 6.00000
197.2 1.41421i 0 −2.00000 4.24264i 0 0 2.82843i 0 6.00000
 $$n$$: 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.e 2
3.b odd 2 1 inner 882.3.b.e 2
7.b odd 2 1 882.3.b.b 2
7.c even 3 2 882.3.s.a 4
7.d odd 6 2 126.3.s.a 4
21.c even 2 1 882.3.b.b 2
21.g even 6 2 126.3.s.a 4
21.h odd 6 2 882.3.s.a 4
28.f even 6 2 1008.3.dc.b 4
84.j odd 6 2 1008.3.dc.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.s.a 4 7.d odd 6 2
126.3.s.a 4 21.g even 6 2
882.3.b.b 2 7.b odd 2 1
882.3.b.b 2 21.c even 2 1
882.3.b.e 2 1.a even 1 1 trivial
882.3.b.e 2 3.b odd 2 1 inner
882.3.s.a 4 7.c even 3 2
882.3.s.a 4 21.h odd 6 2
1008.3.dc.b 4 28.f even 6 2
1008.3.dc.b 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 $$S_{3}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{2} + 18$$ $$T_{11}^{2} + 162$$ $$T_{13} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$18 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$162 + T^{2}$$
$13$ $$( -1 + T )^{2}$$
$17$ $$288 + T^{2}$$
$19$ $$( 23 + T )^{2}$$
$23$ $$288 + T^{2}$$
$29$ $$1152 + T^{2}$$
$31$ $$( 47 + T )^{2}$$
$37$ $$( 55 + T )^{2}$$
$41$ $$2178 + T^{2}$$
$43$ $$( -23 + T )^{2}$$
$47$ $$18 + T^{2}$$
$53$ $$2592 + T^{2}$$
$59$ $$7200 + T^{2}$$
$61$ $$( 104 + T )^{2}$$
$67$ $$( 97 + T )^{2}$$
$71$ $$9522 + T^{2}$$
$73$ $$( 65 + T )^{2}$$
$79$ $$( -113 + T )^{2}$$
$83$ $$882 + T^{2}$$
$89$ $$18432 + T^{2}$$
$97$ $$( 104 + T )^{2}$$