# Properties

 Label 882.2.a.n Level $882$ Weight $2$ Character orbit 882.a Self dual yes Analytic conductor $7.043$ 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: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 882.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.04280545828$$ Analytic rank: $$0$$ 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: 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + 2 \beta q^{5} - q^{8} +O(q^{10})$$ $$q - q^{2} + q^{4} + 2 \beta q^{5} - q^{8} -2 \beta q^{10} + 2 q^{11} + q^{16} -\beta q^{17} + 5 \beta q^{19} + 2 \beta q^{20} -2 q^{22} + 4 q^{23} + 3 q^{25} -2 q^{29} -6 \beta q^{31} - q^{32} + \beta q^{34} + 10 q^{37} -5 \beta q^{38} -2 \beta q^{40} -7 \beta q^{41} + 2 q^{43} + 2 q^{44} -4 q^{46} + 2 \beta q^{47} -3 q^{50} + 2 q^{53} + 4 \beta q^{55} + 2 q^{58} -\beta q^{59} -2 \beta q^{61} + 6 \beta q^{62} + q^{64} + 12 q^{67} -\beta q^{68} + 12 q^{71} + \beta q^{73} -10 q^{74} + 5 \beta q^{76} -4 q^{79} + 2 \beta q^{80} + 7 \beta q^{82} + 7 \beta q^{83} -4 q^{85} -2 q^{86} -2 q^{88} -5 \beta q^{89} + 4 q^{92} -2 \beta q^{94} + 20 q^{95} -7 \beta q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + O(q^{10})$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 4 q^{11} + 2 q^{16} - 4 q^{22} + 8 q^{23} + 6 q^{25} - 4 q^{29} - 2 q^{32} + 20 q^{37} + 4 q^{43} + 4 q^{44} - 8 q^{46} - 6 q^{50} + 4 q^{53} + 4 q^{58} + 2 q^{64} + 24 q^{67} + 24 q^{71} - 20 q^{74} - 8 q^{79} - 8 q^{85} - 4 q^{86} - 4 q^{88} + 8 q^{92} + 40 q^{95} + O(q^{100})$$

## 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
−1.00000 0 1.00000 −2.82843 0 0 −1.00000 0 2.82843
1.2 −1.00000 0 1.00000 2.82843 0 0 −1.00000 0 −2.82843
 $$n$$: 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 882.2.a.n 2
3.b odd 2 1 98.2.a.b 2
4.b odd 2 1 7056.2.a.cl 2
7.b odd 2 1 inner 882.2.a.n 2
7.c even 3 2 882.2.g.l 4
7.d odd 6 2 882.2.g.l 4
12.b even 2 1 784.2.a.l 2
15.d odd 2 1 2450.2.a.bj 2
15.e even 4 2 2450.2.c.v 4
21.c even 2 1 98.2.a.b 2
21.g even 6 2 98.2.c.c 4
21.h odd 6 2 98.2.c.c 4
24.f even 2 1 3136.2.a.bm 2
24.h odd 2 1 3136.2.a.bn 2
28.d even 2 1 7056.2.a.cl 2
84.h odd 2 1 784.2.a.l 2
84.j odd 6 2 784.2.i.m 4
84.n even 6 2 784.2.i.m 4
105.g even 2 1 2450.2.a.bj 2
105.k odd 4 2 2450.2.c.v 4
168.e odd 2 1 3136.2.a.bm 2
168.i even 2 1 3136.2.a.bn 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.2.a.b 2 3.b odd 2 1
98.2.a.b 2 21.c even 2 1
98.2.c.c 4 21.g even 6 2
98.2.c.c 4 21.h odd 6 2
784.2.a.l 2 12.b even 2 1
784.2.a.l 2 84.h odd 2 1
784.2.i.m 4 84.j odd 6 2
784.2.i.m 4 84.n even 6 2
882.2.a.n 2 1.a even 1 1 trivial
882.2.a.n 2 7.b odd 2 1 inner
882.2.g.l 4 7.c even 3 2
882.2.g.l 4 7.d odd 6 2
2450.2.a.bj 2 15.d odd 2 1
2450.2.a.bj 2 105.g even 2 1
2450.2.c.v 4 15.e even 4 2
2450.2.c.v 4 105.k odd 4 2
3136.2.a.bm 2 24.f even 2 1
3136.2.a.bm 2 168.e odd 2 1
3136.2.a.bn 2 24.h odd 2 1
3136.2.a.bn 2 168.i even 2 1
7056.2.a.cl 2 4.b odd 2 1
7056.2.a.cl 2 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(882))$$:

 $$T_{5}^{2} - 8$$ $$T_{11} - 2$$ $$T_{13}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$-8 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -2 + T )^{2}$$
$13$ $$T^{2}$$
$17$ $$-2 + T^{2}$$
$19$ $$-50 + T^{2}$$
$23$ $$( -4 + T )^{2}$$
$29$ $$( 2 + T )^{2}$$
$31$ $$-72 + T^{2}$$
$37$ $$( -10 + T )^{2}$$
$41$ $$-98 + T^{2}$$
$43$ $$( -2 + T )^{2}$$
$47$ $$-8 + T^{2}$$
$53$ $$( -2 + T )^{2}$$
$59$ $$-2 + T^{2}$$
$61$ $$-8 + T^{2}$$
$67$ $$( -12 + T )^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$-2 + T^{2}$$
$79$ $$( 4 + T )^{2}$$
$83$ $$-98 + T^{2}$$
$89$ $$-50 + T^{2}$$
$97$ $$-98 + T^{2}$$