# Properties

 Label 882.2.a.m Level 882 Weight 2 Character orbit 882.a Self dual yes Analytic conductor 7.043 Analytic rank 1 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + \beta q^{5} - q^{8} +O(q^{10})$$ $$q - q^{2} + q^{4} + \beta q^{5} - q^{8} -\beta q^{10} -4 q^{11} -3 \beta q^{13} + q^{16} -5 \beta q^{17} + 4 \beta q^{19} + \beta q^{20} + 4 q^{22} -8 q^{23} -3 q^{25} + 3 \beta q^{26} -2 q^{29} - q^{32} + 5 \beta q^{34} + 4 q^{37} -4 \beta q^{38} -\beta q^{40} + 7 \beta q^{41} -4 q^{43} -4 q^{44} + 8 q^{46} + 4 \beta q^{47} + 3 q^{50} -3 \beta q^{52} -4 q^{53} -4 \beta q^{55} + 2 q^{58} -8 \beta q^{59} -\beta q^{61} + q^{64} -6 q^{65} -12 q^{67} -5 \beta q^{68} + 11 \beta q^{73} -4 q^{74} + 4 \beta q^{76} -16 q^{79} + \beta q^{80} -7 \beta q^{82} -4 \beta q^{83} -10 q^{85} + 4 q^{86} + 4 q^{88} + 5 \beta q^{89} -8 q^{92} -4 \beta q^{94} + 8 q^{95} -5 \beta q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} - 2q^{8} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} - 2q^{8} - 8q^{11} + 2q^{16} + 8q^{22} - 16q^{23} - 6q^{25} - 4q^{29} - 2q^{32} + 8q^{37} - 8q^{43} - 8q^{44} + 16q^{46} + 6q^{50} - 8q^{53} + 4q^{58} + 2q^{64} - 12q^{65} - 24q^{67} - 8q^{74} - 32q^{79} - 20q^{85} + 8q^{86} + 8q^{88} - 16q^{92} + 16q^{95} + O(q^{100})$$

## 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 −1.41421 0 0 −1.00000 0 1.41421
1.2 −1.00000 0 1.00000 1.41421 0 0 −1.00000 0 −1.41421
 $$n$$: 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.m 2
3.b odd 2 1 882.2.a.o yes 2
4.b odd 2 1 7056.2.a.cs 2
7.b odd 2 1 inner 882.2.a.m 2
7.c even 3 2 882.2.g.m 4
7.d odd 6 2 882.2.g.m 4
12.b even 2 1 7056.2.a.ci 2
21.c even 2 1 882.2.a.o yes 2
21.g even 6 2 882.2.g.k 4
21.h odd 6 2 882.2.g.k 4
28.d even 2 1 7056.2.a.cs 2
84.h odd 2 1 7056.2.a.ci 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.2.a.m 2 1.a even 1 1 trivial
882.2.a.m 2 7.b odd 2 1 inner
882.2.a.o yes 2 3.b odd 2 1
882.2.a.o yes 2 21.c even 2 1
882.2.g.k 4 21.g even 6 2
882.2.g.k 4 21.h odd 6 2
882.2.g.m 4 7.c even 3 2
882.2.g.m 4 7.d odd 6 2
7056.2.a.ci 2 12.b even 2 1
7056.2.a.ci 2 84.h odd 2 1
7056.2.a.cs 2 4.b odd 2 1
7056.2.a.cs 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(882))$$:

 $$T_{5}^{2} - 2$$ $$T_{11} + 4$$ $$T_{13}^{2} - 18$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ 1
$5$ $$1 + 8 T^{2} + 25 T^{4}$$
$7$ 1
$11$ $$( 1 + 4 T + 11 T^{2} )^{2}$$
$13$ $$1 + 8 T^{2} + 169 T^{4}$$
$17$ $$1 - 16 T^{2} + 289 T^{4}$$
$19$ $$1 + 6 T^{2} + 361 T^{4}$$
$23$ $$( 1 + 8 T + 23 T^{2} )^{2}$$
$29$ $$( 1 + 2 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 31 T^{2} )^{2}$$
$37$ $$( 1 - 4 T + 37 T^{2} )^{2}$$
$41$ $$1 - 16 T^{2} + 1681 T^{4}$$
$43$ $$( 1 + 4 T + 43 T^{2} )^{2}$$
$47$ $$1 + 62 T^{2} + 2209 T^{4}$$
$53$ $$( 1 + 4 T + 53 T^{2} )^{2}$$
$59$ $$1 - 10 T^{2} + 3481 T^{4}$$
$61$ $$1 + 120 T^{2} + 3721 T^{4}$$
$67$ $$( 1 + 12 T + 67 T^{2} )^{2}$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$1 - 96 T^{2} + 5329 T^{4}$$
$79$ $$( 1 + 16 T + 79 T^{2} )^{2}$$
$83$ $$1 + 134 T^{2} + 6889 T^{4}$$
$89$ $$1 + 128 T^{2} + 7921 T^{4}$$
$97$ $$1 + 144 T^{2} + 9409 T^{4}$$