# Properties

 Label 8820.2.a.bf Level $8820$ Weight $2$ Character orbit 8820.a Self dual yes Analytic conductor $70.428$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8820.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$70.4280545828$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 420) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 3\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{5} +O(q^{10})$$ $$q - q^{5} + \beta q^{11} + ( 1 - \beta ) q^{13} + \beta q^{17} + 7 q^{19} + \beta q^{23} + q^{25} + ( 6 - \beta ) q^{29} + ( 1 + 2 \beta ) q^{31} + ( -1 + \beta ) q^{37} -\beta q^{41} + ( -1 + \beta ) q^{43} -6 q^{47} -2 \beta q^{53} -\beta q^{55} + ( -6 - \beta ) q^{59} + ( 4 - 2 \beta ) q^{61} + ( -1 + \beta ) q^{65} + ( -1 - \beta ) q^{67} + 3 \beta q^{71} + ( -5 - \beta ) q^{73} + 11 q^{79} + ( -6 - \beta ) q^{83} -\beta q^{85} + ( -6 - \beta ) q^{89} -7 q^{95} + ( -8 + 2 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} + O(q^{10})$$ $$2 q - 2 q^{5} + 2 q^{13} + 14 q^{19} + 2 q^{25} + 12 q^{29} + 2 q^{31} - 2 q^{37} - 2 q^{43} - 12 q^{47} - 12 q^{59} + 8 q^{61} - 2 q^{65} - 2 q^{67} - 10 q^{73} + 22 q^{79} - 12 q^{83} - 12 q^{89} - 14 q^{95} - 16 q^{97} + O(q^{100})$$

## 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
0 0 0 −1.00000 0 0 0 0 0
1.2 0 0 0 −1.00000 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8820.2.a.bf 2
3.b odd 2 1 2940.2.a.p 2
7.b odd 2 1 8820.2.a.bk 2
7.d odd 6 2 1260.2.s.e 4
21.c even 2 1 2940.2.a.r 2
21.g even 6 2 420.2.q.d 4
21.h odd 6 2 2940.2.q.q 4
84.j odd 6 2 1680.2.bg.t 4
105.p even 6 2 2100.2.q.k 4
105.w odd 12 4 2100.2.bc.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.d 4 21.g even 6 2
1260.2.s.e 4 7.d odd 6 2
1680.2.bg.t 4 84.j odd 6 2
2100.2.q.k 4 105.p even 6 2
2100.2.bc.f 8 105.w odd 12 4
2940.2.a.p 2 3.b odd 2 1
2940.2.a.r 2 21.c even 2 1
2940.2.q.q 4 21.h odd 6 2
8820.2.a.bf 2 1.a even 1 1 trivial
8820.2.a.bk 2 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8820))$$:

 $$T_{11}^{2} - 18$$ $$T_{13}^{2} - 2 T_{13} - 17$$ $$T_{17}^{2} - 18$$ $$T_{31}^{2} - 2 T_{31} - 71$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$T^{2}$$
$11$ $$-18 + T^{2}$$
$13$ $$-17 - 2 T + T^{2}$$
$17$ $$-18 + T^{2}$$
$19$ $$( -7 + T )^{2}$$
$23$ $$-18 + T^{2}$$
$29$ $$18 - 12 T + T^{2}$$
$31$ $$-71 - 2 T + T^{2}$$
$37$ $$-17 + 2 T + T^{2}$$
$41$ $$-18 + T^{2}$$
$43$ $$-17 + 2 T + T^{2}$$
$47$ $$( 6 + T )^{2}$$
$53$ $$-72 + T^{2}$$
$59$ $$18 + 12 T + T^{2}$$
$61$ $$-56 - 8 T + T^{2}$$
$67$ $$-17 + 2 T + T^{2}$$
$71$ $$-162 + T^{2}$$
$73$ $$7 + 10 T + T^{2}$$
$79$ $$( -11 + T )^{2}$$
$83$ $$18 + 12 T + T^{2}$$
$89$ $$18 + 12 T + T^{2}$$
$97$ $$-8 + 16 T + T^{2}$$