# Properties

 Label 8820.2.a.w Level $8820$ Weight $2$ Character orbit 8820.a Self dual yes Analytic conductor $70.428$ Analytic rank $0$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{5} + O(q^{10})$$ $$q + q^{5} + 2q^{11} - 6q^{13} - 2q^{17} + 9q^{23} + q^{25} - 3q^{29} + 2q^{31} + 8q^{37} - 5q^{41} + q^{43} - 8q^{47} - 4q^{53} + 2q^{55} + 8q^{59} + 7q^{61} - 6q^{65} - 3q^{67} - 8q^{71} + 14q^{73} + 4q^{79} + q^{83} - 2q^{85} - 13q^{89} - 10q^{97} + O(q^{100})$$

## 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
 0
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.w 1
3.b odd 2 1 980.2.a.a 1
7.b odd 2 1 8820.2.a.k 1
7.c even 3 2 1260.2.s.b 2
12.b even 2 1 3920.2.a.bi 1
15.d odd 2 1 4900.2.a.v 1
15.e even 4 2 4900.2.e.c 2
21.c even 2 1 980.2.a.i 1
21.g even 6 2 980.2.i.a 2
21.h odd 6 2 140.2.i.b 2
84.h odd 2 1 3920.2.a.d 1
84.n even 6 2 560.2.q.a 2
105.g even 2 1 4900.2.a.a 1
105.k odd 4 2 4900.2.e.b 2
105.o odd 6 2 700.2.i.a 2
105.x even 12 4 700.2.r.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.i.b 2 21.h odd 6 2
560.2.q.a 2 84.n even 6 2
700.2.i.a 2 105.o odd 6 2
700.2.r.c 4 105.x even 12 4
980.2.a.a 1 3.b odd 2 1
980.2.a.i 1 21.c even 2 1
980.2.i.a 2 21.g even 6 2
1260.2.s.b 2 7.c even 3 2
3920.2.a.d 1 84.h odd 2 1
3920.2.a.bi 1 12.b even 2 1
4900.2.a.a 1 105.g even 2 1
4900.2.a.v 1 15.d odd 2 1
4900.2.e.b 2 105.k odd 4 2
4900.2.e.c 2 15.e even 4 2
8820.2.a.k 1 7.b odd 2 1
8820.2.a.w 1 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{11} - 2$$ $$T_{13} + 6$$ $$T_{17} + 2$$ $$T_{31} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-1 + T$$
$7$ $$T$$
$11$ $$-2 + T$$
$13$ $$6 + T$$
$17$ $$2 + T$$
$19$ $$T$$
$23$ $$-9 + T$$
$29$ $$3 + T$$
$31$ $$-2 + T$$
$37$ $$-8 + T$$
$41$ $$5 + T$$
$43$ $$-1 + T$$
$47$ $$8 + T$$
$53$ $$4 + T$$
$59$ $$-8 + T$$
$61$ $$-7 + T$$
$67$ $$3 + T$$
$71$ $$8 + T$$
$73$ $$-14 + T$$
$79$ $$-4 + T$$
$83$ $$-1 + T$$
$89$ $$13 + T$$
$97$ $$10 + T$$