# Properties

 Label 8820.2.a.z 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 1260) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{5} + O(q^{10})$$ $$q + q^{5} + 4q^{11} - 2q^{17} + 6q^{19} + 6q^{23} + q^{25} - 2q^{31} + 2q^{37} - 2q^{41} + 4q^{43} - 8q^{47} + 10q^{53} + 4q^{55} - 4q^{59} + 2q^{61} + 12q^{67} + 8q^{71} - 8q^{73} - 8q^{79} + 4q^{83} - 2q^{85} - 10q^{89} + 6q^{95} + 4q^{97} + O(q^{100})$$

## 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.z 1
3.b odd 2 1 8820.2.a.c 1
7.b odd 2 1 1260.2.a.b 1
21.c even 2 1 1260.2.a.f yes 1
28.d even 2 1 5040.2.a.m 1
35.c odd 2 1 6300.2.a.bd 1
35.f even 4 2 6300.2.k.o 2
84.h odd 2 1 5040.2.a.bo 1
105.g even 2 1 6300.2.a.q 1
105.k odd 4 2 6300.2.k.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.a.b 1 7.b odd 2 1
1260.2.a.f yes 1 21.c even 2 1
5040.2.a.m 1 28.d even 2 1
5040.2.a.bo 1 84.h odd 2 1
6300.2.a.q 1 105.g even 2 1
6300.2.a.bd 1 35.c odd 2 1
6300.2.k.b 2 105.k odd 4 2
6300.2.k.o 2 35.f even 4 2
8820.2.a.c 1 3.b odd 2 1
8820.2.a.z 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8820))$$:

 $$T_{11} - 4$$ $$T_{13}$$ $$T_{17} + 2$$ $$T_{31} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-1 + T$$
$7$ $$T$$
$11$ $$-4 + T$$
$13$ $$T$$
$17$ $$2 + T$$
$19$ $$-6 + T$$
$23$ $$-6 + T$$
$29$ $$T$$
$31$ $$2 + T$$
$37$ $$-2 + T$$
$41$ $$2 + T$$
$43$ $$-4 + T$$
$47$ $$8 + T$$
$53$ $$-10 + T$$
$59$ $$4 + T$$
$61$ $$-2 + T$$
$67$ $$-12 + T$$
$71$ $$-8 + T$$
$73$ $$8 + T$$
$79$ $$8 + T$$
$83$ $$-4 + T$$
$89$ $$10 + T$$
$97$ $$-4 + T$$