# Properties

 Label 8820.2.a.b 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 420) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{5} + O(q^{10})$$ $$q - q^{5} - 6q^{11} + 4q^{13} + 6q^{17} - 2q^{19} + q^{25} - 6q^{29} + 10q^{31} + 2q^{37} - 6q^{41} - 4q^{43} + 12q^{53} + 6q^{55} - 14q^{61} - 4q^{65} - 4q^{67} - 6q^{71} + 4q^{73} - 16q^{79} - 12q^{83} - 6q^{85} + 6q^{89} + 2q^{95} + 16q^{97} + O(q^{100})$$

## 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.b 1
3.b odd 2 1 2940.2.a.f 1
7.b odd 2 1 1260.2.a.i 1
21.c even 2 1 420.2.a.c 1
21.g even 6 2 2940.2.q.e 2
21.h odd 6 2 2940.2.q.i 2
28.d even 2 1 5040.2.a.bc 1
35.c odd 2 1 6300.2.a.a 1
35.f even 4 2 6300.2.k.a 2
84.h odd 2 1 1680.2.a.a 1
105.g even 2 1 2100.2.a.d 1
105.k odd 4 2 2100.2.k.j 2
168.e odd 2 1 6720.2.a.ch 1
168.i even 2 1 6720.2.a.x 1
420.o odd 2 1 8400.2.a.cj 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.a.c 1 21.c even 2 1
1260.2.a.i 1 7.b odd 2 1
1680.2.a.a 1 84.h odd 2 1
2100.2.a.d 1 105.g even 2 1
2100.2.k.j 2 105.k odd 4 2
2940.2.a.f 1 3.b odd 2 1
2940.2.q.e 2 21.g even 6 2
2940.2.q.i 2 21.h odd 6 2
5040.2.a.bc 1 28.d even 2 1
6300.2.a.a 1 35.c odd 2 1
6300.2.k.a 2 35.f even 4 2
6720.2.a.x 1 168.i even 2 1
6720.2.a.ch 1 168.e odd 2 1
8400.2.a.cj 1 420.o odd 2 1
8820.2.a.b 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8820))$$:

 $$T_{11} + 6$$ $$T_{13} - 4$$ $$T_{17} - 6$$ $$T_{31} - 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$1 + T$$
$7$ $$T$$
$11$ $$6 + T$$
$13$ $$-4 + T$$
$17$ $$-6 + T$$
$19$ $$2 + T$$
$23$ $$T$$
$29$ $$6 + T$$
$31$ $$-10 + T$$
$37$ $$-2 + T$$
$41$ $$6 + T$$
$43$ $$4 + T$$
$47$ $$T$$
$53$ $$-12 + T$$
$59$ $$T$$
$61$ $$14 + T$$
$67$ $$4 + T$$
$71$ $$6 + T$$
$73$ $$-4 + T$$
$79$ $$16 + T$$
$83$ $$12 + T$$
$89$ $$-6 + T$$
$97$ $$-16 + T$$