# Properties

 Label 8470.2.a.z Level $8470$ Weight $2$ Character orbit 8470.a Self dual yes Analytic conductor $67.633$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8470.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$67.6332905120$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 770) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{5} - q^{7} + q^{8} - 3q^{9} + O(q^{10})$$ $$q + q^{2} + q^{4} + q^{5} - q^{7} + q^{8} - 3q^{9} + q^{10} + 6q^{13} - q^{14} + q^{16} + 2q^{17} - 3q^{18} + 4q^{19} + q^{20} - 4q^{23} + q^{25} + 6q^{26} - q^{28} - 6q^{29} + q^{32} + 2q^{34} - q^{35} - 3q^{36} - 2q^{37} + 4q^{38} + q^{40} + 6q^{41} + 4q^{43} - 3q^{45} - 4q^{46} + 4q^{47} + q^{49} + q^{50} + 6q^{52} - 2q^{53} - q^{56} - 6q^{58} + 12q^{59} + 2q^{61} + 3q^{63} + q^{64} + 6q^{65} - 8q^{67} + 2q^{68} - q^{70} - 8q^{71} - 3q^{72} + 10q^{73} - 2q^{74} + 4q^{76} + 8q^{79} + q^{80} + 9q^{81} + 6q^{82} + 12q^{83} + 2q^{85} + 4q^{86} + 10q^{89} - 3q^{90} - 6q^{91} - 4q^{92} + 4q^{94} + 4q^{95} - 6q^{97} + q^{98} + O(q^{100})$$

## 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
1.00000 0 1.00000 1.00000 0 −1.00000 1.00000 −3.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$1$$
$$11$$ $$-1$$

## 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 8470.2.a.z 1
11.b odd 2 1 770.2.a.d 1
33.d even 2 1 6930.2.a.x 1
44.c even 2 1 6160.2.a.e 1
55.d odd 2 1 3850.2.a.s 1
55.e even 4 2 3850.2.c.m 2
77.b even 2 1 5390.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.d 1 11.b odd 2 1
3850.2.a.s 1 55.d odd 2 1
3850.2.c.m 2 55.e even 4 2
5390.2.a.j 1 77.b even 2 1
6160.2.a.e 1 44.c even 2 1
6930.2.a.x 1 33.d even 2 1
8470.2.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8470))$$:

 $$T_{3}$$ $$T_{13} - 6$$ $$T_{17} - 2$$ $$T_{19} - 4$$

## Hecke characteristic polynomials

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