# Properties

 Label 8470.2.a.x Level $8470$ Weight $2$ Character orbit 8470.a Self dual yes Analytic conductor $67.633$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - q^{5} + q^{7} + q^{8} - 3q^{9} + O(q^{10})$$ $$q + q^{2} + q^{4} - q^{5} + q^{7} + q^{8} - 3q^{9} - q^{10} - 4q^{13} + q^{14} + q^{16} + 4q^{17} - 3q^{18} + 4q^{19} - q^{20} - 2q^{23} + q^{25} - 4q^{26} + q^{28} - 4q^{29} + 2q^{31} + q^{32} + 4q^{34} - q^{35} - 3q^{36} + 6q^{37} + 4q^{38} - q^{40} - 10q^{41} - 4q^{43} + 3q^{45} - 2q^{46} + q^{49} + q^{50} - 4q^{52} - 10q^{53} + q^{56} - 4q^{58} + 6q^{59} - 2q^{61} + 2q^{62} - 3q^{63} + q^{64} + 4q^{65} + 14q^{67} + 4q^{68} - q^{70} - 12q^{71} - 3q^{72} + 6q^{74} + 4q^{76} + 8q^{79} - q^{80} + 9q^{81} - 10q^{82} - 16q^{83} - 4q^{85} - 4q^{86} - 14q^{89} + 3q^{90} - 4q^{91} - 2q^{92} - 4q^{95} - 14q^{97} + q^{98} + O(q^{100})$$

## 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.x yes 1
11.b odd 2 1 8470.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8470.2.a.i 1 11.b odd 2 1
8470.2.a.x yes 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8470))$$:

 $$T_{3}$$ $$T_{13} + 4$$ $$T_{17} - 4$$ $$T_{19} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$T$$
$5$ $$1 + T$$
$7$ $$-1 + T$$
$11$ $$T$$
$13$ $$4 + T$$
$17$ $$-4 + T$$
$19$ $$-4 + T$$
$23$ $$2 + T$$
$29$ $$4 + T$$
$31$ $$-2 + T$$
$37$ $$-6 + T$$
$41$ $$10 + T$$
$43$ $$4 + T$$
$47$ $$T$$
$53$ $$10 + T$$
$59$ $$-6 + T$$
$61$ $$2 + T$$
$67$ $$-14 + T$$
$71$ $$12 + T$$
$73$ $$T$$
$79$ $$-8 + T$$
$83$ $$16 + T$$
$89$ $$14 + T$$
$97$ $$14 + T$$