# Properties

 Label 8470.2.a.w 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

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - q^{7} + q^{8} - 2 q^{9} + O(q^{10})$$ $$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - q^{7} + q^{8} - 2 q^{9} - q^{10} - q^{12} + q^{13} - q^{14} + q^{15} + q^{16} + 4 q^{17} - 2 q^{18} - 3 q^{19} - q^{20} + q^{21} - 3 q^{23} - q^{24} + q^{25} + q^{26} + 5 q^{27} - q^{28} + 2 q^{29} + q^{30} + q^{32} + 4 q^{34} + q^{35} - 2 q^{36} + 8 q^{37} - 3 q^{38} - q^{39} - q^{40} + 6 q^{41} + q^{42} - 10 q^{43} + 2 q^{45} - 3 q^{46} - 4 q^{47} - q^{48} + q^{49} + q^{50} - 4 q^{51} + q^{52} + 6 q^{53} + 5 q^{54} - q^{56} + 3 q^{57} + 2 q^{58} + 3 q^{59} + q^{60} - 14 q^{61} + 2 q^{63} + q^{64} - q^{65} - 2 q^{67} + 4 q^{68} + 3 q^{69} + q^{70} - 12 q^{71} - 2 q^{72} - 6 q^{73} + 8 q^{74} - q^{75} - 3 q^{76} - q^{78} + 17 q^{79} - q^{80} + q^{81} + 6 q^{82} - 3 q^{83} + q^{84} - 4 q^{85} - 10 q^{86} - 2 q^{87} + 8 q^{89} + 2 q^{90} - q^{91} - 3 q^{92} - 4 q^{94} + 3 q^{95} - q^{96} - 2 q^{97} + q^{98} + O(q^{100})$$

## 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 −1.00000 1.00000 −1.00000 −1.00000 −1.00000 1.00000 −2.00000 −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.w yes 1
11.b odd 2 1 8470.2.a.g 1

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

 $$T_{3} + 1$$ $$T_{13} - 1$$ $$T_{17} - 4$$ $$T_{19} + 3$$

## Hecke characteristic polynomials

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