# Properties

 Label 8470.2.a.a 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

 $$f(q)$$ $$=$$ $$q - q^{2} - 3q^{3} + q^{4} + q^{5} + 3q^{6} - q^{7} - q^{8} + 6q^{9} + O(q^{10})$$ $$q - q^{2} - 3q^{3} + q^{4} + q^{5} + 3q^{6} - q^{7} - q^{8} + 6q^{9} - q^{10} - 3q^{12} - q^{13} + q^{14} - 3q^{15} + q^{16} - 6q^{18} + 7q^{19} + q^{20} + 3q^{21} + q^{23} + 3q^{24} + q^{25} + q^{26} - 9q^{27} - q^{28} - 8q^{29} + 3q^{30} - 4q^{31} - q^{32} - q^{35} + 6q^{36} + 2q^{37} - 7q^{38} + 3q^{39} - q^{40} + 6q^{41} - 3q^{42} + 6q^{43} + 6q^{45} - q^{46} - 12q^{47} - 3q^{48} + q^{49} - q^{50} - q^{52} - 12q^{53} + 9q^{54} + q^{56} - 21q^{57} + 8q^{58} + 3q^{59} - 3q^{60} + 6q^{61} + 4q^{62} - 6q^{63} + q^{64} - q^{65} + 8q^{67} - 3q^{69} + q^{70} - 8q^{71} - 6q^{72} - 16q^{73} - 2q^{74} - 3q^{75} + 7q^{76} - 3q^{78} + 9q^{79} + q^{80} + 9q^{81} - 6q^{82} + 13q^{83} + 3q^{84} - 6q^{86} + 24q^{87} + 6q^{89} - 6q^{90} + q^{91} + q^{92} + 12q^{93} + 12q^{94} + 7q^{95} + 3q^{96} - 8q^{97} - q^{98} + O(q^{100})$$

## 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 −3.00000 1.00000 1.00000 3.00000 −1.00000 −1.00000 6.00000 −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.a 1
11.b odd 2 1 8470.2.a.q yes 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8470.2.a.a 1 1.a even 1 1 trivial
8470.2.a.q yes 1 11.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8470))$$:

 $$T_{3} + 3$$ $$T_{13} + 1$$ $$T_{17}$$ $$T_{19} - 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$3 + T$$
$5$ $$-1 + T$$
$7$ $$1 + T$$
$11$ $$T$$
$13$ $$1 + T$$
$17$ $$T$$
$19$ $$-7 + T$$
$23$ $$-1 + T$$
$29$ $$8 + T$$
$31$ $$4 + T$$
$37$ $$-2 + T$$
$41$ $$-6 + T$$
$43$ $$-6 + T$$
$47$ $$12 + T$$
$53$ $$12 + T$$
$59$ $$-3 + T$$
$61$ $$-6 + T$$
$67$ $$-8 + T$$
$71$ $$8 + T$$
$73$ $$16 + T$$
$79$ $$-9 + T$$
$83$ $$-13 + T$$
$89$ $$-6 + T$$
$97$ $$8 + T$$