# Properties

 Label 8470.2.a.f 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: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} - 2q^{3} + q^{4} + q^{5} + 2q^{6} + q^{7} - q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} - 2q^{3} + q^{4} + q^{5} + 2q^{6} + q^{7} - q^{8} + q^{9} - q^{10} - 2q^{12} + 2q^{13} - q^{14} - 2q^{15} + q^{16} + 3q^{17} - q^{18} - q^{19} + q^{20} - 2q^{21} - 6q^{23} + 2q^{24} + q^{25} - 2q^{26} + 4q^{27} + q^{28} + 6q^{29} + 2q^{30} - 4q^{31} - q^{32} - 3q^{34} + q^{35} + q^{36} + 2q^{37} + q^{38} - 4q^{39} - q^{40} + 2q^{42} - 7q^{43} + q^{45} + 6q^{46} - 2q^{48} + q^{49} - q^{50} - 6q^{51} + 2q^{52} + 9q^{53} - 4q^{54} - q^{56} + 2q^{57} - 6q^{58} + 3q^{59} - 2q^{60} - 7q^{61} + 4q^{62} + q^{63} + q^{64} + 2q^{65} + 11q^{67} + 3q^{68} + 12q^{69} - q^{70} - 3q^{71} - q^{72} + 11q^{73} - 2q^{74} - 2q^{75} - q^{76} + 4q^{78} + 11q^{79} + q^{80} - 11q^{81} - 12q^{83} - 2q^{84} + 3q^{85} + 7q^{86} - 12q^{87} - q^{90} + 2q^{91} - 6q^{92} + 8q^{93} - q^{95} + 2q^{96} - q^{97} - q^{98} + O(q^{100})$$

## 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 −2.00000 1.00000 1.00000 2.00000 1.00000 −1.00000 1.00000 −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.f 1
11.b odd 2 1 8470.2.a.u yes 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8470.2.a.f 1 1.a even 1 1 trivial
8470.2.a.u 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8470))$$:

 $$T_{3} + 2$$ $$T_{13} - 2$$ $$T_{17} - 3$$ $$T_{19} + 1$$

## Hecke characteristic polynomials

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