# Properties

 Label 8470.2.a.bd 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + q^{8} - 2q^{9} + O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + q^{8} - 2q^{9} - q^{10} + q^{12} + 5q^{13} + q^{14} - q^{15} + q^{16} + 6q^{17} - 2q^{18} + 5q^{19} - q^{20} + q^{21} - 3q^{23} + q^{24} + q^{25} + 5q^{26} - 5q^{27} + q^{28} - q^{30} - 10q^{31} + q^{32} + 6q^{34} - q^{35} - 2q^{36} + 2q^{37} + 5q^{38} + 5q^{39} - q^{40} + q^{42} + 8q^{43} + 2q^{45} - 3q^{46} - 6q^{47} + q^{48} + q^{49} + q^{50} + 6q^{51} + 5q^{52} + 6q^{53} - 5q^{54} + q^{56} + 5q^{57} + 9q^{59} - q^{60} - 10q^{61} - 10q^{62} - 2q^{63} + q^{64} - 5q^{65} + 14q^{67} + 6q^{68} - 3q^{69} - q^{70} - 2q^{72} + 2q^{73} + 2q^{74} + q^{75} + 5q^{76} + 5q^{78} - q^{79} - q^{80} + q^{81} + 9q^{83} + q^{84} - 6q^{85} + 8q^{86} + 2q^{90} + 5q^{91} - 3q^{92} - 10q^{93} - 6q^{94} - 5q^{95} + q^{96} + 8q^{97} + q^{98} + O(q^{100})$$

## 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 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.bd yes 1
11.b odd 2 1 8470.2.a.k 1

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

 $$T_{3} - 1$$ $$T_{13} - 5$$ $$T_{17} - 6$$ $$T_{19} - 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$-1 + T$$
$5$ $$1 + T$$
$7$ $$-1 + T$$
$11$ $$T$$
$13$ $$-5 + T$$
$17$ $$-6 + T$$
$19$ $$-5 + T$$
$23$ $$3 + T$$
$29$ $$T$$
$31$ $$10 + T$$
$37$ $$-2 + T$$
$41$ $$T$$
$43$ $$-8 + T$$
$47$ $$6 + T$$
$53$ $$-6 + T$$
$59$ $$-9 + T$$
$61$ $$10 + T$$
$67$ $$-14 + T$$
$71$ $$T$$
$73$ $$-2 + T$$
$79$ $$1 + T$$
$83$ $$-9 + T$$
$89$ $$T$$
$97$ $$-8 + T$$