# Properties

 Label 8470.2.a.o 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

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

## 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.o 1
11.b odd 2 1 8470.2.a.be yes 1

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$-1 + T$$
$5$ $$-1 + T$$
$7$ $$-1 + T$$
$11$ $$T$$
$13$ $$-5 + T$$
$17$ $$T$$
$19$ $$-5 + T$$
$23$ $$-9 + T$$
$29$ $$6 + T$$
$31$ $$-8 + T$$
$37$ $$-8 + T$$
$41$ $$-6 + T$$
$43$ $$10 + T$$
$47$ $$T$$
$53$ $$-6 + T$$
$59$ $$3 + T$$
$61$ $$-2 + T$$
$67$ $$10 + T$$
$71$ $$-12 + T$$
$73$ $$10 + T$$
$79$ $$13 + T$$
$83$ $$3 + T$$
$89$ $$-12 + T$$
$97$ $$10 + T$$