# Properties

 Label 8470.2.a.r 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: no (minimal twist has level 770) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} - 2q^{3} + q^{4} - q^{5} - 2q^{6} - q^{7} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} - 2q^{3} + q^{4} - q^{5} - 2q^{6} - q^{7} + q^{8} + q^{9} - q^{10} - 2q^{12} - 2q^{13} - q^{14} + 2q^{15} + q^{16} + 6q^{17} + q^{18} - 2q^{19} - q^{20} + 2q^{21} - 6q^{23} - 2q^{24} + q^{25} - 2q^{26} + 4q^{27} - q^{28} + 2q^{30} + 8q^{31} + q^{32} + 6q^{34} + q^{35} + q^{36} - 4q^{37} - 2q^{38} + 4q^{39} - q^{40} - 12q^{41} + 2q^{42} + 4q^{43} - q^{45} - 6q^{46} + 12q^{47} - 2q^{48} + q^{49} + q^{50} - 12q^{51} - 2q^{52} + 4q^{54} - q^{56} + 4q^{57} + 2q^{60} - 2q^{61} + 8q^{62} - q^{63} + q^{64} + 2q^{65} + 8q^{67} + 6q^{68} + 12q^{69} + q^{70} + 12q^{71} + q^{72} - 2q^{73} - 4q^{74} - 2q^{75} - 2q^{76} + 4q^{78} - 14q^{79} - q^{80} - 11q^{81} - 12q^{82} - 12q^{83} + 2q^{84} - 6q^{85} + 4q^{86} + 6q^{89} - q^{90} + 2q^{91} - 6q^{92} - 16q^{93} + 12q^{94} + 2q^{95} - 2q^{96} + 8q^{97} + q^{98} + O(q^{100})$$

## 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 −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.r 1
11.b odd 2 1 770.2.a.a 1
33.d even 2 1 6930.2.a.bm 1
44.c even 2 1 6160.2.a.k 1
55.d odd 2 1 3850.2.a.ba 1
55.e even 4 2 3850.2.c.o 2
77.b even 2 1 5390.2.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.a 1 11.b odd 2 1
3850.2.a.ba 1 55.d odd 2 1
3850.2.c.o 2 55.e even 4 2
5390.2.a.r 1 77.b even 2 1
6160.2.a.k 1 44.c even 2 1
6930.2.a.bm 1 33.d even 2 1
8470.2.a.r 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8470))$$:

 $$T_{3} + 2$$ $$T_{13} + 2$$ $$T_{17} - 6$$ $$T_{19} + 2$$

## Hecke characteristic polynomials

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