# Properties

 Label 8470.2.a.bo Level $8470$ Weight $2$ Character orbit 8470.a Self dual yes Analytic conductor $67.633$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 770) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + \beta q^{3} + q^{4} - q^{5} -\beta q^{6} + q^{7} - q^{8} + 5 q^{9} +O(q^{10})$$ $$q - q^{2} + \beta q^{3} + q^{4} - q^{5} -\beta q^{6} + q^{7} - q^{8} + 5 q^{9} + q^{10} + \beta q^{12} -2 q^{13} - q^{14} -\beta q^{15} + q^{16} + ( -2 - 2 \beta ) q^{17} -5 q^{18} + \beta q^{19} - q^{20} + \beta q^{21} -\beta q^{23} -\beta q^{24} + q^{25} + 2 q^{26} + 2 \beta q^{27} + q^{28} + ( -6 + \beta ) q^{29} + \beta q^{30} -2 \beta q^{31} - q^{32} + ( 2 + 2 \beta ) q^{34} - q^{35} + 5 q^{36} + ( 6 - \beta ) q^{37} -\beta q^{38} -2 \beta q^{39} + q^{40} + ( -2 + \beta ) q^{41} -\beta q^{42} + ( 4 + 2 \beta ) q^{43} -5 q^{45} + \beta q^{46} -8 q^{47} + \beta q^{48} + q^{49} - q^{50} + ( -16 - 2 \beta ) q^{51} -2 q^{52} + ( -2 - 3 \beta ) q^{53} -2 \beta q^{54} - q^{56} + 8 q^{57} + ( 6 - \beta ) q^{58} -8 q^{59} -\beta q^{60} + ( -2 + 4 \beta ) q^{61} + 2 \beta q^{62} + 5 q^{63} + q^{64} + 2 q^{65} + ( 4 + 4 \beta ) q^{67} + ( -2 - 2 \beta ) q^{68} -8 q^{69} + q^{70} -2 \beta q^{71} -5 q^{72} + ( -2 - 2 \beta ) q^{73} + ( -6 + \beta ) q^{74} + \beta q^{75} + \beta q^{76} + 2 \beta q^{78} + ( 8 - \beta ) q^{79} - q^{80} + q^{81} + ( 2 - \beta ) q^{82} + ( 8 - 2 \beta ) q^{83} + \beta q^{84} + ( 2 + 2 \beta ) q^{85} + ( -4 - 2 \beta ) q^{86} + ( 8 - 6 \beta ) q^{87} + ( -6 - 2 \beta ) q^{89} + 5 q^{90} -2 q^{91} -\beta q^{92} -16 q^{93} + 8 q^{94} -\beta q^{95} -\beta q^{96} + ( 2 - \beta ) q^{97} - q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + 10 q^{9} + O(q^{10})$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + 10 q^{9} + 2 q^{10} - 4 q^{13} - 2 q^{14} + 2 q^{16} - 4 q^{17} - 10 q^{18} - 2 q^{20} + 2 q^{25} + 4 q^{26} + 2 q^{28} - 12 q^{29} - 2 q^{32} + 4 q^{34} - 2 q^{35} + 10 q^{36} + 12 q^{37} + 2 q^{40} - 4 q^{41} + 8 q^{43} - 10 q^{45} - 16 q^{47} + 2 q^{49} - 2 q^{50} - 32 q^{51} - 4 q^{52} - 4 q^{53} - 2 q^{56} + 16 q^{57} + 12 q^{58} - 16 q^{59} - 4 q^{61} + 10 q^{63} + 2 q^{64} + 4 q^{65} + 8 q^{67} - 4 q^{68} - 16 q^{69} + 2 q^{70} - 10 q^{72} - 4 q^{73} - 12 q^{74} + 16 q^{79} - 2 q^{80} + 2 q^{81} + 4 q^{82} + 16 q^{83} + 4 q^{85} - 8 q^{86} + 16 q^{87} - 12 q^{89} + 10 q^{90} - 4 q^{91} - 32 q^{93} + 16 q^{94} + 4 q^{97} - 2 q^{98} + O(q^{100})$$

## 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
 −1.41421 1.41421
−1.00000 −2.82843 1.00000 −1.00000 2.82843 1.00000 −1.00000 5.00000 1.00000
1.2 −1.00000 2.82843 1.00000 −1.00000 −2.82843 1.00000 −1.00000 5.00000 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.bo 2
11.b odd 2 1 770.2.a.i 2
33.d even 2 1 6930.2.a.br 2
44.c even 2 1 6160.2.a.ba 2
55.d odd 2 1 3850.2.a.bi 2
55.e even 4 2 3850.2.c.u 4
77.b even 2 1 5390.2.a.bt 2

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

 $$T_{3}^{2} - 8$$ $$T_{13} + 2$$ $$T_{17}^{2} + 4 T_{17} - 28$$ $$T_{19}^{2} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$-8 + T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$( 2 + T )^{2}$$
$17$ $$-28 + 4 T + T^{2}$$
$19$ $$-8 + T^{2}$$
$23$ $$-8 + T^{2}$$
$29$ $$28 + 12 T + T^{2}$$
$31$ $$-32 + T^{2}$$
$37$ $$28 - 12 T + T^{2}$$
$41$ $$-4 + 4 T + T^{2}$$
$43$ $$-16 - 8 T + T^{2}$$
$47$ $$( 8 + T )^{2}$$
$53$ $$-68 + 4 T + T^{2}$$
$59$ $$( 8 + T )^{2}$$
$61$ $$-124 + 4 T + T^{2}$$
$67$ $$-112 - 8 T + T^{2}$$
$71$ $$-32 + T^{2}$$
$73$ $$-28 + 4 T + T^{2}$$
$79$ $$56 - 16 T + T^{2}$$
$83$ $$32 - 16 T + T^{2}$$
$89$ $$4 + 12 T + T^{2}$$
$97$ $$-4 - 4 T + T^{2}$$