# Properties

 Label 8470.2.a.br Level $8470$ Weight $2$ Character orbit 8470.a Self dual yes Analytic conductor $67.633$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + ( 1 + \beta ) q^{3} + q^{4} - q^{5} + ( -1 - \beta ) q^{6} - q^{7} - q^{8} + ( 1 + 2 \beta ) q^{9} +O(q^{10})$$ $$q - q^{2} + ( 1 + \beta ) q^{3} + q^{4} - q^{5} + ( -1 - \beta ) q^{6} - q^{7} - q^{8} + ( 1 + 2 \beta ) q^{9} + q^{10} + ( 1 + \beta ) q^{12} + ( -2 + 2 \beta ) q^{13} + q^{14} + ( -1 - \beta ) q^{15} + q^{16} + 2 \beta q^{17} + ( -1 - 2 \beta ) q^{18} + ( -5 - \beta ) q^{19} - q^{20} + ( -1 - \beta ) q^{21} + ( -3 - 3 \beta ) q^{23} + ( -1 - \beta ) q^{24} + q^{25} + ( 2 - 2 \beta ) q^{26} + 4 q^{27} - q^{28} + ( 3 + \beta ) q^{29} + ( 1 + \beta ) q^{30} + 2 q^{31} - q^{32} -2 \beta q^{34} + q^{35} + ( 1 + 2 \beta ) q^{36} + ( -1 + \beta ) q^{37} + ( 5 + \beta ) q^{38} + 4 q^{39} + q^{40} + ( -3 + 3 \beta ) q^{41} + ( 1 + \beta ) q^{42} -2 q^{43} + ( -1 - 2 \beta ) q^{45} + ( 3 + 3 \beta ) q^{46} + 4 \beta q^{47} + ( 1 + \beta ) q^{48} + q^{49} - q^{50} + ( 6 + 2 \beta ) q^{51} + ( -2 + 2 \beta ) q^{52} + ( -9 + \beta ) q^{53} -4 q^{54} + q^{56} + ( -8 - 6 \beta ) q^{57} + ( -3 - \beta ) q^{58} + 4 \beta q^{59} + ( -1 - \beta ) q^{60} + ( -2 + 4 \beta ) q^{61} -2 q^{62} + ( -1 - 2 \beta ) q^{63} + q^{64} + ( 2 - 2 \beta ) q^{65} -4 q^{67} + 2 \beta q^{68} + ( -12 - 6 \beta ) q^{69} - q^{70} + ( 6 + 2 \beta ) q^{71} + ( -1 - 2 \beta ) q^{72} + ( 4 + 6 \beta ) q^{73} + ( 1 - \beta ) q^{74} + ( 1 + \beta ) q^{75} + ( -5 - \beta ) q^{76} -4 q^{78} + ( 7 + 3 \beta ) q^{79} - q^{80} + ( 1 - 2 \beta ) q^{81} + ( 3 - 3 \beta ) q^{82} + ( 6 + 6 \beta ) q^{83} + ( -1 - \beta ) q^{84} -2 \beta q^{85} + 2 q^{86} + ( 6 + 4 \beta ) q^{87} + 2 \beta q^{89} + ( 1 + 2 \beta ) q^{90} + ( 2 - 2 \beta ) q^{91} + ( -3 - 3 \beta ) q^{92} + ( 2 + 2 \beta ) q^{93} -4 \beta q^{94} + ( 5 + \beta ) q^{95} + ( -1 - \beta ) q^{96} + ( -1 + 9 \beta ) q^{97} - q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + 2q^{10} + 2q^{12} - 4q^{13} + 2q^{14} - 2q^{15} + 2q^{16} - 2q^{18} - 10q^{19} - 2q^{20} - 2q^{21} - 6q^{23} - 2q^{24} + 2q^{25} + 4q^{26} + 8q^{27} - 2q^{28} + 6q^{29} + 2q^{30} + 4q^{31} - 2q^{32} + 2q^{35} + 2q^{36} - 2q^{37} + 10q^{38} + 8q^{39} + 2q^{40} - 6q^{41} + 2q^{42} - 4q^{43} - 2q^{45} + 6q^{46} + 2q^{48} + 2q^{49} - 2q^{50} + 12q^{51} - 4q^{52} - 18q^{53} - 8q^{54} + 2q^{56} - 16q^{57} - 6q^{58} - 2q^{60} - 4q^{61} - 4q^{62} - 2q^{63} + 2q^{64} + 4q^{65} - 8q^{67} - 24q^{69} - 2q^{70} + 12q^{71} - 2q^{72} + 8q^{73} + 2q^{74} + 2q^{75} - 10q^{76} - 8q^{78} + 14q^{79} - 2q^{80} + 2q^{81} + 6q^{82} + 12q^{83} - 2q^{84} + 4q^{86} + 12q^{87} + 2q^{90} + 4q^{91} - 6q^{92} + 4q^{93} + 10q^{95} - 2q^{96} - 2q^{97} - 2q^{98} + O(q^{100})$$

## 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
 −1.73205 1.73205
−1.00000 −0.732051 1.00000 −1.00000 0.732051 −1.00000 −1.00000 −2.46410 1.00000
1.2 −1.00000 2.73205 1.00000 −1.00000 −2.73205 −1.00000 −1.00000 4.46410 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.br 2
11.b odd 2 1 770.2.a.j 2
33.d even 2 1 6930.2.a.bv 2
44.c even 2 1 6160.2.a.t 2
55.d odd 2 1 3850.2.a.bd 2
55.e even 4 2 3850.2.c.x 4
77.b even 2 1 5390.2.a.bs 2

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

 $$T_{3}^{2} - 2 T_{3} - 2$$ $$T_{13}^{2} + 4 T_{13} - 8$$ $$T_{17}^{2} - 12$$ $$T_{19}^{2} + 10 T_{19} + 22$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$-2 - 2 T + T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$-8 + 4 T + T^{2}$$
$17$ $$-12 + T^{2}$$
$19$ $$22 + 10 T + T^{2}$$
$23$ $$-18 + 6 T + T^{2}$$
$29$ $$6 - 6 T + T^{2}$$
$31$ $$( -2 + T )^{2}$$
$37$ $$-2 + 2 T + T^{2}$$
$41$ $$-18 + 6 T + T^{2}$$
$43$ $$( 2 + T )^{2}$$
$47$ $$-48 + T^{2}$$
$53$ $$78 + 18 T + T^{2}$$
$59$ $$-48 + T^{2}$$
$61$ $$-44 + 4 T + T^{2}$$
$67$ $$( 4 + T )^{2}$$
$71$ $$24 - 12 T + T^{2}$$
$73$ $$-92 - 8 T + T^{2}$$
$79$ $$22 - 14 T + T^{2}$$
$83$ $$-72 - 12 T + T^{2}$$
$89$ $$-12 + T^{2}$$
$97$ $$-242 + 2 T + T^{2}$$