# Properties

 Label 8470.2.a.bi 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: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{7} - q^{8} -2 q^{9} +O(q^{10})$$ $$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{7} - q^{8} -2 q^{9} - q^{10} - q^{12} + ( -1 + \beta ) q^{13} + q^{14} - q^{15} + q^{16} + ( 4 - \beta ) q^{17} + 2 q^{18} + ( 1 + 2 \beta ) q^{19} + q^{20} + q^{21} + ( -1 + \beta ) q^{23} + q^{24} + q^{25} + ( 1 - \beta ) q^{26} + 5 q^{27} - q^{28} -2 \beta q^{29} + q^{30} + ( -4 - \beta ) q^{31} - q^{32} + ( -4 + \beta ) q^{34} - q^{35} -2 q^{36} + ( -2 + 2 \beta ) q^{37} + ( -1 - 2 \beta ) q^{38} + ( 1 - \beta ) q^{39} - q^{40} + ( -2 - 3 \beta ) q^{41} - q^{42} + ( 2 - 3 \beta ) q^{43} -2 q^{45} + ( 1 - \beta ) q^{46} + ( -4 - \beta ) q^{47} - q^{48} + q^{49} - q^{50} + ( -4 + \beta ) q^{51} + ( -1 + \beta ) q^{52} + ( -4 + \beta ) q^{53} -5 q^{54} + q^{56} + ( -1 - 2 \beta ) q^{57} + 2 \beta q^{58} + ( 3 - 2 \beta ) q^{59} - q^{60} + ( 2 + 2 \beta ) q^{61} + ( 4 + \beta ) q^{62} + 2 q^{63} + q^{64} + ( -1 + \beta ) q^{65} + ( -4 - 3 \beta ) q^{67} + ( 4 - \beta ) q^{68} + ( 1 - \beta ) q^{69} + q^{70} + 8 q^{71} + 2 q^{72} + ( 4 + \beta ) q^{73} + ( 2 - 2 \beta ) q^{74} - q^{75} + ( 1 + 2 \beta ) q^{76} + ( -1 + \beta ) q^{78} + ( 11 + \beta ) q^{79} + q^{80} + q^{81} + ( 2 + 3 \beta ) q^{82} + q^{83} + q^{84} + ( 4 - \beta ) q^{85} + ( -2 + 3 \beta ) q^{86} + 2 \beta q^{87} + ( -14 + \beta ) q^{89} + 2 q^{90} + ( 1 - \beta ) q^{91} + ( -1 + \beta ) q^{92} + ( 4 + \beta ) q^{93} + ( 4 + \beta ) q^{94} + ( 1 + 2 \beta ) q^{95} + q^{96} + ( -8 - 2 \beta ) q^{97} - q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} - 2q^{7} - 2q^{8} - 4q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} - 2q^{7} - 2q^{8} - 4q^{9} - 2q^{10} - 2q^{12} - 2q^{13} + 2q^{14} - 2q^{15} + 2q^{16} + 8q^{17} + 4q^{18} + 2q^{19} + 2q^{20} + 2q^{21} - 2q^{23} + 2q^{24} + 2q^{25} + 2q^{26} + 10q^{27} - 2q^{28} + 2q^{30} - 8q^{31} - 2q^{32} - 8q^{34} - 2q^{35} - 4q^{36} - 4q^{37} - 2q^{38} + 2q^{39} - 2q^{40} - 4q^{41} - 2q^{42} + 4q^{43} - 4q^{45} + 2q^{46} - 8q^{47} - 2q^{48} + 2q^{49} - 2q^{50} - 8q^{51} - 2q^{52} - 8q^{53} - 10q^{54} + 2q^{56} - 2q^{57} + 6q^{59} - 2q^{60} + 4q^{61} + 8q^{62} + 4q^{63} + 2q^{64} - 2q^{65} - 8q^{67} + 8q^{68} + 2q^{69} + 2q^{70} + 16q^{71} + 4q^{72} + 8q^{73} + 4q^{74} - 2q^{75} + 2q^{76} - 2q^{78} + 22q^{79} + 2q^{80} + 2q^{81} + 4q^{82} + 2q^{83} + 2q^{84} + 8q^{85} - 4q^{86} - 28q^{89} + 4q^{90} + 2q^{91} - 2q^{92} + 8q^{93} + 8q^{94} + 2q^{95} + 2q^{96} - 16q^{97} - 2q^{98} + O(q^{100})$$

## 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 −1.00000 1.00000 1.00000 1.00000 −1.00000 −1.00000 −2.00000 −1.00000
1.2 −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.bi 2
11.b odd 2 1 8470.2.a.bw yes 2

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

 $$T_{3} + 1$$ $$T_{13}^{2} + 2 T_{13} - 11$$ $$T_{17}^{2} - 8 T_{17} + 4$$ $$T_{19}^{2} - 2 T_{19} - 47$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$-11 + 2 T + T^{2}$$
$17$ $$4 - 8 T + T^{2}$$
$19$ $$-47 - 2 T + T^{2}$$
$23$ $$-11 + 2 T + T^{2}$$
$29$ $$-48 + T^{2}$$
$31$ $$4 + 8 T + T^{2}$$
$37$ $$-44 + 4 T + T^{2}$$
$41$ $$-104 + 4 T + T^{2}$$
$43$ $$-104 - 4 T + T^{2}$$
$47$ $$4 + 8 T + T^{2}$$
$53$ $$4 + 8 T + T^{2}$$
$59$ $$-39 - 6 T + T^{2}$$
$61$ $$-44 - 4 T + T^{2}$$
$67$ $$-92 + 8 T + T^{2}$$
$71$ $$( -8 + T )^{2}$$
$73$ $$4 - 8 T + T^{2}$$
$79$ $$109 - 22 T + T^{2}$$
$83$ $$( -1 + T )^{2}$$
$89$ $$184 + 28 T + T^{2}$$
$97$ $$16 + 16 T + T^{2}$$