# Properties

 Label 8470.2.a.bz 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

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

## 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
 −1.73205 1.73205
1.00000 −1.73205 1.00000 −1.00000 −1.73205 1.00000 1.00000 0 −1.00000
1.2 1.00000 1.73205 1.00000 −1.00000 1.73205 1.00000 1.00000 0 −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.bz yes 2
11.b odd 2 1 8470.2.a.bm 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8470.2.a.bm 2 11.b odd 2 1
8470.2.a.bz yes 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8470))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$-3 + T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$13 + 8 T + T^{2}$$
$17$ $$-2 - 2 T + T^{2}$$
$19$ $$1 + 4 T + T^{2}$$
$23$ $$( 5 + T )^{2}$$
$29$ $$( -8 + T )^{2}$$
$31$ $$-2 + 2 T + T^{2}$$
$37$ $$( -6 + T )^{2}$$
$41$ $$-26 + 2 T + T^{2}$$
$43$ $$46 + 14 T + T^{2}$$
$47$ $$6 + 6 T + T^{2}$$
$53$ $$-26 + 2 T + T^{2}$$
$59$ $$69 + 18 T + T^{2}$$
$61$ $$-32 - 8 T + T^{2}$$
$67$ $$-74 + 2 T + T^{2}$$
$71$ $$-48 + T^{2}$$
$73$ $$-138 + 6 T + T^{2}$$
$79$ $$( 7 + T )^{2}$$
$83$ $$( -5 + T )^{2}$$
$89$ $$-50 + 10 T + T^{2}$$
$97$ $$-32 - 8 T + T^{2}$$