# Properties

 Label 8470.2.a.cd Level $8470$ Weight $2$ Character orbit 8470.a Self dual yes Analytic conductor $67.633$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Learn more

## 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: 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.

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

## 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.cd yes 2
11.b odd 2 1 8470.2.a.bs 2

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$-2 - 2 T + T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$( 4 + T )^{2}$$
$17$ $$4 + 8 T + T^{2}$$
$19$ $$-2 + 2 T + T^{2}$$
$23$ $$-18 - 6 T + T^{2}$$
$29$ $$-74 + 2 T + T^{2}$$
$31$ $$4 - 8 T + T^{2}$$
$37$ $$-2 + 2 T + T^{2}$$
$41$ $$-18 - 6 T + T^{2}$$
$43$ $$-44 + 4 T + T^{2}$$
$47$ $$( -6 + T )^{2}$$
$53$ $$46 - 14 T + T^{2}$$
$59$ $$-104 + 4 T + T^{2}$$
$61$ $$( -8 + T )^{2}$$
$67$ $$-188 - 4 T + T^{2}$$
$71$ $$16 - 16 T + T^{2}$$
$73$ $$( -6 + T )^{2}$$
$79$ $$-74 - 2 T + T^{2}$$
$83$ $$16 + 16 T + T^{2}$$
$89$ $$-44 + 4 T + T^{2}$$
$97$ $$-2 - 2 T + T^{2}$$
show more
show less