# Properties

 Label 8470.2.a.ck Level $8470$ Weight $2$ Character orbit 8470.a Self dual yes Analytic conductor $67.633$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.316.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{2}$$ $$=$$ $$-\nu^{2} + 2 \nu + 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{1} + 3$$

## 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
 0.470683 −1.81361 2.34292
1.00000 −2.77846 1.00000 −1.00000 −2.77846 1.00000 1.00000 4.71982 −1.00000
1.2 1.00000 0.289169 1.00000 −1.00000 0.289169 1.00000 1.00000 −2.91638 −1.00000
1.3 1.00000 2.48929 1.00000 −1.00000 2.48929 1.00000 1.00000 3.19656 −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.ck yes 3
11.b odd 2 1 8470.2.a.cg 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8470.2.a.cg 3 11.b odd 2 1
8470.2.a.ck yes 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8470))$$:

 $$T_{3}^{3} - 7 T_{3} + 2$$ $$T_{13}^{3} - 7 T_{13} + 2$$ $$T_{17}^{3} - 11 T_{17}^{2} + 24 T_{17} + 32$$ $$T_{19}^{3} - 5 T_{19}^{2} - 21 T_{19} + 17$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{3}$$
$3$ $$2 - 7 T + T^{3}$$
$5$ $$( 1 + T )^{3}$$
$7$ $$( -1 + T )^{3}$$
$11$ $$T^{3}$$
$13$ $$2 - 7 T + T^{3}$$
$17$ $$32 + 24 T - 11 T^{2} + T^{3}$$
$19$ $$17 - 21 T - 5 T^{2} + T^{3}$$
$23$ $$-106 - 27 T + 4 T^{2} + T^{3}$$
$29$ $$-128 - 64 T - 2 T^{2} + T^{3}$$
$31$ $$T^{3}$$
$37$ $$-8 - 68 T + 2 T^{2} + T^{3}$$
$41$ $$-16 - 28 T + T^{3}$$
$43$ $$548 - 88 T - 7 T^{2} + T^{3}$$
$47$ $$128 - 48 T - 8 T^{2} + T^{3}$$
$53$ $$-16 - 16 T - T^{2} + T^{3}$$
$59$ $$83 - 13 T - 7 T^{2} + T^{3}$$
$61$ $$316 - 13 T^{2} + T^{3}$$
$67$ $$-992 - 120 T + 9 T^{2} + T^{3}$$
$71$ $$-64 + 64 T - 17 T^{2} + T^{3}$$
$73$ $$16 + 88 T - 19 T^{2} + T^{3}$$
$79$ $$-41 - T + 9 T^{2} + T^{3}$$
$83$ $$8 + 5 T - 6 T^{2} + T^{3}$$
$89$ $$32 + 20 T - 12 T^{2} + T^{3}$$
$97$ $$-128 - 16 T + 7 T^{2} + T^{3}$$