# Properties

 Label 8470.2.a.ci Level $8470$ Weight $2$ Character orbit 8470.a Self dual yes Analytic conductor $67.633$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

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## 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: 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + ( 1 + \beta_{2} ) q^{3} + q^{4} + q^{5} + ( -1 - \beta_{2} ) q^{6} + q^{7} - q^{8} + ( 3 - \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q - q^{2} + ( 1 + \beta_{2} ) q^{3} + q^{4} + q^{5} + ( -1 - \beta_{2} ) q^{6} + q^{7} - q^{8} + ( 3 - \beta_{1} + \beta_{2} ) q^{9} - q^{10} + ( 1 + \beta_{2} ) q^{12} + ( -\beta_{1} + \beta_{2} ) q^{13} - q^{14} + ( 1 + \beta_{2} ) q^{15} + q^{16} + 2 \beta_{2} q^{17} + ( -3 + \beta_{1} - \beta_{2} ) q^{18} + ( 3 - 2 \beta_{1} + \beta_{2} ) q^{19} + q^{20} + ( 1 + \beta_{2} ) q^{21} + ( 3 - \beta_{2} ) q^{23} + ( -1 - \beta_{2} ) q^{24} + q^{25} + ( \beta_{1} - \beta_{2} ) q^{26} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{27} + q^{28} + ( 1 + 3 \beta_{2} ) q^{29} + ( -1 - \beta_{2} ) q^{30} + ( -4 + \beta_{1} + \beta_{2} ) q^{31} - q^{32} -2 \beta_{2} q^{34} + q^{35} + ( 3 - \beta_{1} + \beta_{2} ) q^{36} + ( 1 - \beta_{2} ) q^{37} + ( -3 + 2 \beta_{1} - \beta_{2} ) q^{38} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{39} - q^{40} + ( -3 + 2 \beta_{1} - 3 \beta_{2} ) q^{41} + ( -1 - \beta_{2} ) q^{42} + ( -4 + 3 \beta_{1} - \beta_{2} ) q^{43} + ( 3 - \beta_{1} + \beta_{2} ) q^{45} + ( -3 + \beta_{2} ) q^{46} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{47} + ( 1 + \beta_{2} ) q^{48} + q^{49} - q^{50} + ( 10 - 2 \beta_{1} ) q^{51} + ( -\beta_{1} + \beta_{2} ) q^{52} + ( 5 + 3 \beta_{2} ) q^{53} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{54} - q^{56} + ( 6 - 3 \beta_{1} + 7 \beta_{2} ) q^{57} + ( -1 - 3 \beta_{2} ) q^{58} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{59} + ( 1 + \beta_{2} ) q^{60} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{61} + ( 4 - \beta_{1} - \beta_{2} ) q^{62} + ( 3 - \beta_{1} + \beta_{2} ) q^{63} + q^{64} + ( -\beta_{1} + \beta_{2} ) q^{65} -4 \beta_{2} q^{67} + 2 \beta_{2} q^{68} + ( -2 + \beta_{1} + 3 \beta_{2} ) q^{69} - q^{70} + ( -10 + 2 \beta_{1} ) q^{71} + ( -3 + \beta_{1} - \beta_{2} ) q^{72} + ( 4 - 2 \beta_{2} ) q^{73} + ( -1 + \beta_{2} ) q^{74} + ( 1 + \beta_{2} ) q^{75} + ( 3 - 2 \beta_{1} + \beta_{2} ) q^{76} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{78} + ( -7 + 2 \beta_{1} - \beta_{2} ) q^{79} + q^{80} + ( 3 - \beta_{1} + 5 \beta_{2} ) q^{81} + ( 3 - 2 \beta_{1} + 3 \beta_{2} ) q^{82} + ( 2 - 2 \beta_{2} ) q^{83} + ( 1 + \beta_{2} ) q^{84} + 2 \beta_{2} q^{85} + ( 4 - 3 \beta_{1} + \beta_{2} ) q^{86} + ( 16 - 3 \beta_{1} + \beta_{2} ) q^{87} + ( 4 + 2 \beta_{1} - 4 \beta_{2} ) q^{89} + ( -3 + \beta_{1} - \beta_{2} ) q^{90} + ( -\beta_{1} + \beta_{2} ) q^{91} + ( 3 - \beta_{2} ) q^{92} + ( 2 - 6 \beta_{2} ) q^{93} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{94} + ( 3 - 2 \beta_{1} + \beta_{2} ) q^{95} + ( -1 - \beta_{2} ) q^{96} + ( -1 + 2 \beta_{1} + 3 \beta_{2} ) q^{97} - q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} - 2 q^{6} + 3 q^{7} - 3 q^{8} + 7 q^{9} + O(q^{10})$$ $$3 q - 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} - 2 q^{6} + 3 q^{7} - 3 q^{8} + 7 q^{9} - 3 q^{10} + 2 q^{12} - 2 q^{13} - 3 q^{14} + 2 q^{15} + 3 q^{16} - 2 q^{17} - 7 q^{18} + 6 q^{19} + 3 q^{20} + 2 q^{21} + 10 q^{23} - 2 q^{24} + 3 q^{25} + 2 q^{26} + 8 q^{27} + 3 q^{28} - 2 q^{30} - 12 q^{31} - 3 q^{32} + 2 q^{34} + 3 q^{35} + 7 q^{36} + 4 q^{37} - 6 q^{38} + 8 q^{39} - 3 q^{40} - 4 q^{41} - 2 q^{42} - 8 q^{43} + 7 q^{45} - 10 q^{46} + 2 q^{48} + 3 q^{49} - 3 q^{50} + 28 q^{51} - 2 q^{52} + 12 q^{53} - 8 q^{54} - 3 q^{56} + 8 q^{57} - 4 q^{59} + 2 q^{60} + 6 q^{61} + 12 q^{62} + 7 q^{63} + 3 q^{64} - 2 q^{65} + 4 q^{67} - 2 q^{68} - 8 q^{69} - 3 q^{70} - 28 q^{71} - 7 q^{72} + 14 q^{73} - 4 q^{74} + 2 q^{75} + 6 q^{76} - 8 q^{78} - 18 q^{79} + 3 q^{80} + 3 q^{81} + 4 q^{82} + 8 q^{83} + 2 q^{84} - 2 q^{85} + 8 q^{86} + 44 q^{87} + 18 q^{89} - 7 q^{90} - 2 q^{91} + 10 q^{92} + 12 q^{93} + 6 q^{95} - 2 q^{96} - 4 q^{97} - 3 q^{98} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} + \nu - 3$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 6$$$$)/2$$

## 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 2.34292 −1.81361
−1.00000 −2.24914 1.00000 1.00000 2.24914 1.00000 −1.00000 2.05863 −1.00000
1.2 −1.00000 1.14637 1.00000 1.00000 −1.14637 1.00000 −1.00000 −1.68585 −1.00000
1.3 −1.00000 3.10278 1.00000 1.00000 −3.10278 1.00000 −1.00000 6.62721 −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.ci 3
11.b odd 2 1 770.2.a.m 3
33.d even 2 1 6930.2.a.ce 3
44.c even 2 1 6160.2.a.bf 3
55.d odd 2 1 3850.2.a.bt 3
55.e even 4 2 3850.2.c.ba 6
77.b even 2 1 5390.2.a.ca 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.m 3 11.b odd 2 1
3850.2.a.bt 3 55.d odd 2 1
3850.2.c.ba 6 55.e even 4 2
5390.2.a.ca 3 77.b even 2 1
6160.2.a.bf 3 44.c even 2 1
6930.2.a.ce 3 33.d even 2 1
8470.2.a.ci 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8470))$$:

 $$T_{3}^{3} - 2 T_{3}^{2} - 6 T_{3} + 8$$ $$T_{13}^{3} + 2 T_{13}^{2} - 16 T_{13} - 16$$ $$T_{17}^{3} + 2 T_{17}^{2} - 28 T_{17} + 8$$ $$T_{19}^{3} - 6 T_{19}^{2} - 46 T_{19} + 232$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{3}$$
$3$ $$8 - 6 T - 2 T^{2} + T^{3}$$
$5$ $$( -1 + T )^{3}$$
$7$ $$( -1 + T )^{3}$$
$11$ $$T^{3}$$
$13$ $$-16 - 16 T + 2 T^{2} + T^{3}$$
$17$ $$8 - 28 T + 2 T^{2} + T^{3}$$
$19$ $$232 - 46 T - 6 T^{2} + T^{3}$$
$23$ $$-16 + 26 T - 10 T^{2} + T^{3}$$
$29$ $$92 - 66 T + T^{3}$$
$31$ $$-32 + 20 T + 12 T^{2} + T^{3}$$
$37$ $$4 - 2 T - 4 T^{2} + T^{3}$$
$41$ $$164 - 90 T + 4 T^{2} + T^{3}$$
$43$ $$-848 - 108 T + 8 T^{2} + T^{3}$$
$47$ $$-128 - 112 T + T^{3}$$
$53$ $$292 - 18 T - 12 T^{2} + T^{3}$$
$59$ $$-128 - 64 T + 4 T^{2} + T^{3}$$
$61$ $$344 - 100 T - 6 T^{2} + T^{3}$$
$67$ $$-64 - 112 T - 4 T^{2} + T^{3}$$
$71$ $$64 + 200 T + 28 T^{2} + T^{3}$$
$73$ $$8 + 36 T - 14 T^{2} + T^{3}$$
$79$ $$-256 + 50 T + 18 T^{2} + T^{3}$$
$83$ $$32 - 8 T - 8 T^{2} + T^{3}$$
$89$ $$1208 - 28 T - 18 T^{2} + T^{3}$$
$97$ $$316 - 154 T + 4 T^{2} + T^{3}$$
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