# Properties

 Label 8470.2.a.cm 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: 3.3.733.1 Defining polynomial: $$x^{3} - x^{2} - 7 x + 8$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 7 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} + \nu - 5$$ $$\beta_{2}$$ $$=$$ $$-\nu^{2} + \nu + 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1} + 10$$$$)/2$$

## 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
 2.51820 −2.69639 1.17819
1.00000 −2.85952 1.00000 1.00000 −2.85952 −1.00000 1.00000 5.17687 1.00000
1.2 1.00000 1.42586 1.00000 1.00000 1.42586 −1.00000 1.00000 −0.966927 1.00000
1.3 1.00000 3.43366 1.00000 1.00000 3.43366 −1.00000 1.00000 8.79005 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.cm yes 3
11.b odd 2 1 8470.2.a.cj 3

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

 $$T_{3}^{3} - 2 T_{3}^{2} - 9 T_{3} + 14$$ $$T_{13}^{3} + 4 T_{13}^{2} - 19 T_{13} - 50$$ $$T_{17}^{3} + T_{17}^{2} - 10 T_{17} + 4$$ $$T_{19}^{3} - 3 T_{19}^{2} - 37 T_{19} - 49$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{3}$$
$3$ $$14 - 9 T - 2 T^{2} + T^{3}$$
$5$ $$( -1 + T )^{3}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$T^{3}$$
$13$ $$-50 - 19 T + 4 T^{2} + T^{3}$$
$17$ $$4 - 10 T + T^{2} + T^{3}$$
$19$ $$-49 - 37 T - 3 T^{2} + T^{3}$$
$23$ $$-10 + 23 T - 10 T^{2} + T^{3}$$
$29$ $$( -2 + T )^{3}$$
$31$ $$16 - 28 T + 6 T^{2} + T^{3}$$
$37$ $$160 - 8 T - 10 T^{2} + T^{3}$$
$41$ $$-80 + 36 T + 14 T^{2} + T^{3}$$
$43$ $$-4 + 6 T + 7 T^{2} + T^{3}$$
$47$ $$160 - 28 T - 6 T^{2} + T^{3}$$
$53$ $$52 - 66 T - 9 T^{2} + T^{3}$$
$59$ $$655 - 61 T - 11 T^{2} + T^{3}$$
$61$ $$4 - 52 T + 3 T^{2} + T^{3}$$
$67$ $$-196 + 110 T - 19 T^{2} + T^{3}$$
$71$ $$16 + 20 T - 17 T^{2} + T^{3}$$
$73$ $$4 + 6 T - 7 T^{2} + T^{3}$$
$79$ $$-5 - 13 T + 9 T^{2} + T^{3}$$
$83$ $$4 - 5 T - 4 T^{2} + T^{3}$$
$89$ $$-16 - 28 T - 6 T^{2} + T^{3}$$
$97$ $$-548 - 160 T - 5 T^{2} + T^{3}$$