# Properties

 Label 8470.2.a.bj 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: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + ( -1 - \beta ) q^{3} + q^{4} + q^{5} + ( 1 + \beta ) q^{6} + q^{7} - q^{8} + ( 3 + 2 \beta ) q^{9} +O(q^{10})$$ $$q - q^{2} + ( -1 - \beta ) q^{3} + q^{4} + q^{5} + ( 1 + \beta ) q^{6} + q^{7} - q^{8} + ( 3 + 2 \beta ) q^{9} - q^{10} + ( -1 - \beta ) q^{12} -2 \beta q^{13} - q^{14} + ( -1 - \beta ) q^{15} + q^{16} -2 q^{17} + ( -3 - 2 \beta ) q^{18} + ( 3 + \beta ) q^{19} + q^{20} + ( -1 - \beta ) q^{21} + ( -5 - \beta ) q^{23} + ( 1 + \beta ) q^{24} + q^{25} + 2 \beta q^{26} + ( -10 - 2 \beta ) q^{27} + q^{28} + ( -1 + \beta ) q^{29} + ( 1 + \beta ) q^{30} + 4 \beta q^{31} - q^{32} + 2 q^{34} + q^{35} + ( 3 + 2 \beta ) q^{36} + ( -1 - \beta ) q^{37} + ( -3 - \beta ) q^{38} + ( 10 + 2 \beta ) q^{39} - q^{40} + ( 1 + 3 \beta ) q^{41} + ( 1 + \beta ) q^{42} + 8 q^{43} + ( 3 + 2 \beta ) q^{45} + ( 5 + \beta ) q^{46} + ( -4 + 2 \beta ) q^{47} + ( -1 - \beta ) q^{48} + q^{49} - q^{50} + ( 2 + 2 \beta ) q^{51} -2 \beta q^{52} + ( -3 + \beta ) q^{53} + ( 10 + 2 \beta ) q^{54} - q^{56} + ( -8 - 4 \beta ) q^{57} + ( 1 - \beta ) q^{58} -8 q^{59} + ( -1 - \beta ) q^{60} + ( -10 - 2 \beta ) q^{61} -4 \beta q^{62} + ( 3 + 2 \beta ) q^{63} + q^{64} -2 \beta q^{65} -6 q^{67} -2 q^{68} + ( 10 + 6 \beta ) q^{69} - q^{70} -8 q^{71} + ( -3 - 2 \beta ) q^{72} + ( 4 + 2 \beta ) q^{73} + ( 1 + \beta ) q^{74} + ( -1 - \beta ) q^{75} + ( 3 + \beta ) q^{76} + ( -10 - 2 \beta ) q^{78} + ( -3 - \beta ) q^{79} + q^{80} + ( 11 + 6 \beta ) q^{81} + ( -1 - 3 \beta ) q^{82} -8 q^{83} + ( -1 - \beta ) q^{84} -2 q^{85} -8 q^{86} -4 q^{87} + ( -6 - 4 \beta ) q^{89} + ( -3 - 2 \beta ) q^{90} -2 \beta q^{91} + ( -5 - \beta ) q^{92} + ( -20 - 4 \beta ) q^{93} + ( 4 - 2 \beta ) q^{94} + ( 3 + \beta ) q^{95} + ( 1 + \beta ) q^{96} + ( -5 + 3 \beta ) q^{97} - q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 2q^{7} - 2q^{8} + 6q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 2q^{7} - 2q^{8} + 6q^{9} - 2q^{10} - 2q^{12} - 2q^{14} - 2q^{15} + 2q^{16} - 4q^{17} - 6q^{18} + 6q^{19} + 2q^{20} - 2q^{21} - 10q^{23} + 2q^{24} + 2q^{25} - 20q^{27} + 2q^{28} - 2q^{29} + 2q^{30} - 2q^{32} + 4q^{34} + 2q^{35} + 6q^{36} - 2q^{37} - 6q^{38} + 20q^{39} - 2q^{40} + 2q^{41} + 2q^{42} + 16q^{43} + 6q^{45} + 10q^{46} - 8q^{47} - 2q^{48} + 2q^{49} - 2q^{50} + 4q^{51} - 6q^{53} + 20q^{54} - 2q^{56} - 16q^{57} + 2q^{58} - 16q^{59} - 2q^{60} - 20q^{61} + 6q^{63} + 2q^{64} - 12q^{67} - 4q^{68} + 20q^{69} - 2q^{70} - 16q^{71} - 6q^{72} + 8q^{73} + 2q^{74} - 2q^{75} + 6q^{76} - 20q^{78} - 6q^{79} + 2q^{80} + 22q^{81} - 2q^{82} - 16q^{83} - 2q^{84} - 4q^{85} - 16q^{86} - 8q^{87} - 12q^{89} - 6q^{90} - 10q^{92} - 40q^{93} + 8q^{94} + 6q^{95} + 2q^{96} - 10q^{97} - 2q^{98} + O(q^{100})$$

## 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
 1.61803 −0.618034
−1.00000 −3.23607 1.00000 1.00000 3.23607 1.00000 −1.00000 7.47214 −1.00000
1.2 −1.00000 1.23607 1.00000 1.00000 −1.23607 1.00000 −1.00000 −1.47214 −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.bj 2
11.b odd 2 1 8470.2.a.bv yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8470.2.a.bj 2 1.a even 1 1 trivial
8470.2.a.bv yes 2 11.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8470))$$:

 $$T_{3}^{2} + 2 T_{3} - 4$$ $$T_{13}^{2} - 20$$ $$T_{17} + 2$$ $$T_{19}^{2} - 6 T_{19} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$-4 + 2 T + T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$-20 + T^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$4 - 6 T + T^{2}$$
$23$ $$20 + 10 T + T^{2}$$
$29$ $$-4 + 2 T + T^{2}$$
$31$ $$-80 + T^{2}$$
$37$ $$-4 + 2 T + T^{2}$$
$41$ $$-44 - 2 T + T^{2}$$
$43$ $$( -8 + T )^{2}$$
$47$ $$-4 + 8 T + T^{2}$$
$53$ $$4 + 6 T + T^{2}$$
$59$ $$( 8 + T )^{2}$$
$61$ $$80 + 20 T + T^{2}$$
$67$ $$( 6 + T )^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$-4 - 8 T + T^{2}$$
$79$ $$4 + 6 T + T^{2}$$
$83$ $$( 8 + T )^{2}$$
$89$ $$-44 + 12 T + T^{2}$$
$97$ $$-20 + 10 T + T^{2}$$