# Properties

 Label 8470.2.a.cb 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: $$1$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −0.618034 1.61803
1.00000 −0.618034 1.00000 −1.00000 −0.618034 −1.00000 1.00000 −2.61803 −1.00000
1.2 1.00000 1.61803 1.00000 −1.00000 1.61803 −1.00000 1.00000 −0.381966 −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.cb 2
11.b odd 2 1 8470.2.a.bp 2
11.d odd 10 2 770.2.n.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.n.d 4 11.d odd 10 2
8470.2.a.bp 2 11.b odd 2 1
8470.2.a.cb 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8470))$$:

 $$T_{3}^{2} - T_{3} - 1$$ $$T_{13}^{2} - 6 T_{13} + 4$$ $$T_{17}^{2} + 3 T_{17} + 1$$ $$T_{19}^{2} + 9 T_{19} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$-1 - T + T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$4 - 6 T + T^{2}$$
$17$ $$1 + 3 T + T^{2}$$
$19$ $$9 + 9 T + T^{2}$$
$23$ $$-4 - 8 T + T^{2}$$
$29$ $$-20 + T^{2}$$
$31$ $$-16 + 4 T + T^{2}$$
$37$ $$76 + 18 T + T^{2}$$
$41$ $$-11 - T + T^{2}$$
$43$ $$-19 + 7 T + T^{2}$$
$47$ $$44 - 14 T + T^{2}$$
$53$ $$20 + 10 T + T^{2}$$
$59$ $$45 - 15 T + T^{2}$$
$61$ $$4 + 14 T + T^{2}$$
$67$ $$19 + 9 T + T^{2}$$
$71$ $$-76 - 4 T + T^{2}$$
$73$ $$-11 - T + T^{2}$$
$79$ $$( 4 + T )^{2}$$
$83$ $$19 - 11 T + T^{2}$$
$89$ $$121 + 23 T + T^{2}$$
$97$ $$89 + 19 T + T^{2}$$