# Properties

 Label 8470.2.a.bt 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.

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

## 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 0.381966 1.00000 −1.00000 −0.381966 −1.00000 −1.00000 −2.85410 1.00000
1.2 −1.00000 2.61803 1.00000 −1.00000 −2.61803 −1.00000 −1.00000 3.85410 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.bt 2
11.b odd 2 1 8470.2.a.cf 2
11.c even 5 2 770.2.n.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.n.b 4 11.c even 5 2
8470.2.a.bt 2 1.a even 1 1 trivial
8470.2.a.cf 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8470))$$:

 $$T_{3}^{2} - 3 T_{3} + 1$$ $$T_{13} + 2$$ $$T_{17}^{2} - T_{17} - 1$$ $$T_{19}^{2} + 7 T_{19} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$1 - 3 T + T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$( 2 + T )^{2}$$
$17$ $$-1 - T + T^{2}$$
$19$ $$1 + 7 T + T^{2}$$
$23$ $$( -6 + T )^{2}$$
$29$ $$-4 + 2 T + T^{2}$$
$31$ $$-4 - 2 T + T^{2}$$
$37$ $$-16 + 4 T + T^{2}$$
$41$ $$-55 - 5 T + T^{2}$$
$43$ $$-9 - 3 T + T^{2}$$
$47$ $$44 - 14 T + T^{2}$$
$53$ $$-4 + 2 T + T^{2}$$
$59$ $$41 + 13 T + T^{2}$$
$61$ $$44 - 16 T + T^{2}$$
$67$ $$-31 - T + T^{2}$$
$71$ $$-36 + 6 T + T^{2}$$
$73$ $$99 + 21 T + T^{2}$$
$79$ $$-4 - 8 T + T^{2}$$
$83$ $$-55 + 5 T + T^{2}$$
$89$ $$25 - 15 T + T^{2}$$
$97$ $$-101 - T + T^{2}$$