# Properties

 Label 8470.2.a.bu 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: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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 $$\beta = \sqrt{33}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −2.37228 3.37228
−1.00000 2.00000 1.00000 1.00000 −2.00000 −1.00000 −1.00000 1.00000 −1.00000
1.2 −1.00000 2.00000 1.00000 1.00000 −2.00000 −1.00000 −1.00000 1.00000 −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.bu 2
11.b odd 2 1 770.2.a.k 2
33.d even 2 1 6930.2.a.bo 2
44.c even 2 1 6160.2.a.r 2
55.d odd 2 1 3850.2.a.bc 2
55.e even 4 2 3850.2.c.y 4
77.b even 2 1 5390.2.a.bq 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.k 2 11.b odd 2 1
3850.2.a.bc 2 55.d odd 2 1
3850.2.c.y 4 55.e even 4 2
5390.2.a.bq 2 77.b even 2 1
6160.2.a.r 2 44.c even 2 1
6930.2.a.bo 2 33.d even 2 1
8470.2.a.bu 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8470))$$:

 $$T_{3} - 2$$ $$T_{13}^{2} + 2 T_{13} - 32$$ $$T_{17}^{2} - 2 T_{17} - 32$$ $$T_{19}^{2} - 2 T_{19} - 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( -2 + T )^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$-32 + 2 T + T^{2}$$
$17$ $$-32 - 2 T + T^{2}$$
$19$ $$-32 - 2 T + T^{2}$$
$23$ $$-32 + 2 T + T^{2}$$
$29$ $$-24 + 6 T + T^{2}$$
$31$ $$-32 + 2 T + T^{2}$$
$37$ $$-8 + 10 T + T^{2}$$
$41$ $$( -4 + T )^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$-32 + 2 T + T^{2}$$
$53$ $$16 + 14 T + T^{2}$$
$59$ $$-24 + 6 T + T^{2}$$
$61$ $$16 + 14 T + T^{2}$$
$67$ $$( 4 + T )^{2}$$
$71$ $$( 4 + T )^{2}$$
$73$ $$-8 + 10 T + T^{2}$$
$79$ $$-32 + 2 T + T^{2}$$
$83$ $$( 8 + T )^{2}$$
$89$ $$-116 - 8 T + T^{2}$$
$97$ $$88 + 22 T + T^{2}$$