# Properties

 Label 8470.2.a.co Level $8470$ Weight $2$ Character orbit 8470.a Self dual yes Analytic conductor $67.633$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.4400.1 Defining polynomial: $$x^{4} - 7 x^{2} + 11$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 7 x^{2} + 11$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 4 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 4 \beta_{1}$$

## 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
 −2.14896 −1.54336 1.54336 2.14896
−1.00000 −3.14896 1.00000 1.00000 3.14896 −1.00000 −1.00000 6.91596 −1.00000
1.2 −1.00000 −2.54336 1.00000 1.00000 2.54336 −1.00000 −1.00000 3.46869 −1.00000
1.3 −1.00000 0.543362 1.00000 1.00000 −0.543362 −1.00000 −1.00000 −2.70476 −1.00000
1.4 −1.00000 1.14896 1.00000 1.00000 −1.14896 −1.00000 −1.00000 −1.67989 −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.co 4
11.b odd 2 1 8470.2.a.cs 4
11.c even 5 2 770.2.n.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.n.f 8 11.c even 5 2
8470.2.a.co 4 1.a even 1 1 trivial
8470.2.a.cs 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8470))$$:

 $$T_{3}^{4} + 4 T_{3}^{3} - T_{3}^{2} - 10 T_{3} + 5$$ $$T_{13}^{4} - 14 T_{13}^{3} + 54 T_{13}^{2} - 220$$ $$T_{17}^{4} - 12 T_{17}^{3} + 41 T_{17}^{2} - 30 T_{17} - 25$$ $$T_{19}^{4} + 2 T_{19}^{3} - 29 T_{19}^{2} - 10 T_{19} + 145$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{4}$$
$3$ $$5 - 10 T - T^{2} + 4 T^{3} + T^{4}$$
$5$ $$( -1 + T )^{4}$$
$7$ $$( 1 + T )^{4}$$
$11$ $$T^{4}$$
$13$ $$-220 + 54 T^{2} - 14 T^{3} + T^{4}$$
$17$ $$-25 - 30 T + 41 T^{2} - 12 T^{3} + T^{4}$$
$19$ $$145 - 10 T - 29 T^{2} + 2 T^{3} + T^{4}$$
$23$ $$176 - 32 T^{2} + T^{4}$$
$29$ $$-164 + 440 T - 38 T^{2} - 10 T^{3} + T^{4}$$
$31$ $$-4684 - 1128 T + 14 T^{2} + 18 T^{3} + T^{4}$$
$37$ $$-1100 + 440 T + 46 T^{2} - 18 T^{3} + T^{4}$$
$41$ $$839 - 708 T + 203 T^{2} - 24 T^{3} + T^{4}$$
$43$ $$971 + 280 T - 53 T^{2} - 10 T^{3} + T^{4}$$
$47$ $$-1616 - 736 T - 68 T^{2} + 8 T^{3} + T^{4}$$
$53$ $$-2804 + 500 T + 78 T^{2} - 20 T^{3} + T^{4}$$
$59$ $$-155 - 290 T - 21 T^{2} + 14 T^{3} + T^{4}$$
$61$ $$-284 + 156 T + 16 T^{2} - 14 T^{3} + T^{4}$$
$67$ $$-29 - 50 T - 23 T^{2} + T^{4}$$
$71$ $$-556 - 152 T + 38 T^{2} + 14 T^{3} + T^{4}$$
$73$ $$-26249 + 3580 T + 103 T^{2} - 30 T^{3} + T^{4}$$
$79$ $$1420 + 300 T - 74 T^{2} - 8 T^{3} + T^{4}$$
$83$ $$-1055 + 850 T - 129 T^{2} - 8 T^{3} + T^{4}$$
$89$ $$-2749 - 1396 T - 169 T^{2} + 4 T^{3} + T^{4}$$
$97$ $$-1429 - 760 T - 77 T^{2} + 10 T^{3} + T^{4}$$