# Properties

 Label 8470.2.a.cz Level $8470$ Weight $2$ Character orbit 8470.a Self dual yes Analytic conductor $67.633$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.6.10784448.1 Defining polynomial: $$x^{6} - 11 x^{4} - 4 x^{3} + 31 x^{2} + 22 x - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 11 x^{4} - 4 x^{3} + 31 x^{2} + 22 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} - 6 \nu - 2$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{4} - \nu^{3} + 7 \nu^{2} + 6 \nu - 4$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{4} + \nu^{3} - 5 \nu^{2} - 8 \nu - 4$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{4} + \nu^{3} + 7 \nu^{2} - 2 \nu - 8$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{5} - \nu^{4} - 17 \nu^{3} + 3 \nu^{2} + 32 \nu + 6$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} - \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_{1} + 8$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{4} - 3 \beta_{2} - 2 \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$($$$$7 \beta_{4} + 14 \beta_{3} + 3 \beta_{2} - 9 \beta_{1} + 44$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$2 \beta_{5} + 37 \beta_{4} + 4 \beta_{3} - 35 \beta_{2} - 21 \beta_{1} + 38$$$$)/2$$

## 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
 0.0816388 2.40765 −1.84763 −0.930827 2.58124 −2.29207
−1.00000 −1.34292 1.00000 1.00000 1.34292 −1.00000 −1.00000 −1.19656 −1.00000
1.2 −1.00000 −1.34292 1.00000 1.00000 1.34292 −1.00000 −1.00000 −1.19656 −1.00000
1.3 −1.00000 0.529317 1.00000 1.00000 −0.529317 −1.00000 −1.00000 −2.71982 −1.00000
1.4 −1.00000 0.529317 1.00000 1.00000 −0.529317 −1.00000 −1.00000 −2.71982 −1.00000
1.5 −1.00000 2.81361 1.00000 1.00000 −2.81361 −1.00000 −1.00000 4.91638 −1.00000
1.6 −1.00000 2.81361 1.00000 1.00000 −2.81361 −1.00000 −1.00000 4.91638 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.cz 6
11.b odd 2 1 8470.2.a.df yes 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8470.2.a.cz 6 1.a even 1 1 trivial
8470.2.a.df yes 6 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8470))$$:

 $$T_{3}^{3} - 2 T_{3}^{2} - 3 T_{3} + 2$$ $$T_{13}^{6} - 42 T_{13}^{4} + 441 T_{13}^{2} - 108$$ $$T_{17}^{3} + T_{17}^{2} - 24 T_{17} + 38$$ $$T_{19}^{6} - 41 T_{19}^{4} - 80 T_{19}^{3} + 283 T_{19}^{2} + 944 T_{19} + 709$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{6}$$
$3$ $$( 2 - 3 T - 2 T^{2} + T^{3} )^{2}$$
$5$ $$( -1 + T )^{6}$$
$7$ $$( 1 + T )^{6}$$
$11$ $$T^{6}$$
$13$ $$-108 + 441 T^{2} - 42 T^{4} + T^{6}$$
$17$ $$( 38 - 24 T + T^{2} + T^{3} )^{2}$$
$19$ $$709 + 944 T + 283 T^{2} - 80 T^{3} - 41 T^{4} + T^{6}$$
$23$ $$2932 - 2280 T + 109 T^{2} + 268 T^{3} - 50 T^{4} - 4 T^{5} + T^{6}$$
$29$ $$256 + 640 T - 352 T^{2} - 336 T^{3} - 36 T^{4} + 8 T^{5} + T^{6}$$
$31$ $$( 8 - 6 T - 2 T^{2} + T^{3} )^{2}$$
$37$ $$-512 - 384 T + 400 T^{2} + 112 T^{3} - 56 T^{4} - 4 T^{5} + T^{6}$$
$41$ $$-32 - 432 T - 876 T^{2} - 320 T^{3} + 12 T^{5} + T^{6}$$
$43$ $$1012 + 536 T - 1388 T^{2} + 568 T^{3} - 59 T^{4} - 6 T^{5} + T^{6}$$
$47$ $$-26864 - 23456 T - 124 T^{2} + 1488 T^{3} - 60 T^{4} - 16 T^{5} + T^{6}$$
$53$ $$436 + 144 T - 660 T^{2} + 248 T^{3} + 9 T^{4} - 12 T^{5} + T^{6}$$
$59$ $$-19679 + 20650 T - 6961 T^{2} + 588 T^{3} + 111 T^{4} - 22 T^{5} + T^{6}$$
$61$ $$-5888 + 3328 T + 4464 T^{2} - 48 T^{3} - 175 T^{4} - 4 T^{5} + T^{6}$$
$67$ $$45748 + 6272 T - 11028 T^{2} + 1824 T^{3} + 17 T^{4} - 20 T^{5} + T^{6}$$
$71$ $$-402176 - 113536 T + 12528 T^{2} + 3240 T^{3} - 247 T^{4} - 14 T^{5} + T^{6}$$
$73$ $$-3788 + 6040 T - 1604 T^{2} - 736 T^{3} + 25 T^{4} + 18 T^{5} + T^{6}$$
$79$ $$-197027 - 141536 T - 33241 T^{2} - 2112 T^{3} + 219 T^{4} + 32 T^{5} + T^{6}$$
$83$ $$-25136 - 13376 T + 6081 T^{2} + 1680 T^{3} - 130 T^{4} - 16 T^{5} + T^{6}$$
$89$ $$529504 - 336944 T + 40452 T^{2} + 2352 T^{3} - 424 T^{4} - 4 T^{5} + T^{6}$$
$97$ $$46912 + 55776 T + 14488 T^{2} - 188 T^{3} - 251 T^{4} - 4 T^{5} + T^{6}$$