# Properties

 Label 8470.2.a.cw Level $8470$ Weight $2$ Character orbit 8470.a Self dual yes Analytic conductor $67.633$ Analytic rank $1$ 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: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.19898000.1 Defining polynomial: $$x^{6} - x^{5} - 10 x^{4} + 7 x^{3} + 24 x^{2} - 15 x - 5$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 10 x^{4} + 7 x^{3} + 24 x^{2} - 15 x - 5$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{4} - 2 \nu^{3} - 5 \nu^{2} + 8 \nu - 1$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{4} + 2 \nu^{3} + 7 \nu^{2} - 8 \nu - 7$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{5} - 2 \nu^{4} - 5 \nu^{3} + 8 \nu^{2} - \nu$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{5} + 2 \nu^{4} + 7 \nu^{3} - 10 \nu^{2} - 9 \nu + 6$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 5 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{5} + 2 \beta_{4} + 7 \beta_{3} + 9 \beta_{2} + 2 \beta_{1} + 23$$ $$\nu^{5}$$ $$=$$ $$9 \beta_{5} + 11 \beta_{4} + 11 \beta_{3} + 15 \beta_{2} + 30 \beta_{1} + 19$$

## 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
 2.86564 1.68692 0.935683 −0.245893 −2.05906 −2.18328
−1.00000 −2.86564 1.00000 1.00000 2.86564 1.00000 −1.00000 5.21187 −1.00000
1.2 −1.00000 −1.68692 1.00000 1.00000 1.68692 1.00000 −1.00000 −0.154300 −1.00000
1.3 −1.00000 −0.935683 1.00000 1.00000 0.935683 1.00000 −1.00000 −2.12450 −1.00000
1.4 −1.00000 0.245893 1.00000 1.00000 −0.245893 1.00000 −1.00000 −2.93954 −1.00000
1.5 −1.00000 2.05906 1.00000 1.00000 −2.05906 1.00000 −1.00000 1.23973 −1.00000
1.6 −1.00000 2.18328 1.00000 1.00000 −2.18328 1.00000 −1.00000 1.76673 −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.cw 6
11.b odd 2 1 8470.2.a.dc 6
11.c even 5 2 770.2.n.j 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.n.j 12 11.c even 5 2
8470.2.a.cw 6 1.a even 1 1 trivial
8470.2.a.dc 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8470))$$:

 $$T_{3}^{6} + T_{3}^{5} - 10 T_{3}^{4} - 7 T_{3}^{3} + 24 T_{3}^{2} + 15 T_{3} - 5$$ $$T_{13}^{6} + 9 T_{13}^{5} + 15 T_{13}^{4} - 44 T_{13}^{3} - 138 T_{13}^{2} - 88 T_{13} + 4$$ $$T_{17}^{6} + 9 T_{17}^{5} - 50 T_{17}^{4} - 581 T_{17}^{3} - 42 T_{17}^{2} + 8421 T_{17} + 14431$$ $$T_{19}^{6} + 12 T_{19}^{5} + 23 T_{19}^{4} - 144 T_{19}^{3} - 635 T_{19}^{2} - 846 T_{19} - 356$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{6}$$
$3$ $$-5 + 15 T + 24 T^{2} - 7 T^{3} - 10 T^{4} + T^{5} + T^{6}$$
$5$ $$( -1 + T )^{6}$$
$7$ $$( -1 + T )^{6}$$
$11$ $$T^{6}$$
$13$ $$4 - 88 T - 138 T^{2} - 44 T^{3} + 15 T^{4} + 9 T^{5} + T^{6}$$
$17$ $$14431 + 8421 T - 42 T^{2} - 581 T^{3} - 50 T^{4} + 9 T^{5} + T^{6}$$
$19$ $$-356 - 846 T - 635 T^{2} - 144 T^{3} + 23 T^{4} + 12 T^{5} + T^{6}$$
$23$ $$-2624 + 2496 T + 2288 T^{2} + 160 T^{3} - 92 T^{4} - 4 T^{5} + T^{6}$$
$29$ $$1796 + 1040 T - 590 T^{2} - 252 T^{3} + 35 T^{4} + 15 T^{5} + T^{6}$$
$31$ $$80 + 440 T - 724 T^{2} + 312 T^{3} - 22 T^{4} - 8 T^{5} + T^{6}$$
$37$ $$2416 + 2056 T - 180 T^{2} - 384 T^{3} - 62 T^{4} + 4 T^{5} + T^{6}$$
$41$ $$23056 - 23796 T + 6313 T^{2} + 10 T^{3} - 147 T^{4} + 4 T^{5} + T^{6}$$
$43$ $$-45004 - 42150 T - 11235 T^{2} - 202 T^{3} + 255 T^{4} + 30 T^{5} + T^{6}$$
$47$ $$-4400 + 2400 T + 580 T^{2} - 260 T^{3} - 39 T^{4} + 7 T^{5} + T^{6}$$
$53$ $$-1216 + 832 T + 2236 T^{2} - 68 T^{3} - 130 T^{4} + 6 T^{5} + T^{6}$$
$59$ $$-284 - 386 T + 713 T^{2} + 76 T^{3} - 145 T^{4} - 4 T^{5} + T^{6}$$
$61$ $$-6416 - 24024 T + 44 T^{2} + 1592 T^{3} - 104 T^{4} - 14 T^{5} + T^{6}$$
$67$ $$-44 - 994 T - 1783 T^{2} + 682 T^{3} + 15 T^{4} - 18 T^{5} + T^{6}$$
$71$ $$12724 + 27672 T - 14562 T^{2} + 1800 T^{3} + 67 T^{4} - 23 T^{5} + T^{6}$$
$73$ $$55 + 315 T + 594 T^{2} + 481 T^{3} + 170 T^{4} + 23 T^{5} + T^{6}$$
$79$ $$26356 - 8016 T - 13730 T^{2} - 2336 T^{3} + 3 T^{4} + 21 T^{5} + T^{6}$$
$83$ $$80351 - 45717 T - 23604 T^{2} - 2543 T^{3} + 78 T^{4} + 25 T^{5} + T^{6}$$
$89$ $$-2420 + 2310 T + 569 T^{2} - 444 T^{3} - 35 T^{4} + 18 T^{5} + T^{6}$$
$97$ $$-89129 - 42979 T + 15946 T^{2} + 2399 T^{3} - 350 T^{4} - 7 T^{5} + T^{6}$$