# Properties

 Label 8470.2.a.de 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: 6.6.13298000.1 Defining polynomial: $$x^{6} - x^{5} - 10 x^{4} + 3 x^{3} + 26 x^{2} + 13 x - 1$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 10 x^{4} + 3 x^{3} + 26 x^{2} + 13 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + 4 \nu^{4} + 8 \nu^{3} - 22 \nu^{2} - 20 \nu - 3$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{5} + 3 \nu^{4} + 16 \nu^{3} - 14 \nu^{2} - 30 \nu - 6$$$$)/5$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{5} + 3 \nu^{4} + 21 \nu^{3} - 14 \nu^{2} - 60 \nu - 16$$$$)/5$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{5} - 7 \nu^{4} - 24 \nu^{3} + 41 \nu^{2} + 45 \nu - 11$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{3} + \beta_{2} + \beta_{1} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{4} - \beta_{3} + 6 \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$6 \beta_{5} + 5 \beta_{3} + 8 \beta_{2} + 8 \beta_{1} + 24$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{5} + 8 \beta_{4} - 10 \beta_{3} + 5 \beta_{2} + 38 \beta_{1} + 21$$

## 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
 −2.27063 2.79700 −1.24667 2.42266 0.0677009 −0.770059
1.00000 −2.80837 1.00000 1.00000 −2.80837 1.00000 1.00000 4.88695 1.00000
1.2 1.00000 −2.64424 1.00000 1.00000 −2.64424 1.00000 1.00000 3.99201 1.00000
1.3 1.00000 −0.418877 1.00000 1.00000 −0.418877 1.00000 1.00000 −2.82454 1.00000
1.4 1.00000 1.17142 1.00000 1.00000 1.17142 1.00000 1.00000 −1.62777 1.00000
1.5 1.00000 2.44508 1.00000 1.00000 2.44508 1.00000 1.00000 2.97843 1.00000
1.6 1.00000 3.25498 1.00000 1.00000 3.25498 1.00000 1.00000 7.59492 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.de 6
11.b odd 2 1 8470.2.a.cy 6
11.c even 5 2 770.2.n.i 12

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

 $$T_{3}^{6} - T_{3}^{5} - 16 T_{3}^{4} + 13 T_{3}^{3} + 66 T_{3}^{2} - 45 T_{3} - 29$$ $$T_{13}^{6} - 2 T_{13}^{5} - 32 T_{13}^{4} + 94 T_{13}^{3} + 160 T_{13}^{2} - 704 T_{13} + 484$$ $$T_{17}^{6} - 7 T_{17}^{5} - 20 T_{17}^{4} + 183 T_{17}^{3} - 308 T_{17}^{2} + 139 T_{17} + 11$$ $$T_{19}^{6} - 11 T_{19}^{5} + 20 T_{19}^{4} + 177 T_{19}^{3} - 896 T_{19}^{2} + 1445 T_{19} - 725$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{6}$$
$3$ $$-29 - 45 T + 66 T^{2} + 13 T^{3} - 16 T^{4} - T^{5} + T^{6}$$
$5$ $$( -1 + T )^{6}$$
$7$ $$( -1 + T )^{6}$$
$11$ $$T^{6}$$
$13$ $$484 - 704 T + 160 T^{2} + 94 T^{3} - 32 T^{4} - 2 T^{5} + T^{6}$$
$17$ $$11 + 139 T - 308 T^{2} + 183 T^{3} - 20 T^{4} - 7 T^{5} + T^{6}$$
$19$ $$-725 + 1445 T - 896 T^{2} + 177 T^{3} + 20 T^{4} - 11 T^{5} + T^{6}$$
$23$ $$-1856 + 480 T + 768 T^{2} - 184 T^{3} - 56 T^{4} + 6 T^{5} + T^{6}$$
$29$ $$3020 - 3340 T + 776 T^{2} + 178 T^{3} - 62 T^{4} - 2 T^{5} + T^{6}$$
$31$ $$-284 - 468 T + 316 T^{2} + 58 T^{3} - 42 T^{4} + T^{6}$$
$37$ $$-4 + 276 T - 4320 T^{2} - 1334 T^{3} - 52 T^{4} + 14 T^{5} + T^{6}$$
$41$ $$3649 - 225 T - 1254 T^{2} + 351 T^{3} + 16 T^{4} - 13 T^{5} + T^{6}$$
$43$ $$-20719 + 24251 T - 9780 T^{2} + 1415 T^{3} + 20 T^{4} - 19 T^{5} + T^{6}$$
$47$ $$49856 + 9824 T - 12768 T^{2} + 1928 T^{3} + 40 T^{4} - 22 T^{5} + T^{6}$$
$53$ $$-244 + 1880 T + 800 T^{2} - 338 T^{3} - 50 T^{4} + 10 T^{5} + T^{6}$$
$59$ $$95 + 175 T - 124 T^{2} - 293 T^{3} - 72 T^{4} + 7 T^{5} + T^{6}$$
$61$ $$-112156 + 92484 T - 25280 T^{2} + 2478 T^{3} + 44 T^{4} - 22 T^{5} + T^{6}$$
$67$ $$-5821 + 2633 T + 6014 T^{2} + 1179 T^{3} - 226 T^{4} - 5 T^{5} + T^{6}$$
$71$ $$-13036 - 5160 T + 3516 T^{2} + 1086 T^{3} - 174 T^{4} - 8 T^{5} + T^{6}$$
$73$ $$71779 - 46377 T - 1552 T^{2} + 2443 T^{3} - 170 T^{4} - 13 T^{5} + T^{6}$$
$79$ $$-848020 + 198380 T + 19424 T^{2} - 3822 T^{3} - 250 T^{4} + 16 T^{5} + T^{6}$$
$83$ $$-53629 - 2471 T + 6724 T^{2} - 27 T^{3} - 194 T^{4} + 5 T^{5} + T^{6}$$
$89$ $$-362975 + 55835 T + 31116 T^{2} - 179 T^{3} - 338 T^{4} - T^{5} + T^{6}$$
$97$ $$-4649 - 10511 T - 8088 T^{2} - 2437 T^{3} - 240 T^{4} + 3 T^{5} + T^{6}$$