# Properties

 Label 8470.2.a.db 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: 6.6.4642000.1 Defining polynomial: $$x^{6} - x^{5} - 8 x^{4} + 5 x^{3} + 14 x^{2} - 9 x - 1$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 8 x^{4} + 5 x^{3} + 14 x^{2} - 9 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} - 4 \nu^{2} + 6 \nu - 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{5} - 2 \nu^{4} - 4 \nu^{3} + 6 \nu^{2} - \nu$$ $$\beta_{4}$$ $$=$$ $$-\nu^{5} + \nu^{4} + 6 \nu^{3} - \nu^{2} - 5 \nu - 2$$ $$\beta_{5}$$ $$=$$ $$-\nu^{5} + 2 \nu^{4} + 5 \nu^{3} - 7 \nu^{2} - 4 \nu + 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{3} + \beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + 5 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{5} + 6 \beta_{4} + 8 \beta_{3} + 7 \beta_{2} + 4 \beta_{1} + 15$$ $$\nu^{5}$$ $$=$$ $$8 \beta_{5} + 10 \beta_{4} + 19 \beta_{3} + 12 \beta_{2} + 29 \beta_{1} + 16$$

## 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
 2.73934 1.32077 0.803425 −0.0970464 −1.84175 −1.92474
1.00000 −2.73934 1.00000 1.00000 −2.73934 −1.00000 1.00000 4.50401 1.00000
1.2 1.00000 −1.32077 1.00000 1.00000 −1.32077 −1.00000 1.00000 −1.25558 1.00000
1.3 1.00000 −0.803425 1.00000 1.00000 −0.803425 −1.00000 1.00000 −2.35451 1.00000
1.4 1.00000 0.0970464 1.00000 1.00000 0.0970464 −1.00000 1.00000 −2.99058 1.00000
1.5 1.00000 1.84175 1.00000 1.00000 1.84175 −1.00000 1.00000 0.392057 1.00000
1.6 1.00000 1.92474 1.00000 1.00000 1.92474 −1.00000 1.00000 0.704605 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.db 6
11.b odd 2 1 8470.2.a.cv 6
11.c even 5 2 770.2.n.g 12

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

 $$T_{3}^{6} + T_{3}^{5} - 8 T_{3}^{4} - 5 T_{3}^{3} + 14 T_{3}^{2} + 9 T_{3} - 1$$ $$T_{13}^{6} + 6 T_{13}^{5} - 28 T_{13}^{4} - 214 T_{13}^{3} - 68 T_{13}^{2} + 1208 T_{13} + 1516$$ $$T_{17}^{6} + 21 T_{17}^{5} + 138 T_{17}^{4} + 151 T_{17}^{3} - 1274 T_{17}^{2} - 1937 T_{17} + 4609$$ $$T_{19}^{6} - 3 T_{19}^{5} - 56 T_{19}^{4} + 149 T_{19}^{3} + 652 T_{19}^{2} - 891 T_{19} - 941$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{6}$$
$3$ $$-1 + 9 T + 14 T^{2} - 5 T^{3} - 8 T^{4} + T^{5} + T^{6}$$
$5$ $$( -1 + T )^{6}$$
$7$ $$( 1 + T )^{6}$$
$11$ $$T^{6}$$
$13$ $$1516 + 1208 T - 68 T^{2} - 214 T^{3} - 28 T^{4} + 6 T^{5} + T^{6}$$
$17$ $$4609 - 1937 T - 1274 T^{2} + 151 T^{3} + 138 T^{4} + 21 T^{5} + T^{6}$$
$19$ $$-941 - 891 T + 652 T^{2} + 149 T^{3} - 56 T^{4} - 3 T^{5} + T^{6}$$
$23$ $$576 + 288 T - 704 T^{2} - 312 T^{3} - 8 T^{4} + 10 T^{5} + T^{6}$$
$29$ $$-24676 + 21524 T + 2036 T^{2} - 1074 T^{3} - 102 T^{4} + 10 T^{5} + T^{6}$$
$31$ $$164 - 380 T + 272 T^{2} - 34 T^{3} - 26 T^{4} + 4 T^{5} + T^{6}$$
$37$ $$10796 + 8844 T + 932 T^{2} - 546 T^{3} - 108 T^{4} + 2 T^{5} + T^{6}$$
$41$ $$-15829 + 1815 T + 4552 T^{2} - 613 T^{3} - 134 T^{4} + 7 T^{5} + T^{6}$$
$43$ $$2979 - 783 T - 1514 T^{2} - 211 T^{3} + 78 T^{4} + 19 T^{5} + T^{6}$$
$47$ $$704 - 288 T - 896 T^{2} + 424 T^{3} - 24 T^{4} - 10 T^{5} + T^{6}$$
$53$ $$6764 + 5768 T - 6004 T^{2} - 1674 T^{3} - 42 T^{4} + 16 T^{5} + T^{6}$$
$59$ $$-881 - 4361 T + 3272 T^{2} - 13 T^{3} - 152 T^{4} + 3 T^{5} + T^{6}$$
$61$ $$-2650964 - 170580 T + 55392 T^{2} + 2362 T^{3} - 404 T^{4} - 8 T^{5} + T^{6}$$
$67$ $$-103099 - 76739 T - 18804 T^{2} - 1197 T^{3} + 172 T^{4} + 27 T^{5} + T^{6}$$
$71$ $$-171684 - 600 T + 13384 T^{2} + 98 T^{3} - 266 T^{4} - 4 T^{5} + T^{6}$$
$73$ $$-5931 + 5469 T + 1322 T^{2} - 603 T^{3} - 72 T^{4} + 13 T^{5} + T^{6}$$
$79$ $$342676 + 255420 T + 17684 T^{2} - 3918 T^{3} - 286 T^{4} + 14 T^{5} + T^{6}$$
$83$ $$43699 - 73813 T - 388 T^{2} + 5683 T^{3} + 898 T^{4} + 51 T^{5} + T^{6}$$
$89$ $$-89321 + 26647 T + 22746 T^{2} + 909 T^{3} - 372 T^{4} - T^{5} + T^{6}$$
$97$ $$401 - 4093 T - 7166 T^{2} + 2845 T^{3} - 242 T^{4} - 7 T^{5} + T^{6}$$