# Properties

 Label 8470.2.a.dh Level $8470$ Weight $2$ Character orbit 8470.a Self dual yes Analytic conductor $67.633$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 16 x^{6} + 69 x^{4} - 10 x^{3} - 70 x^{2} + 10 x + 5$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + \beta_{1} q^{3} + q^{4} - q^{5} + \beta_{1} q^{6} - q^{7} + q^{8} + ( 1 + \beta_{5} - \beta_{6} ) q^{9} +O(q^{10})$$ $$q + q^{2} + \beta_{1} q^{3} + q^{4} - q^{5} + \beta_{1} q^{6} - q^{7} + q^{8} + ( 1 + \beta_{5} - \beta_{6} ) q^{9} - q^{10} + \beta_{1} q^{12} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} ) q^{13} - q^{14} -\beta_{1} q^{15} + q^{16} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{7} ) q^{17} + ( 1 + \beta_{5} - \beta_{6} ) q^{18} + ( -1 + 2 \beta_{1} - 3 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{19} - q^{20} -\beta_{1} q^{21} + ( 2 - 2 \beta_{2} ) q^{23} + \beta_{1} q^{24} + q^{25} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} ) q^{26} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{27} - q^{28} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{29} -\beta_{1} q^{30} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{31} + q^{32} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{7} ) q^{34} + q^{35} + ( 1 + \beta_{5} - \beta_{6} ) q^{36} + ( -2 + \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{37} + ( -1 + 2 \beta_{1} - 3 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{38} + ( 7 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{39} - q^{40} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{41} -\beta_{1} q^{42} + ( -1 + \beta_{2} + 2 \beta_{4} + \beta_{6} ) q^{43} + ( -1 - \beta_{5} + \beta_{6} ) q^{45} + ( 2 - 2 \beta_{2} ) q^{46} + ( -1 + 3 \beta_{3} - \beta_{4} + 2 \beta_{6} ) q^{47} + \beta_{1} q^{48} + q^{49} + q^{50} + ( -2 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{51} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} ) q^{52} + ( -3 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{6} - \beta_{7} ) q^{53} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{54} - q^{56} + ( 4 + \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{57} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{58} + ( -2 + 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{59} -\beta_{1} q^{60} + ( 6 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{61} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{62} + ( -1 - \beta_{5} + \beta_{6} ) q^{63} + q^{64} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} ) q^{65} + ( 4 - 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} ) q^{67} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{7} ) q^{68} + ( 8 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{69} + q^{70} + ( 2 + \beta_{1} + \beta_{2} - 2 \beta_{7} ) q^{71} + ( 1 + \beta_{5} - \beta_{6} ) q^{72} + ( -3 + \beta_{2} - \beta_{4} - \beta_{6} - 3 \beta_{7} ) q^{73} + ( -2 + \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{74} + \beta_{1} q^{75} + ( -1 + 2 \beta_{1} - 3 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{76} + ( 7 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{78} + ( 6 + 2 \beta_{1} - \beta_{2} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{79} - q^{80} + ( 5 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{6} ) q^{81} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{82} + ( 6 + \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{83} -\beta_{1} q^{84} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} ) q^{85} + ( -1 + \beta_{2} + 2 \beta_{4} + \beta_{6} ) q^{86} + ( -3 + \beta_{1} - 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{87} + ( -1 + \beta_{1} - \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{7} ) q^{89} + ( -1 - \beta_{5} + \beta_{6} ) q^{90} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} ) q^{91} + ( 2 - 2 \beta_{2} ) q^{92} + ( 1 - 3 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{93} + ( -1 + 3 \beta_{3} - \beta_{4} + 2 \beta_{6} ) q^{94} + ( 1 - 2 \beta_{1} + 3 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{95} + \beta_{1} q^{96} + ( 1 - 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{97} + q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{2} + 8q^{4} - 8q^{5} - 8q^{7} + 8q^{8} + 8q^{9} + O(q^{10})$$ $$8q + 8q^{2} + 8q^{4} - 8q^{5} - 8q^{7} + 8q^{8} + 8q^{9} - 8q^{10} - q^{13} - 8q^{14} + 8q^{16} + 6q^{17} + 8q^{18} + 5q^{19} - 8q^{20} + 10q^{23} + 8q^{25} - q^{26} - 8q^{28} + 3q^{29} - 8q^{31} + 8q^{32} + 6q^{34} + 8q^{35} + 8q^{36} - 6q^{37} + 5q^{38} + 35q^{39} - 8q^{40} + 11q^{41} - 5q^{43} - 8q^{45} + 10q^{46} - 15q^{47} + 8q^{49} + 8q^{50} - 6q^{51} - q^{52} - 16q^{53} - 8q^{56} + 38q^{57} + 3q^{58} - 9q^{59} + 32q^{61} - 8q^{62} - 8q^{63} + 8q^{64} + q^{65} + 33q^{67} + 6q^{68} - 22q^{69} + 8q^{70} + 11q^{71} + 8q^{72} - 34q^{73} - 6q^{74} + 5q^{76} + 35q^{78} + 31q^{79} - 8q^{80} + 20q^{81} + 11q^{82} + 50q^{83} - 6q^{85} - 5q^{86} - 12q^{87} + q^{89} - 8q^{90} + q^{91} + 10q^{92} + 26q^{93} - 15q^{94} - 5q^{95} - 4q^{97} + 8q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 16 x^{6} + 69 x^{4} - 10 x^{3} - 70 x^{2} + 10 x + 5$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-3 \nu^{6} + \nu^{5} + 34 \nu^{4} - 3 \nu^{3} - 70 \nu^{2} + 17 \nu + 29$$$$)/34$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + 19 \nu^{5} - \nu^{4} - 103 \nu^{3} + 13 \nu^{2} + 140 \nu - 27$$$$)/34$$ $$\beta_{4}$$ $$=$$ $$($$$$4 \nu^{7} - 5 \nu^{6} - 63 \nu^{5} + 72 \nu^{4} + 271 \nu^{3} - 282 \nu^{2} - 237 \nu + 111$$$$)/34$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{7} - \nu^{6} + 46 \nu^{5} + 14 \nu^{4} - 174 \nu^{3} - 7 \nu^{2} + 97 \nu - 26$$$$)/17$$ $$\beta_{6}$$ $$=$$ $$($$$$-3 \nu^{7} - \nu^{6} + 46 \nu^{5} + 14 \nu^{4} - 174 \nu^{3} - 24 \nu^{2} + 97 \nu + 42$$$$)/17$$ $$\beta_{7}$$ $$=$$ $$($$$$9 \nu^{7} + 6 \nu^{6} - 139 \nu^{5} - 93 \nu^{4} + 559 \nu^{3} + 261 \nu^{2} - 512 \nu - 87$$$$)/34$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{6} + \beta_{5} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 7 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-8 \beta_{6} + 9 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + 2 \beta_{1} + 32$$ $$\nu^{5}$$ $$=$$ $$12 \beta_{7} + 10 \beta_{6} + 13 \beta_{5} + 13 \beta_{4} + 22 \beta_{3} - 13 \beta_{2} + 56 \beta_{1} + 12$$ $$\nu^{6}$$ $$=$$ $$3 \beta_{7} - 65 \beta_{6} + 82 \beta_{5} + 26 \beta_{4} + 29 \beta_{3} - 60 \beta_{2} + 40 \beta_{1} + 283$$ $$\nu^{7}$$ $$=$$ $$125 \beta_{7} + 82 \beta_{6} + 148 \beta_{5} + 142 \beta_{4} + 279 \beta_{3} - 140 \beta_{2} + 481 \beta_{1} + 221$$

## 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
 −2.90474 −2.53932 −1.12455 −0.211079 0.365778 1.22201 2.00431 3.18761
1.00000 −2.90474 1.00000 −1.00000 −2.90474 −1.00000 1.00000 5.43751 −1.00000
1.2 1.00000 −2.53932 1.00000 −1.00000 −2.53932 −1.00000 1.00000 3.44815 −1.00000
1.3 1.00000 −1.12455 1.00000 −1.00000 −1.12455 −1.00000 1.00000 −1.73538 −1.00000
1.4 1.00000 −0.211079 1.00000 −1.00000 −0.211079 −1.00000 1.00000 −2.95545 −1.00000
1.5 1.00000 0.365778 1.00000 −1.00000 0.365778 −1.00000 1.00000 −2.86621 −1.00000
1.6 1.00000 1.22201 1.00000 −1.00000 1.22201 −1.00000 1.00000 −1.50670 −1.00000
1.7 1.00000 2.00431 1.00000 −1.00000 2.00431 −1.00000 1.00000 1.01724 −1.00000
1.8 1.00000 3.18761 1.00000 −1.00000 3.18761 −1.00000 1.00000 7.16083 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 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.dh 8
11.b odd 2 1 8470.2.a.dg 8
11.c even 5 2 770.2.n.k 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.n.k 16 11.c even 5 2
8470.2.a.dg 8 11.b odd 2 1
8470.2.a.dh 8 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8470))$$:

 $$T_{3}^{8} - 16 T_{3}^{6} + 69 T_{3}^{4} - 10 T_{3}^{3} - 70 T_{3}^{2} + 10 T_{3} + 5$$ $$T_{13}^{8} + \cdots$$ $$T_{17}^{8} - \cdots$$ $$T_{19}^{8} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{8}$$
$3$ $$5 + 10 T - 70 T^{2} - 10 T^{3} + 69 T^{4} - 16 T^{6} + T^{8}$$
$5$ $$( 1 + T )^{8}$$
$7$ $$( 1 + T )^{8}$$
$11$ $$T^{8}$$
$13$ $$-12764 - 39548 T - 14908 T^{2} + 4850 T^{3} + 2216 T^{4} - 140 T^{5} - 87 T^{6} + T^{7} + T^{8}$$
$17$ $$109 + 556 T - 2152 T^{2} - 3264 T^{3} + 749 T^{4} + 350 T^{5} - 62 T^{6} - 6 T^{7} + T^{8}$$
$19$ $$-70900 + 158450 T - 33635 T^{2} - 15235 T^{3} + 3534 T^{4} + 485 T^{5} - 106 T^{6} - 5 T^{7} + T^{8}$$
$23$ $$1280 - 5760 T + 8000 T^{2} - 3200 T^{3} - 576 T^{4} + 480 T^{5} - 36 T^{6} - 10 T^{7} + T^{8}$$
$29$ $$-16636 + 24912 T - 3908 T^{2} - 7922 T^{3} + 2734 T^{4} + 310 T^{5} - 113 T^{6} - 3 T^{7} + T^{8}$$
$31$ $$135344 + 277528 T + 190172 T^{2} + 45924 T^{3} + 204 T^{4} - 1350 T^{5} - 120 T^{6} + 8 T^{7} + T^{8}$$
$37$ $$3280 + 1080 T - 7140 T^{2} - 1020 T^{3} + 3584 T^{4} - 442 T^{5} - 130 T^{6} + 6 T^{7} + T^{8}$$
$41$ $$21296 + 1026300 T - 180387 T^{2} - 101649 T^{3} + 9544 T^{4} + 2279 T^{5} - 204 T^{6} - 11 T^{7} + T^{8}$$
$43$ $$-591484 + 612198 T - 169261 T^{2} - 14663 T^{3} + 10984 T^{4} - 393 T^{5} - 188 T^{6} + 5 T^{7} + T^{8}$$
$47$ $$614336 + 972960 T + 504544 T^{2} + 83560 T^{3} - 4944 T^{4} - 2410 T^{5} - 101 T^{6} + 15 T^{7} + T^{8}$$
$53$ $$10229824 - 7182736 T - 563892 T^{2} + 301404 T^{3} + 17024 T^{4} - 3970 T^{5} - 232 T^{6} + 16 T^{7} + T^{8}$$
$59$ $$10961956 - 1447614 T - 806503 T^{2} + 87723 T^{3} + 21090 T^{4} - 1613 T^{5} - 238 T^{6} + 9 T^{7} + T^{8}$$
$61$ $$58256 - 131832 T - 2684 T^{2} + 28128 T^{3} - 4756 T^{4} - 962 T^{5} + 336 T^{6} - 32 T^{7} + T^{8}$$
$67$ $$162964 - 333930 T - 13175 T^{2} + 80879 T^{3} - 19566 T^{4} + 435 T^{5} + 306 T^{6} - 33 T^{7} + T^{8}$$
$71$ $$44 - 524 T - 3240 T^{2} - 750 T^{3} + 5786 T^{4} + 708 T^{5} - 143 T^{6} - 11 T^{7} + T^{8}$$
$73$ $$-7365731 + 3020566 T + 1777518 T^{2} - 32644 T^{3} - 64771 T^{4} - 5470 T^{5} + 168 T^{6} + 34 T^{7} + T^{8}$$
$79$ $$15119876 - 8618540 T + 804008 T^{2} + 362726 T^{3} - 93746 T^{4} + 6944 T^{5} + 81 T^{6} - 31 T^{7} + T^{8}$$
$83$ $$-13602031 + 14467992 T - 5572566 T^{2} + 961370 T^{3} - 58159 T^{4} - 4336 T^{5} + 880 T^{6} - 50 T^{7} + T^{8}$$
$89$ $$108020 - 22690 T - 62945 T^{2} + 14135 T^{3} + 6756 T^{4} - 539 T^{5} - 208 T^{6} - T^{7} + T^{8}$$
$97$ $$-968759 - 1541370 T - 87842 T^{2} + 187326 T^{3} + 31449 T^{4} - 1716 T^{5} - 384 T^{6} + 4 T^{7} + T^{8}$$