Properties

 Label 8470.2.a.cu 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: 6.6.38498000.1 Defining polynomial: $$x^{6} - x^{5} - 12 x^{4} + 5 x^{3} + 38 x^{2} - x - 19$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 12 x^{4} + 5 x^{3} + 38 x^{2} - x - 19$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{4} - 2 \nu^{3} - 6 \nu^{2} + 8 \nu + 1$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} - 2 \nu^{4} - 6 \nu^{3} + 8 \nu^{2} + \nu$$$$)/5$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{4} + 2 \nu^{3} + 11 \nu^{2} - 13 \nu - 21$$$$)/5$$ $$\beta_{5}$$ $$=$$ $$($$$$-2 \nu^{5} + 4 \nu^{4} + 17 \nu^{3} - 21 \nu^{2} - 32 \nu + 10$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{2} + \beta_{1} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + 7 \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{5} + 8 \beta_{4} + 4 \beta_{3} + 13 \beta_{2} + 12 \beta_{1} + 27$$ $$\nu^{5}$$ $$=$$ $$10 \beta_{5} + 14 \beta_{4} + 25 \beta_{3} + 24 \beta_{2} + 57 \beta_{1} + 34$$

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
 −2.29803 −1.95171 −0.824369 0.752765 2.15804 3.16330
−1.00000 −3.29803 1.00000 −1.00000 3.29803 −1.00000 −1.00000 7.87703 1.00000
1.2 −1.00000 −2.95171 1.00000 −1.00000 2.95171 −1.00000 −1.00000 5.71258 1.00000
1.3 −1.00000 −1.82437 1.00000 −1.00000 1.82437 −1.00000 −1.00000 0.328323 1.00000
1.4 −1.00000 −0.247235 1.00000 −1.00000 0.247235 −1.00000 −1.00000 −2.93888 1.00000
1.5 −1.00000 1.15804 1.00000 −1.00000 −1.15804 −1.00000 −1.00000 −1.65894 1.00000
1.6 −1.00000 2.16330 1.00000 −1.00000 −2.16330 −1.00000 −1.00000 1.67988 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.cu 6
11.b odd 2 1 8470.2.a.da 6
11.d odd 10 2 770.2.n.h 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.n.h 12 11.d odd 10 2
8470.2.a.cu 6 1.a even 1 1 trivial
8470.2.a.da 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8470))$$:

 $$T_{3}^{6} + 5 T_{3}^{5} - 2 T_{3}^{4} - 33 T_{3}^{3} - 14 T_{3}^{2} + 43 T_{3} + 11$$ $$T_{13}^{6} - 38 T_{13}^{4} + 34 T_{13}^{3} + 316 T_{13}^{2} - 256 T_{13} - 556$$ $$T_{17}^{6} - 15 T_{17}^{5} + 36 T_{17}^{4} + 305 T_{17}^{3} - 1076 T_{17}^{2} - 725 T_{17} + 545$$ $$T_{19}^{6} - 13 T_{19}^{5} + 40 T_{19}^{4} + 103 T_{19}^{3} - 692 T_{19}^{2} + 963 T_{19} - 401$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{6}$$
$3$ $$11 + 43 T - 14 T^{2} - 33 T^{3} - 2 T^{4} + 5 T^{5} + T^{6}$$
$5$ $$( 1 + T )^{6}$$
$7$ $$( 1 + T )^{6}$$
$11$ $$T^{6}$$
$13$ $$-556 - 256 T + 316 T^{2} + 34 T^{3} - 38 T^{4} + T^{6}$$
$17$ $$545 - 725 T - 1076 T^{2} + 305 T^{3} + 36 T^{4} - 15 T^{5} + T^{6}$$
$19$ $$-401 + 963 T - 692 T^{2} + 103 T^{3} + 40 T^{4} - 13 T^{5} + T^{6}$$
$23$ $$3520 - 4640 T + 1664 T^{2} + 24 T^{3} - 80 T^{4} + 2 T^{5} + T^{6}$$
$29$ $$-484 + 196 T + 568 T^{2} + 70 T^{3} - 52 T^{4} - 4 T^{5} + T^{6}$$
$31$ $$2420 + 7700 T + 2476 T^{2} - 262 T^{3} - 116 T^{4} + 2 T^{5} + T^{6}$$
$37$ $$-6844 + 412 T + 1176 T^{2} - 22 T^{3} - 62 T^{4} + T^{6}$$
$41$ $$-3509 - 3795 T - 642 T^{2} + 357 T^{3} + 46 T^{4} - 17 T^{5} + T^{6}$$
$43$ $$-29 + 449 T - 396 T^{2} - 197 T^{3} + 36 T^{4} + 15 T^{5} + T^{6}$$
$47$ $$-17216 + 10272 T + 608 T^{2} - 888 T^{3} + 18 T^{5} + T^{6}$$
$53$ $$1804 + 2956 T - 7460 T^{2} + 1974 T^{3} + 8 T^{4} - 22 T^{5} + T^{6}$$
$59$ $$-54121 + 32603 T + 11228 T^{2} - 1197 T^{3} - 220 T^{4} + 7 T^{5} + T^{6}$$
$61$ $$-22084 + 52680 T - 22760 T^{2} + 3162 T^{3} - 40 T^{4} - 20 T^{5} + T^{6}$$
$67$ $$-15679 - 28301 T - 14630 T^{2} - 1465 T^{3} + 170 T^{4} + 29 T^{5} + T^{6}$$
$71$ $$44 - 2248 T + 3648 T^{2} - 150 T^{3} - 168 T^{4} + 2 T^{5} + T^{6}$$
$73$ $$-292471 - 40435 T + 20468 T^{2} + 211 T^{3} - 266 T^{4} + T^{5} + T^{6}$$
$79$ $$-49484 - 14280 T + 3368 T^{2} + 882 T^{3} - 84 T^{4} - 12 T^{5} + T^{6}$$
$83$ $$1711 - 2057 T - 6264 T^{2} - 323 T^{3} + 348 T^{4} - 35 T^{5} + T^{6}$$
$89$ $$370271 + 15771 T - 25312 T^{2} - 3011 T^{3} + 120 T^{4} + 29 T^{5} + T^{6}$$
$97$ $$2383105 + 626365 T + 11284 T^{2} - 7043 T^{3} - 320 T^{4} + 19 T^{5} + T^{6}$$