Properties

 Label 8470.2.a.cr Level $8470$ Weight $2$ Character orbit 8470.a Self dual yes Analytic conductor $67.633$ Analytic rank $1$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\zeta_{24})^+$$ Defining polynomial: $$x^{4} - 4 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

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
 −1.93185 0.517638 −0.517638 1.93185
1.00000 −2.41421 1.00000 1.00000 −2.41421 −1.00000 1.00000 2.82843 1.00000
1.2 1.00000 −2.41421 1.00000 1.00000 −2.41421 −1.00000 1.00000 2.82843 1.00000
1.3 1.00000 0.414214 1.00000 1.00000 0.414214 −1.00000 1.00000 −2.82843 1.00000
1.4 1.00000 0.414214 1.00000 1.00000 0.414214 −1.00000 1.00000 −2.82843 1.00000
 $$n$$: e.g. 2-40 or 990-1000 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.cr yes 4
11.b odd 2 1 8470.2.a.cp 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8470.2.a.cp 4 11.b odd 2 1
8470.2.a.cr yes 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8470))$$:

 $$T_{3}^{2} + 2 T_{3} - 1$$ $$T_{13}^{4} - 22 T_{13}^{2} + 48 T_{13} - 23$$ $$T_{17}^{2} - 4 T_{17} - 2$$ $$T_{19}^{4} + 12 T_{19}^{3} + 26 T_{19}^{2} - 60 T_{19} - 167$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{4}$$
$3$ $$( -1 + 2 T + T^{2} )^{2}$$
$5$ $$( -1 + T )^{4}$$
$7$ $$( 1 + T )^{4}$$
$11$ $$T^{4}$$
$13$ $$-23 + 48 T - 22 T^{2} + T^{4}$$
$17$ $$( -2 - 4 T + T^{2} )^{2}$$
$19$ $$-167 - 60 T + 26 T^{2} + 12 T^{3} + T^{4}$$
$23$ $$1 - 8 T - 10 T^{2} + 8 T^{3} + T^{4}$$
$29$ $$16 - 32 T - 16 T^{2} + 8 T^{3} + T^{4}$$
$31$ $$2116 - 124 T^{2} + T^{4}$$
$37$ $$-800 - 160 T + 56 T^{2} + 16 T^{3} + T^{4}$$
$41$ $$-2396 + 976 T - 84 T^{2} - 8 T^{3} + T^{4}$$
$43$ $$-284 - 304 T - 60 T^{2} + 8 T^{3} + T^{4}$$
$47$ $$772 - 544 T - 4 T^{2} + 16 T^{3} + T^{4}$$
$53$ $$4 - 192 T - 76 T^{2} + T^{4}$$
$59$ $$73 - 136 T - 18 T^{2} + 8 T^{3} + T^{4}$$
$61$ $$-128 - 128 T - 16 T^{2} + 8 T^{3} + T^{4}$$
$67$ $$900 - 132 T^{2} + T^{4}$$
$71$ $$-3008 - 1984 T - 192 T^{2} + 8 T^{3} + T^{4}$$
$73$ $$( 2 - 4 T + T^{2} )^{2}$$
$79$ $$-1031 + 72 T + 158 T^{2} + 24 T^{3} + T^{4}$$
$83$ $$-5807 + 2632 T - 282 T^{2} - 8 T^{3} + T^{4}$$
$89$ $$676 - 148 T^{2} + T^{4}$$
$97$ $$-2672 - 1504 T - 144 T^{2} + 8 T^{3} + T^{4}$$