# Properties

 Label 5070.2.b.s Level $5070$ Weight $2$ Character orbit 5070.b Analytic conductor $40.484$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$40.4841538248$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.153664.1 Defining polynomial: $$x^{6} + 5 x^{4} + 6 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{2} - q^{3} - q^{4} + \beta_{5} q^{5} -\beta_{5} q^{6} + ( \beta_{1} + \beta_{3} ) q^{7} -\beta_{5} q^{8} + q^{9} +O(q^{10})$$ $$q + \beta_{5} q^{2} - q^{3} - q^{4} + \beta_{5} q^{5} -\beta_{5} q^{6} + ( \beta_{1} + \beta_{3} ) q^{7} -\beta_{5} q^{8} + q^{9} - q^{10} + ( -\beta_{1} + 5 \beta_{5} ) q^{11} + q^{12} + ( 1 - \beta_{2} - 2 \beta_{4} ) q^{14} -\beta_{5} q^{15} + q^{16} + ( -1 - \beta_{2} + \beta_{4} ) q^{17} + \beta_{5} q^{18} + ( 5 \beta_{1} - 3 \beta_{3} - 5 \beta_{5} ) q^{19} -\beta_{5} q^{20} + ( -\beta_{1} - \beta_{3} ) q^{21} + ( -5 + \beta_{4} ) q^{22} + ( 5 - 7 \beta_{2} - 3 \beta_{4} ) q^{23} + \beta_{5} q^{24} - q^{25} - q^{27} + ( -\beta_{1} - \beta_{3} ) q^{28} + ( 1 + 3 \beta_{2} + 6 \beta_{4} ) q^{29} + q^{30} + ( 3 \beta_{1} - 4 \beta_{3} - 7 \beta_{5} ) q^{31} + \beta_{5} q^{32} + ( \beta_{1} - 5 \beta_{5} ) q^{33} + ( 2 \beta_{1} - \beta_{3} - 2 \beta_{5} ) q^{34} + ( 1 - \beta_{2} - 2 \beta_{4} ) q^{35} - q^{36} + ( -\beta_{1} - 3 \beta_{3} - 2 \beta_{5} ) q^{37} + ( 2 + 3 \beta_{2} - 2 \beta_{4} ) q^{38} + q^{40} + ( -5 \beta_{1} + 5 \beta_{3} - \beta_{5} ) q^{41} + ( -1 + \beta_{2} + 2 \beta_{4} ) q^{42} + ( 2 \beta_{2} + 3 \beta_{4} ) q^{43} + ( \beta_{1} - 5 \beta_{5} ) q^{44} + \beta_{5} q^{45} + ( 4 \beta_{1} - 7 \beta_{3} - 2 \beta_{5} ) q^{46} + ( 4 \beta_{1} - 2 \beta_{3} + 5 \beta_{5} ) q^{47} - q^{48} + ( 1 + \beta_{2} + 3 \beta_{4} ) q^{49} -\beta_{5} q^{50} + ( 1 + \beta_{2} - \beta_{4} ) q^{51} + ( 2 - 3 \beta_{2} ) q^{53} -\beta_{5} q^{54} + ( -5 + \beta_{4} ) q^{55} + ( -1 + \beta_{2} + 2 \beta_{4} ) q^{56} + ( -5 \beta_{1} + 3 \beta_{3} + 5 \beta_{5} ) q^{57} + ( 3 \beta_{1} + 3 \beta_{3} + 4 \beta_{5} ) q^{58} + ( -3 \beta_{1} - 4 \beta_{3} + 2 \beta_{5} ) q^{59} + \beta_{5} q^{60} + ( -4 - 6 \beta_{2} + \beta_{4} ) q^{61} + ( 3 + 4 \beta_{2} + \beta_{4} ) q^{62} + ( \beta_{1} + \beta_{3} ) q^{63} - q^{64} + ( 5 - \beta_{4} ) q^{66} + ( -\beta_{3} - 2 \beta_{5} ) q^{67} + ( 1 + \beta_{2} - \beta_{4} ) q^{68} + ( -5 + 7 \beta_{2} + 3 \beta_{4} ) q^{69} + ( -\beta_{1} - \beta_{3} ) q^{70} + ( 4 \beta_{1} - 7 \beta_{3} - 8 \beta_{5} ) q^{71} -\beta_{5} q^{72} + ( 7 \beta_{3} - 2 \beta_{5} ) q^{73} + ( -1 + 3 \beta_{2} + 4 \beta_{4} ) q^{74} + q^{75} + ( -5 \beta_{1} + 3 \beta_{3} + 5 \beta_{5} ) q^{76} + ( 8 - 6 \beta_{2} - 11 \beta_{4} ) q^{77} + ( -6 + 5 \beta_{2} + 6 \beta_{4} ) q^{79} + \beta_{5} q^{80} + q^{81} + ( 6 - 5 \beta_{2} ) q^{82} + ( -5 \beta_{1} - 2 \beta_{3} - 3 \beta_{5} ) q^{83} + ( \beta_{1} + \beta_{3} ) q^{84} + ( 2 \beta_{1} - \beta_{3} - 2 \beta_{5} ) q^{85} + ( \beta_{1} + 2 \beta_{3} + 2 \beta_{5} ) q^{86} + ( -1 - 3 \beta_{2} - 6 \beta_{4} ) q^{87} + ( 5 - \beta_{4} ) q^{88} + ( 2 \beta_{1} + 5 \beta_{3} + 2 \beta_{5} ) q^{89} - q^{90} + ( -5 + 7 \beta_{2} + 3 \beta_{4} ) q^{92} + ( -3 \beta_{1} + 4 \beta_{3} + 7 \beta_{5} ) q^{93} + ( -7 + 2 \beta_{2} - 2 \beta_{4} ) q^{94} + ( 2 + 3 \beta_{2} - 2 \beta_{4} ) q^{95} -\beta_{5} q^{96} + ( 3 \beta_{1} - 3 \beta_{3} - 2 \beta_{5} ) q^{97} + ( 2 \beta_{1} + \beta_{3} + 2 \beta_{5} ) q^{98} + ( -\beta_{1} + 5 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 6q^{3} - 6q^{4} + 6q^{9} + O(q^{10})$$ $$6q - 6q^{3} - 6q^{4} + 6q^{9} - 6q^{10} + 6q^{12} + 6q^{16} - 6q^{17} - 28q^{22} + 10q^{23} - 6q^{25} - 6q^{27} + 24q^{29} + 6q^{30} - 6q^{36} + 14q^{38} + 6q^{40} + 10q^{43} - 6q^{48} + 14q^{49} + 6q^{51} + 6q^{53} - 28q^{55} - 34q^{61} + 28q^{62} - 6q^{64} + 28q^{66} + 6q^{68} - 10q^{69} + 8q^{74} + 6q^{75} + 14q^{77} - 14q^{79} + 6q^{81} + 26q^{82} - 24q^{87} + 28q^{88} - 6q^{90} - 10q^{92} - 42q^{94} + 14q^{95} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 3 \nu$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} + 3 \nu^{2} + 1$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} + 4 \nu^{3} + 3 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} - 3 \beta_{2} + 5$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} - 4 \beta_{3} + 9 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$$.

 $$n$$ $$1691$$ $$1861$$ $$4057$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1351.1
 − 0.445042i − 1.80194i 1.24698i − 1.24698i 1.80194i 0.445042i
1.00000i −1.00000 −1.00000 1.00000i 1.00000i 1.69202i 1.00000i 1.00000 −1.00000
1351.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 1.35690i 1.00000i 1.00000 −1.00000
1351.3 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 3.04892i 1.00000i 1.00000 −1.00000
1351.4 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 3.04892i 1.00000i 1.00000 −1.00000
1351.5 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 1.35690i 1.00000i 1.00000 −1.00000
1351.6 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 1.69202i 1.00000i 1.00000 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1351.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5070.2.b.s 6
13.b even 2 1 inner 5070.2.b.s 6
13.d odd 4 1 5070.2.a.bk 3
13.d odd 4 1 5070.2.a.bt yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5070.2.a.bk 3 13.d odd 4 1
5070.2.a.bt yes 3 13.d odd 4 1
5070.2.b.s 6 1.a even 1 1 trivial
5070.2.b.s 6 13.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(5070, [\chi])$$:

 $$T_{7}^{6} + 14 T_{7}^{4} + 49 T_{7}^{2} + 49$$ $$T_{11}^{6} + 70 T_{11}^{4} + 1421 T_{11}^{2} + 8281$$ $$T_{17}^{3} + 3 T_{17}^{2} - 4 T_{17} - 13$$ $$T_{31}^{6} + 126 T_{31}^{4} + 1421 T_{31}^{2} + 49$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{3}$$
$3$ $$( 1 + T )^{6}$$
$5$ $$( 1 + T^{2} )^{3}$$
$7$ $$49 + 49 T^{2} + 14 T^{4} + T^{6}$$
$11$ $$8281 + 1421 T^{2} + 70 T^{4} + T^{6}$$
$13$ $$T^{6}$$
$17$ $$( -13 - 4 T + 3 T^{2} + T^{3} )^{2}$$
$19$ $$41209 + 3626 T^{2} + 105 T^{4} + T^{6}$$
$23$ $$( 419 - 78 T - 5 T^{2} + T^{3} )^{2}$$
$29$ $$( 377 - 15 T - 12 T^{2} + T^{3} )^{2}$$
$31$ $$49 + 1421 T^{2} + 126 T^{4} + T^{6}$$
$37$ $$5041 + 1193 T^{2} + 66 T^{4} + T^{6}$$
$41$ $$19321 + 3618 T^{2} + 173 T^{4} + T^{6}$$
$43$ $$( 41 - 8 T - 5 T^{2} + T^{3} )^{2}$$
$47$ $$41209 + 5635 T^{2} + 203 T^{4} + T^{6}$$
$53$ $$( 13 - 18 T - 3 T^{2} + T^{3} )^{2}$$
$59$ $$8281 + 3626 T^{2} + 189 T^{4} + T^{6}$$
$61$ $$( -601 - 4 T + 17 T^{2} + T^{3} )^{2}$$
$67$ $$1 + 26 T^{2} + 13 T^{4} + T^{6}$$
$71$ $$169 + 1238 T^{2} + 229 T^{4} + T^{6}$$
$73$ $$253009 + 16442 T^{2} + 285 T^{4} + T^{6}$$
$79$ $$( -91 - 56 T + 7 T^{2} + T^{3} )^{2}$$
$83$ $$344569 + 15937 T^{2} + 230 T^{4} + T^{6}$$
$89$ $$85849 + 9502 T^{2} + 185 T^{4} + T^{6}$$
$97$ $$49 + 441 T^{2} + 42 T^{4} + T^{6}$$
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