# Properties

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

# Related objects

## 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.

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

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
 − 1.24698i 1.80194i 0.445042i − 0.445042i − 1.80194i 1.24698i
1.00000i 1.00000 −1.00000 1.00000i 1.00000i 0.801938i 1.00000i 1.00000 −1.00000
1351.2 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 0.554958i 1.00000i 1.00000 −1.00000
1351.3 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 2.24698i 1.00000i 1.00000 −1.00000
1351.4 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 2.24698i 1.00000i 1.00000 −1.00000
1351.5 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 0.554958i 1.00000i 1.00000 −1.00000
1351.6 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 0.801938i 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.x 6
13.b even 2 1 inner 5070.2.b.x 6
13.d odd 4 1 5070.2.a.bn 3
13.d odd 4 1 5070.2.a.by yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5070.2.a.bn 3 13.d odd 4 1
5070.2.a.by yes 3 13.d odd 4 1
5070.2.b.x 6 1.a even 1 1 trivial
5070.2.b.x 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 $$S_{2}^{\mathrm{new}}(5070, [\chi])$$:

 $$T_{7}^{6} + 6 T_{7}^{4} + 5 T_{7}^{2} + 1$$ $$T_{11}^{6} + 14 T_{11}^{4} + 49 T_{11}^{2} + 49$$ $$T_{17}^{3} + T_{17}^{2} - 2 T_{17} - 1$$ $$T_{31}^{6} + 38 T_{31}^{4} + 381 T_{31}^{2} + 841$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{3}$$
$3$ $$( -1 + T )^{6}$$
$5$ $$( 1 + T^{2} )^{3}$$
$7$ $$1 + 5 T^{2} + 6 T^{4} + T^{6}$$
$11$ $$49 + 49 T^{2} + 14 T^{4} + T^{6}$$
$13$ $$T^{6}$$
$17$ $$( -1 - 2 T + T^{2} + T^{3} )^{2}$$
$19$ $$1849 + 810 T^{2} + 69 T^{4} + T^{6}$$
$23$ $$( 1 - 4 T + 3 T^{2} + T^{3} )^{2}$$
$29$ $$( -1 + 5 T - 6 T^{2} + T^{3} )^{2}$$
$31$ $$841 + 381 T^{2} + 38 T^{4} + T^{6}$$
$37$ $$1 + 41 T^{2} + 54 T^{4} + T^{6}$$
$41$ $$169 + 402 T^{2} + 45 T^{4} + T^{6}$$
$43$ $$( 41 - 30 T + T^{2} + T^{3} )^{2}$$
$47$ $$49 + 147 T^{2} + 35 T^{4} + T^{6}$$
$53$ $$( 1 - 8 T + 5 T^{2} + T^{3} )^{2}$$
$59$ $$117649 + 14406 T^{2} + 245 T^{4} + T^{6}$$
$61$ $$( -29 - 64 T - 5 T^{2} + T^{3} )^{2}$$
$67$ $$9409 + 1454 T^{2} + 69 T^{4} + T^{6}$$
$71$ $$175561 + 12158 T^{2} + 229 T^{4} + T^{6}$$
$73$ $$175561 + 15914 T^{2} + 241 T^{4} + T^{6}$$
$79$ $$( -293 + 24 T + 17 T^{2} + T^{3} )^{2}$$
$83$ $$841 + 341 T^{2} + 34 T^{4} + T^{6}$$
$89$ $$90601 + 11270 T^{2} + 217 T^{4} + T^{6}$$
$97$ $$1352569 + 43389 T^{2} + 402 T^{4} + T^{6}$$