# Properties

 Label 5070.2.b.u 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.

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

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 0.445042i 1.80194i − 1.80194i − 0.445042i 1.24698i
1.00000i −1.00000 −1.00000 1.00000i 1.00000i 4.24698i 1.00000i 1.00000 1.00000
1351.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 2.55496i 1.00000i 1.00000 1.00000
1351.3 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 1.19806i 1.00000i 1.00000 1.00000
1351.4 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 1.19806i 1.00000i 1.00000 1.00000
1351.5 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 2.55496i 1.00000i 1.00000 1.00000
1351.6 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 4.24698i 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.u 6
13.b even 2 1 inner 5070.2.b.u 6
13.d odd 4 1 5070.2.a.bm 3
13.d odd 4 1 5070.2.a.br yes 3

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

 $$T_{7}^{6} + 26 T_{7}^{4} + 153 T_{7}^{2} + 169$$ $$T_{11}^{6} + 6 T_{11}^{4} + 5 T_{11}^{2} + 1$$ $$T_{17}^{3} + 3 T_{17}^{2} - 46 T_{17} - 97$$ $$T_{31}^{6} + 14 T_{31}^{4} + 49 T_{31}^{2} + 49$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{3}$$
$3$ $$( 1 + T )^{6}$$
$5$ $$( 1 + T^{2} )^{3}$$
$7$ $$169 + 153 T^{2} + 26 T^{4} + T^{6}$$
$11$ $$1 + 5 T^{2} + 6 T^{4} + T^{6}$$
$13$ $$T^{6}$$
$17$ $$( -97 - 46 T + 3 T^{2} + T^{3} )^{2}$$
$19$ $$729 + 486 T^{2} + 45 T^{4} + T^{6}$$
$23$ $$( -113 + 10 T + 11 T^{2} + T^{3} )^{2}$$
$29$ $$( 127 - 43 T - 2 T^{2} + T^{3} )^{2}$$
$31$ $$49 + 49 T^{2} + 14 T^{4} + T^{6}$$
$37$ $$552049 + 20537 T^{2} + 250 T^{4} + T^{6}$$
$41$ $$841 + 846 T^{2} + 89 T^{4} + T^{6}$$
$43$ $$( -1 - 8 T - 5 T^{2} + T^{3} )^{2}$$
$47$ $$1399489 + 42483 T^{2} + 371 T^{4} + T^{6}$$
$53$ $$( 349 + 48 T - 19 T^{2} + T^{3} )^{2}$$
$59$ $$27889 + 13634 T^{2} + 237 T^{4} + T^{6}$$
$61$ $$( -29 - 22 T + 9 T^{2} + T^{3} )^{2}$$
$67$ $$49729 + 5942 T^{2} + 145 T^{4} + T^{6}$$
$71$ $$169 + 94 T^{2} + 17 T^{4} + T^{6}$$
$73$ $$1849 + 1242 T^{2} + 185 T^{4} + T^{6}$$
$79$ $$( 125 + 150 T + 25 T^{2} + T^{3} )^{2}$$
$83$ $$142129 + 15441 T^{2} + 362 T^{4} + T^{6}$$
$89$ $$3087049 + 74578 T^{2} + 497 T^{4} + T^{6}$$
$97$ $$94249 + 6285 T^{2} + 138 T^{4} + T^{6}$$