# Properties

 Label 5070.2.a.k Level $5070$ Weight $2$ Character orbit 5070.a Self dual yes Analytic conductor $40.484$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$40.4841538248$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - 2q^{7} - q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - 2q^{7} - q^{8} + q^{9} - q^{10} + q^{12} + 2q^{14} + q^{15} + q^{16} - q^{18} - 2q^{19} + q^{20} - 2q^{21} - 6q^{23} - q^{24} + q^{25} + q^{27} - 2q^{28} - q^{30} + 4q^{31} - q^{32} - 2q^{35} + q^{36} - 2q^{37} + 2q^{38} - q^{40} + 6q^{41} + 2q^{42} - 4q^{43} + q^{45} + 6q^{46} + q^{48} - 3q^{49} - q^{50} - 6q^{53} - q^{54} + 2q^{56} - 2q^{57} + q^{60} - 10q^{61} - 4q^{62} - 2q^{63} + q^{64} - 8q^{67} - 6q^{69} + 2q^{70} - q^{72} - 8q^{73} + 2q^{74} + q^{75} - 2q^{76} + 8q^{79} + q^{80} + q^{81} - 6q^{82} + 12q^{83} - 2q^{84} + 4q^{86} - 6q^{89} - q^{90} - 6q^{92} + 4q^{93} - 2q^{95} - q^{96} - 8q^{97} + 3q^{98} + O(q^{100})$$

## 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
 0
−1.00000 1.00000 1.00000 1.00000 −1.00000 −2.00000 −1.00000 1.00000 −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
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$13$$ $$1$$

## 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 5070.2.a.k 1
13.b even 2 1 390.2.a.g 1
13.d odd 4 2 5070.2.b.n 2
39.d odd 2 1 1170.2.a.g 1
52.b odd 2 1 3120.2.a.b 1
65.d even 2 1 1950.2.a.b 1
65.h odd 4 2 1950.2.e.k 2
156.h even 2 1 9360.2.a.bg 1
195.e odd 2 1 5850.2.a.bk 1
195.s even 4 2 5850.2.e.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.g 1 13.b even 2 1
1170.2.a.g 1 39.d odd 2 1
1950.2.a.b 1 65.d even 2 1
1950.2.e.k 2 65.h odd 4 2
3120.2.a.b 1 52.b odd 2 1
5070.2.a.k 1 1.a even 1 1 trivial
5070.2.b.n 2 13.d odd 4 2
5850.2.a.bk 1 195.e odd 2 1
5850.2.e.r 2 195.s even 4 2
9360.2.a.bg 1 156.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5070))$$:

 $$T_{7} + 2$$ $$T_{11}$$ $$T_{17}$$ $$T_{31} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$-1 + T$$
$5$ $$-1 + T$$
$7$ $$2 + T$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T$$
$19$ $$2 + T$$
$23$ $$6 + T$$
$29$ $$T$$
$31$ $$-4 + T$$
$37$ $$2 + T$$
$41$ $$-6 + T$$
$43$ $$4 + T$$
$47$ $$T$$
$53$ $$6 + T$$
$59$ $$T$$
$61$ $$10 + T$$
$67$ $$8 + T$$
$71$ $$T$$
$73$ $$8 + T$$
$79$ $$-8 + T$$
$83$ $$-12 + T$$
$89$ $$6 + T$$
$97$ $$8 + T$$