# Properties

 Label 5070.2.a.x Level $5070$ Weight $2$ Character orbit 5070.a Self dual yes Analytic conductor $40.484$ Analytic rank $0$ 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: $$0$$ 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

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

## 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 5.00000 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.x 1
13.b even 2 1 5070.2.a.i 1
13.d odd 4 2 5070.2.b.l 2
13.e even 6 2 390.2.i.d 2
39.h odd 6 2 1170.2.i.g 2
65.l even 6 2 1950.2.i.i 2
65.r odd 12 4 1950.2.z.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.d 2 13.e even 6 2
1170.2.i.g 2 39.h odd 6 2
1950.2.i.i 2 65.l even 6 2
1950.2.z.j 4 65.r odd 12 4
5070.2.a.i 1 13.b even 2 1
5070.2.a.x 1 1.a even 1 1 trivial
5070.2.b.l 2 13.d odd 4 2

## Hecke kernels

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

 $$T_{7} - 5$$ $$T_{11} - 3$$ $$T_{17} + 8$$ $$T_{31} - 2$$

## Hecke characteristic polynomials

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