# Properties

 Label 5070.2.a.f 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

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

## 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 −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.f 1
13.b even 2 1 5070.2.a.o 1
13.d odd 4 2 5070.2.b.g 2
13.e even 6 2 390.2.i.a 2
39.h odd 6 2 1170.2.i.k 2
65.l even 6 2 1950.2.i.s 2
65.r odd 12 4 1950.2.z.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.a 2 13.e even 6 2
1170.2.i.k 2 39.h odd 6 2
1950.2.i.s 2 65.l even 6 2
1950.2.z.h 4 65.r odd 12 4
5070.2.a.f 1 1.a even 1 1 trivial
5070.2.a.o 1 13.b even 2 1
5070.2.b.g 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5070))$$:

 $$T_{7} + 2$$ $$T_{11} + 1$$ $$T_{17} - 2$$ $$T_{31} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$1 + T$$
$5$ $$-1 + T$$
$7$ $$2 + T$$
$11$ $$1 + T$$
$13$ $$T$$
$17$ $$-2 + T$$
$19$ $$6 + T$$
$23$ $$3 + T$$
$29$ $$1 + T$$
$31$ $$-3 + T$$
$37$ $$-5 + T$$
$41$ $$10 + T$$
$43$ $$-5 + T$$
$47$ $$3 + T$$
$53$ $$-14 + T$$
$59$ $$-5 + T$$
$61$ $$10 + T$$
$67$ $$T$$
$71$ $$4 + T$$
$73$ $$-2 + T$$
$79$ $$-5 + T$$
$83$ $$-6 + T$$
$89$ $$10 + T$$
$97$ $$-10 + T$$