# Properties

 Label 5070.2.a.a 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

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

## 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 0 −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.a 1
13.b even 2 1 390.2.a.f 1
13.d odd 4 2 5070.2.b.d 2
39.d odd 2 1 1170.2.a.a 1
52.b odd 2 1 3120.2.a.w 1
65.d even 2 1 1950.2.a.k 1
65.h odd 4 2 1950.2.e.g 2
156.h even 2 1 9360.2.a.p 1
195.e odd 2 1 5850.2.a.bo 1
195.s even 4 2 5850.2.e.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.f 1 13.b even 2 1
1170.2.a.a 1 39.d odd 2 1
1950.2.a.k 1 65.d even 2 1
1950.2.e.g 2 65.h odd 4 2
3120.2.a.w 1 52.b odd 2 1
5070.2.a.a 1 1.a even 1 1 trivial
5070.2.b.d 2 13.d odd 4 2
5850.2.a.bo 1 195.e odd 2 1
5850.2.e.e 2 195.s even 4 2
9360.2.a.p 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5070))$$:

 $$T_{7}$$ T7 $$T_{11} + 4$$ T11 + 4 $$T_{17} + 6$$ T17 + 6 $$T_{31} - 8$$ T31 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T + 1$$
$5$ $$T + 1$$
$7$ $$T$$
$11$ $$T + 4$$
$13$ $$T$$
$17$ $$T + 6$$
$19$ $$T + 4$$
$23$ $$T - 8$$
$29$ $$T - 6$$
$31$ $$T - 8$$
$37$ $$T - 10$$
$41$ $$T - 6$$
$43$ $$T - 4$$
$47$ $$T$$
$53$ $$T + 10$$
$59$ $$T + 4$$
$61$ $$T + 2$$
$67$ $$T - 12$$
$71$ $$T + 16$$
$73$ $$T + 2$$
$79$ $$T + 16$$
$83$ $$T - 12$$
$89$ $$T + 10$$
$97$ $$T - 6$$