# Properties

 Label 5070.2.a.e 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

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - 2 q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^3 + q^4 + q^5 + q^6 - 2 * q^7 - q^8 + q^9 $$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - 2 q^{7} - q^{8} + q^{9} - q^{10} - 4 q^{11} - q^{12} + 2 q^{14} - q^{15} + q^{16} + 8 q^{17} - q^{18} + 6 q^{19} + q^{20} + 2 q^{21} + 4 q^{22} + 6 q^{23} + q^{24} + q^{25} - q^{27} - 2 q^{28} - 4 q^{29} + q^{30} - q^{32} + 4 q^{33} - 8 q^{34} - 2 q^{35} + q^{36} + 2 q^{37} - 6 q^{38} - q^{40} + 2 q^{41} - 2 q^{42} - 4 q^{43} - 4 q^{44} + q^{45} - 6 q^{46} - q^{48} - 3 q^{49} - q^{50} - 8 q^{51} - 10 q^{53} + q^{54} - 4 q^{55} + 2 q^{56} - 6 q^{57} + 4 q^{58} - 4 q^{59} - q^{60} - 10 q^{61} - 2 q^{63} + q^{64} - 4 q^{66} - 12 q^{67} + 8 q^{68} - 6 q^{69} + 2 q^{70} + 8 q^{71} - q^{72} + 8 q^{73} - 2 q^{74} - q^{75} + 6 q^{76} + 8 q^{77} + 8 q^{79} + q^{80} + q^{81} - 2 q^{82} - 12 q^{83} + 2 q^{84} + 8 q^{85} + 4 q^{86} + 4 q^{87} + 4 q^{88} + 14 q^{89} - q^{90} + 6 q^{92} + 6 q^{95} + q^{96} + 16 q^{97} + 3 q^{98} - 4 q^{99}+O(q^{100})$$ q - q^2 - q^3 + q^4 + q^5 + q^6 - 2 * q^7 - q^8 + q^9 - q^10 - 4 * q^11 - q^12 + 2 * q^14 - q^15 + q^16 + 8 * q^17 - q^18 + 6 * q^19 + q^20 + 2 * q^21 + 4 * q^22 + 6 * q^23 + q^24 + q^25 - q^27 - 2 * q^28 - 4 * q^29 + q^30 - q^32 + 4 * q^33 - 8 * q^34 - 2 * q^35 + q^36 + 2 * q^37 - 6 * q^38 - q^40 + 2 * q^41 - 2 * q^42 - 4 * q^43 - 4 * q^44 + q^45 - 6 * q^46 - q^48 - 3 * q^49 - q^50 - 8 * q^51 - 10 * q^53 + q^54 - 4 * q^55 + 2 * q^56 - 6 * q^57 + 4 * q^58 - 4 * q^59 - q^60 - 10 * q^61 - 2 * q^63 + q^64 - 4 * q^66 - 12 * q^67 + 8 * q^68 - 6 * q^69 + 2 * q^70 + 8 * q^71 - q^72 + 8 * q^73 - 2 * q^74 - q^75 + 6 * q^76 + 8 * q^77 + 8 * q^79 + q^80 + q^81 - 2 * q^82 - 12 * q^83 + 2 * q^84 + 8 * q^85 + 4 * q^86 + 4 * q^87 + 4 * q^88 + 14 * q^89 - q^90 + 6 * q^92 + 6 * q^95 + q^96 + 16 * q^97 + 3 * 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 −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.e 1
13.b even 2 1 390.2.a.e 1
13.d odd 4 2 5070.2.b.e 2
39.d odd 2 1 1170.2.a.e 1
52.b odd 2 1 3120.2.a.o 1
65.d even 2 1 1950.2.a.h 1
65.h odd 4 2 1950.2.e.f 2
156.h even 2 1 9360.2.a.bh 1
195.e odd 2 1 5850.2.a.bi 1
195.s even 4 2 5850.2.e.i 2

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

 $$T_{7} + 2$$ T7 + 2 $$T_{11} + 4$$ T11 + 4 $$T_{17} - 8$$ T17 - 8 $$T_{31}$$ T31

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T + 1$$
$5$ $$T - 1$$
$7$ $$T + 2$$
$11$ $$T + 4$$
$13$ $$T$$
$17$ $$T - 8$$
$19$ $$T - 6$$
$23$ $$T - 6$$
$29$ $$T + 4$$
$31$ $$T$$
$37$ $$T - 2$$
$41$ $$T - 2$$
$43$ $$T + 4$$
$47$ $$T$$
$53$ $$T + 10$$
$59$ $$T + 4$$
$61$ $$T + 10$$
$67$ $$T + 12$$
$71$ $$T - 8$$
$73$ $$T - 8$$
$79$ $$T - 8$$
$83$ $$T + 12$$
$89$ $$T - 14$$
$97$ $$T - 16$$