# Properties

 Label 560.2.a.e Level $560$ Weight $2$ Character orbit 560.a Self dual yes Analytic conductor $4.472$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$560 = 2^{4} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 560.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.47162251319$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{5} + q^{7} - 2 q^{9}+O(q^{10})$$ q + q^3 - q^5 + q^7 - 2 * q^9 $$q + q^{3} - q^{5} + q^{7} - 2 q^{9} + 5 q^{11} + q^{13} - q^{15} + 3 q^{17} + 6 q^{19} + q^{21} + 6 q^{23} + q^{25} - 5 q^{27} - 9 q^{29} + 5 q^{33} - q^{35} + 6 q^{37} + q^{39} + 8 q^{41} - 6 q^{43} + 2 q^{45} - 3 q^{47} + q^{49} + 3 q^{51} - 12 q^{53} - 5 q^{55} + 6 q^{57} - 8 q^{59} - 4 q^{61} - 2 q^{63} - q^{65} + 4 q^{67} + 6 q^{69} - 8 q^{71} + 10 q^{73} + q^{75} + 5 q^{77} + 3 q^{79} + q^{81} + 12 q^{83} - 3 q^{85} - 9 q^{87} - 16 q^{89} + q^{91} - 6 q^{95} + 7 q^{97} - 10 q^{99}+O(q^{100})$$ q + q^3 - q^5 + q^7 - 2 * q^9 + 5 * q^11 + q^13 - q^15 + 3 * q^17 + 6 * q^19 + q^21 + 6 * q^23 + q^25 - 5 * q^27 - 9 * q^29 + 5 * q^33 - q^35 + 6 * q^37 + q^39 + 8 * q^41 - 6 * q^43 + 2 * q^45 - 3 * q^47 + q^49 + 3 * q^51 - 12 * q^53 - 5 * q^55 + 6 * q^57 - 8 * q^59 - 4 * q^61 - 2 * q^63 - q^65 + 4 * q^67 + 6 * q^69 - 8 * q^71 + 10 * q^73 + q^75 + 5 * q^77 + 3 * q^79 + q^81 + 12 * q^83 - 3 * q^85 - 9 * q^87 - 16 * q^89 + q^91 - 6 * q^95 + 7 * q^97 - 10 * 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
0 1.00000 0 −1.00000 0 1.00000 0 −2.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$1$$
$$7$$ $$-1$$

## 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 560.2.a.e 1
3.b odd 2 1 5040.2.a.be 1
4.b odd 2 1 280.2.a.b 1
5.b even 2 1 2800.2.a.i 1
5.c odd 4 2 2800.2.g.m 2
7.b odd 2 1 3920.2.a.r 1
8.b even 2 1 2240.2.a.j 1
8.d odd 2 1 2240.2.a.v 1
12.b even 2 1 2520.2.a.p 1
20.d odd 2 1 1400.2.a.k 1
20.e even 4 2 1400.2.g.e 2
28.d even 2 1 1960.2.a.k 1
28.f even 6 2 1960.2.q.e 2
28.g odd 6 2 1960.2.q.m 2
140.c even 2 1 9800.2.a.n 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.b 1 4.b odd 2 1
560.2.a.e 1 1.a even 1 1 trivial
1400.2.a.k 1 20.d odd 2 1
1400.2.g.e 2 20.e even 4 2
1960.2.a.k 1 28.d even 2 1
1960.2.q.e 2 28.f even 6 2
1960.2.q.m 2 28.g odd 6 2
2240.2.a.j 1 8.b even 2 1
2240.2.a.v 1 8.d odd 2 1
2520.2.a.p 1 12.b even 2 1
2800.2.a.i 1 5.b even 2 1
2800.2.g.m 2 5.c odd 4 2
3920.2.a.r 1 7.b odd 2 1
5040.2.a.be 1 3.b odd 2 1
9800.2.a.n 1 140.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(560))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{11} - 5$$ T11 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T + 1$$
$7$ $$T - 1$$
$11$ $$T - 5$$
$13$ $$T - 1$$
$17$ $$T - 3$$
$19$ $$T - 6$$
$23$ $$T - 6$$
$29$ $$T + 9$$
$31$ $$T$$
$37$ $$T - 6$$
$41$ $$T - 8$$
$43$ $$T + 6$$
$47$ $$T + 3$$
$53$ $$T + 12$$
$59$ $$T + 8$$
$61$ $$T + 4$$
$67$ $$T - 4$$
$71$ $$T + 8$$
$73$ $$T - 10$$
$79$ $$T - 3$$
$83$ $$T - 12$$
$89$ $$T + 16$$
$97$ $$T - 7$$