# Properties

 Label 560.2.a.b 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 35) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} - q^{5} - q^{7} - 2q^{9} + O(q^{10})$$ $$q - q^{3} - q^{5} - q^{7} - 2q^{9} + 3q^{11} + 5q^{13} + q^{15} + 3q^{17} - 2q^{19} + q^{21} + 6q^{23} + q^{25} + 5q^{27} + 3q^{29} + 4q^{31} - 3q^{33} + q^{35} + 2q^{37} - 5q^{39} - 12q^{41} + 10q^{43} + 2q^{45} - 9q^{47} + q^{49} - 3q^{51} + 12q^{53} - 3q^{55} + 2q^{57} + 8q^{61} + 2q^{63} - 5q^{65} + 4q^{67} - 6q^{69} + 2q^{73} - q^{75} - 3q^{77} + q^{79} + q^{81} - 12q^{83} - 3q^{85} - 3q^{87} - 12q^{89} - 5q^{91} - 4q^{93} + 2q^{95} - q^{97} - 6q^{99} + O(q^{100})$$

## 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
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.b 1
3.b odd 2 1 5040.2.a.v 1
4.b odd 2 1 35.2.a.a 1
5.b even 2 1 2800.2.a.z 1
5.c odd 4 2 2800.2.g.l 2
7.b odd 2 1 3920.2.a.ba 1
8.b even 2 1 2240.2.a.u 1
8.d odd 2 1 2240.2.a.k 1
12.b even 2 1 315.2.a.b 1
20.d odd 2 1 175.2.a.b 1
20.e even 4 2 175.2.b.a 2
28.d even 2 1 245.2.a.c 1
28.f even 6 2 245.2.e.b 2
28.g odd 6 2 245.2.e.a 2
44.c even 2 1 4235.2.a.c 1
52.b odd 2 1 5915.2.a.f 1
60.h even 2 1 1575.2.a.f 1
60.l odd 4 2 1575.2.d.c 2
84.h odd 2 1 2205.2.a.e 1
140.c even 2 1 1225.2.a.e 1
140.j odd 4 2 1225.2.b.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.a 1 4.b odd 2 1
175.2.a.b 1 20.d odd 2 1
175.2.b.a 2 20.e even 4 2
245.2.a.c 1 28.d even 2 1
245.2.e.a 2 28.g odd 6 2
245.2.e.b 2 28.f even 6 2
315.2.a.b 1 12.b even 2 1
560.2.a.b 1 1.a even 1 1 trivial
1225.2.a.e 1 140.c even 2 1
1225.2.b.d 2 140.j odd 4 2
1575.2.a.f 1 60.h even 2 1
1575.2.d.c 2 60.l odd 4 2
2205.2.a.e 1 84.h odd 2 1
2240.2.a.k 1 8.d odd 2 1
2240.2.a.u 1 8.b even 2 1
2800.2.a.z 1 5.b even 2 1
2800.2.g.l 2 5.c odd 4 2
3920.2.a.ba 1 7.b odd 2 1
4235.2.a.c 1 44.c even 2 1
5040.2.a.v 1 3.b odd 2 1
5915.2.a.f 1 52.b odd 2 1

## Hecke kernels

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

 $$T_{3} + 1$$ $$T_{11} - 3$$

## Hecke characteristic polynomials

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