# Properties

 Label 4560.2.a.k Level $4560$ Weight $2$ Character orbit 4560.a Self dual yes Analytic conductor $36.412$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4560.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$36.4117833217$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 570) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{5} - 4 q^{7} + q^{9} + O(q^{10})$$ $$q - q^{3} + q^{5} - 4 q^{7} + q^{9} + 4 q^{11} - 2 q^{13} - q^{15} - 2 q^{17} + q^{19} + 4 q^{21} + 8 q^{23} + q^{25} - q^{27} + 6 q^{29} - 4 q^{31} - 4 q^{33} - 4 q^{35} - 10 q^{37} + 2 q^{39} - 2 q^{41} - 12 q^{43} + q^{45} + 9 q^{49} + 2 q^{51} + 6 q^{53} + 4 q^{55} - q^{57} - 10 q^{61} - 4 q^{63} - 2 q^{65} + 4 q^{67} - 8 q^{69} + 8 q^{71} + 2 q^{73} - q^{75} - 16 q^{77} + 12 q^{79} + q^{81} + 8 q^{83} - 2 q^{85} - 6 q^{87} + 6 q^{89} + 8 q^{91} + 4 q^{93} + q^{95} + 18 q^{97} + 4 q^{99} + O(q^{100})$$

## 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
0 −1.00000 0 1.00000 0 −4.00000 0 1.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$-1$$
$$19$$ $$-1$$

## 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 4560.2.a.k 1
4.b odd 2 1 570.2.a.m 1
12.b even 2 1 1710.2.a.f 1
20.d odd 2 1 2850.2.a.a 1
20.e even 4 2 2850.2.d.k 2
60.h even 2 1 8550.2.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.a.m 1 4.b odd 2 1
1710.2.a.f 1 12.b even 2 1
2850.2.a.a 1 20.d odd 2 1
2850.2.d.k 2 20.e even 4 2
4560.2.a.k 1 1.a even 1 1 trivial
8550.2.a.t 1 60.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4560))$$:

 $$T_{7} + 4$$ $$T_{11} - 4$$ $$T_{13} + 2$$ $$T_{17} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$1 + T$$
$5$ $$-1 + T$$
$7$ $$4 + T$$
$11$ $$-4 + T$$
$13$ $$2 + T$$
$17$ $$2 + T$$
$19$ $$-1 + T$$
$23$ $$-8 + T$$
$29$ $$-6 + T$$
$31$ $$4 + T$$
$37$ $$10 + T$$
$41$ $$2 + T$$
$43$ $$12 + T$$
$47$ $$T$$
$53$ $$-6 + T$$
$59$ $$T$$
$61$ $$10 + T$$
$67$ $$-4 + T$$
$71$ $$-8 + T$$
$73$ $$-2 + T$$
$79$ $$-12 + T$$
$83$ $$-8 + T$$
$89$ $$-6 + T$$
$97$ $$-18 + T$$