# Properties

 Label 4560.2.a.s 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

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{5} + q^{9}+O(q^{10})$$ q + q^3 - q^5 + q^9 $$q + q^{3} - q^{5} + q^{9} - 4 q^{11} + 2 q^{13} - q^{15} + 2 q^{17} + q^{19} - 4 q^{23} + q^{25} + q^{27} + 6 q^{29} - 4 q^{31} - 4 q^{33} - 6 q^{37} + 2 q^{39} + 10 q^{41} + 4 q^{43} - q^{45} + 12 q^{47} - 7 q^{49} + 2 q^{51} + 6 q^{53} + 4 q^{55} + q^{57} + 12 q^{59} - 2 q^{61} - 2 q^{65} - 4 q^{67} - 4 q^{69} - 8 q^{71} - 6 q^{73} + q^{75} + 4 q^{79} + q^{81} + 12 q^{83} - 2 q^{85} + 6 q^{87} + 10 q^{89} - 4 q^{93} - q^{95} + 2 q^{97} - 4 q^{99}+O(q^{100})$$ q + q^3 - q^5 + q^9 - 4 * q^11 + 2 * q^13 - q^15 + 2 * q^17 + q^19 - 4 * q^23 + q^25 + q^27 + 6 * q^29 - 4 * q^31 - 4 * q^33 - 6 * q^37 + 2 * q^39 + 10 * q^41 + 4 * q^43 - q^45 + 12 * q^47 - 7 * q^49 + 2 * q^51 + 6 * q^53 + 4 * q^55 + q^57 + 12 * q^59 - 2 * q^61 - 2 * q^65 - 4 * q^67 - 4 * q^69 - 8 * q^71 - 6 * q^73 + q^75 + 4 * q^79 + q^81 + 12 * q^83 - 2 * q^85 + 6 * q^87 + 10 * q^89 - 4 * q^93 - q^95 + 2 * q^97 - 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
0 1.00000 0 −1.00000 0 0 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.s 1
4.b odd 2 1 570.2.a.g 1
12.b even 2 1 1710.2.a.i 1
20.d odd 2 1 2850.2.a.m 1
20.e even 4 2 2850.2.d.i 2
60.h even 2 1 8550.2.a.x 1

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

 $$T_{7}$$ T7 $$T_{11} + 4$$ T11 + 4 $$T_{13} - 2$$ T13 - 2 $$T_{17} - 2$$ T17 - 2

## Hecke characteristic polynomials

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