# Properties

 Label 4560.2.a.q 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

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{5} + 4q^{7} + q^{9} + O(q^{10})$$ $$q - q^{3} + q^{5} + 4q^{7} + q^{9} + 4q^{11} - 6q^{13} - q^{15} - 6q^{17} + q^{19} - 4q^{21} - 4q^{23} + q^{25} - q^{27} + 6q^{29} + 8q^{31} - 4q^{33} + 4q^{35} + 2q^{37} + 6q^{39} + 10q^{41} + 8q^{43} + q^{45} - 12q^{47} + 9q^{49} + 6q^{51} + 2q^{53} + 4q^{55} - q^{57} + 4q^{59} - 2q^{61} + 4q^{63} - 6q^{65} + 12q^{67} + 4q^{69} + 16q^{71} - 14q^{73} - q^{75} + 16q^{77} - 8q^{79} + q^{81} - 6q^{85} - 6q^{87} - 6q^{89} - 24q^{91} - 8q^{93} + q^{95} + 14q^{97} + 4q^{99} + O(q^{100})$$

## 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.q 1
4.b odd 2 1 570.2.a.e 1
12.b even 2 1 1710.2.a.l 1
20.d odd 2 1 2850.2.a.v 1
20.e even 4 2 2850.2.d.a 2
60.h even 2 1 8550.2.a.r 1

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

 $$T_{7} - 4$$ $$T_{11} - 4$$ $$T_{13} + 6$$ $$T_{17} + 6$$

## Hecke characteristic polynomials

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