# Properties

 Label 4560.2.a.w Level $4560$ Weight $2$ Character orbit 4560.a Self dual yes Analytic conductor $36.412$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 285) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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.w 1
4.b odd 2 1 285.2.a.c 1
12.b even 2 1 855.2.a.a 1
20.d odd 2 1 1425.2.a.c 1
20.e even 4 2 1425.2.c.f 2
60.h even 2 1 4275.2.a.j 1
76.d even 2 1 5415.2.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.c 1 4.b odd 2 1
855.2.a.a 1 12.b even 2 1
1425.2.a.c 1 20.d odd 2 1
1425.2.c.f 2 20.e even 4 2
4275.2.a.j 1 60.h even 2 1
4560.2.a.w 1 1.a even 1 1 trivial
5415.2.a.e 1 76.d 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

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