# Properties

 Label 4560.2.a.bc 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

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{5} + 2 q^{7} + q^{9}+O(q^{10})$$ q + q^3 + q^5 + 2 * q^7 + q^9 $$q + q^{3} + q^{5} + 2 q^{7} + q^{9} + 6 q^{13} + q^{15} + 8 q^{17} - q^{19} + 2 q^{21} + 4 q^{23} + q^{25} + q^{27} + 2 q^{29} + 2 q^{31} + 2 q^{35} - 2 q^{37} + 6 q^{39} - 12 q^{41} - 4 q^{43} + q^{45} - 12 q^{47} - 3 q^{49} + 8 q^{51} + 10 q^{53} - q^{57} - 6 q^{59} - 14 q^{61} + 2 q^{63} + 6 q^{65} + 12 q^{67} + 4 q^{69} + 8 q^{71} - 10 q^{73} + q^{75} - 14 q^{79} + q^{81} - 2 q^{83} + 8 q^{85} + 2 q^{87} + 12 q^{91} + 2 q^{93} - q^{95} + 2 q^{97}+O(q^{100})$$ q + q^3 + q^5 + 2 * q^7 + q^9 + 6 * q^13 + q^15 + 8 * q^17 - q^19 + 2 * q^21 + 4 * q^23 + q^25 + q^27 + 2 * q^29 + 2 * q^31 + 2 * q^35 - 2 * q^37 + 6 * q^39 - 12 * q^41 - 4 * q^43 + q^45 - 12 * q^47 - 3 * q^49 + 8 * q^51 + 10 * q^53 - q^57 - 6 * q^59 - 14 * q^61 + 2 * q^63 + 6 * q^65 + 12 * q^67 + 4 * q^69 + 8 * q^71 - 10 * q^73 + q^75 - 14 * q^79 + q^81 - 2 * q^83 + 8 * q^85 + 2 * q^87 + 12 * q^91 + 2 * q^93 - q^95 + 2 * q^97

## 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 2.00000 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.bc 1
4.b odd 2 1 570.2.a.h 1
12.b even 2 1 1710.2.a.c 1
20.d odd 2 1 2850.2.a.n 1
20.e even 4 2 2850.2.d.e 2
60.h even 2 1 8550.2.a.bh 1

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

 $$T_{7} - 2$$ T7 - 2 $$T_{11}$$ T11 $$T_{13} - 6$$ T13 - 6 $$T_{17} - 8$$ T17 - 8

## Hecke characteristic polynomials

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