# Properties

 Label 4560.2.a.bs Level $4560$ Weight $2$ Character orbit 4560.a Self dual yes Analytic conductor $36.412$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.1772.1 Defining polynomial: $$x^{3} - x^{2} - 12 x + 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 2280) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{5} -\beta_{2} q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + q^{5} -\beta_{2} q^{7} + q^{9} + ( -2 - \beta_{2} ) q^{11} + ( 1 - \beta_{1} ) q^{13} - q^{15} + 4 q^{17} - q^{19} + \beta_{2} q^{21} + ( 1 - \beta_{1} + \beta_{2} ) q^{23} + q^{25} - q^{27} -\beta_{2} q^{29} + 2 q^{31} + ( 2 + \beta_{2} ) q^{33} -\beta_{2} q^{35} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{37} + ( -1 + \beta_{1} ) q^{39} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{41} + ( -1 - \beta_{1} ) q^{43} + q^{45} + ( -1 + \beta_{1} - \beta_{2} ) q^{47} + ( 2 - \beta_{1} - \beta_{2} ) q^{49} -4 q^{51} + ( -3 + \beta_{1} - \beta_{2} ) q^{53} + ( -2 - \beta_{2} ) q^{55} + q^{57} + ( -1 - \beta_{1} + 3 \beta_{2} ) q^{59} + ( 5 + \beta_{1} + \beta_{2} ) q^{61} -\beta_{2} q^{63} + ( 1 - \beta_{1} ) q^{65} -4 q^{67} + ( -1 + \beta_{1} - \beta_{2} ) q^{69} + ( 2 + 2 \beta_{1} ) q^{71} + 6 q^{73} - q^{75} + ( 9 - \beta_{1} + \beta_{2} ) q^{77} + ( 1 + \beta_{1} + 3 \beta_{2} ) q^{79} + q^{81} + ( 2 + 2 \beta_{2} ) q^{83} + 4 q^{85} + \beta_{2} q^{87} + ( 3 - \beta_{1} + 2 \beta_{2} ) q^{89} + ( 3 + \beta_{1} - 5 \beta_{2} ) q^{91} -2 q^{93} - q^{95} + ( 11 + \beta_{1} - 2 \beta_{2} ) q^{97} + ( -2 - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{3} + 3q^{5} + 3q^{9} + O(q^{10})$$ $$3q - 3q^{3} + 3q^{5} + 3q^{9} - 6q^{11} + 4q^{13} - 3q^{15} + 12q^{17} - 3q^{19} + 4q^{23} + 3q^{25} - 3q^{27} + 6q^{31} + 6q^{33} - 4q^{37} - 4q^{39} + 2q^{41} - 2q^{43} + 3q^{45} - 4q^{47} + 7q^{49} - 12q^{51} - 10q^{53} - 6q^{55} + 3q^{57} - 2q^{59} + 14q^{61} + 4q^{65} - 12q^{67} - 4q^{69} + 4q^{71} + 18q^{73} - 3q^{75} + 28q^{77} + 2q^{79} + 3q^{81} + 6q^{83} + 12q^{85} + 10q^{89} + 8q^{91} - 6q^{93} - 3q^{95} + 32q^{97} - 6q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 12 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{2} + 3 \nu - 10$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} - \nu - 8$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$3 \beta_{2} + \beta_{1} + 17$$$$)/2$$

## 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
 −3.32803 3.67370 0.654334
0 −1.00000 0 1.00000 0 −3.20191 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −0.911179 0 1.00000 0
1.3 0 −1.00000 0 1.00000 0 4.11309 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.bs 3
4.b odd 2 1 2280.2.a.u 3
12.b even 2 1 6840.2.a.bh 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2280.2.a.u 3 4.b odd 2 1
4560.2.a.bs 3 1.a even 1 1 trivial
6840.2.a.bh 3 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4560))$$:

 $$T_{7}^{3} - 14 T_{7} - 12$$ $$T_{11}^{3} + 6 T_{11}^{2} - 2 T_{11} - 32$$ $$T_{13}^{3} - 4 T_{13}^{2} - 38 T_{13} + 164$$ $$T_{17} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$( -1 + T )^{3}$$
$7$ $$-12 - 14 T + T^{3}$$
$11$ $$-32 - 2 T + 6 T^{2} + T^{3}$$
$13$ $$164 - 38 T - 4 T^{2} + T^{3}$$
$17$ $$( -4 + T )^{3}$$
$19$ $$( 1 + T )^{3}$$
$23$ $$32 - 44 T - 4 T^{2} + T^{3}$$
$29$ $$-12 - 14 T + T^{3}$$
$31$ $$( -2 + T )^{3}$$
$37$ $$180 - 78 T + 4 T^{2} + T^{3}$$
$41$ $$344 - 82 T - 2 T^{2} + T^{3}$$
$43$ $$80 - 42 T + 2 T^{2} + T^{3}$$
$47$ $$-32 - 44 T + 4 T^{2} + T^{3}$$
$53$ $$-96 - 16 T + 10 T^{2} + T^{3}$$
$59$ $$-688 - 144 T + 2 T^{2} + T^{3}$$
$61$ $$144 - 14 T^{2} + T^{3}$$
$67$ $$( 4 + T )^{3}$$
$71$ $$-640 - 168 T - 4 T^{2} + T^{3}$$
$73$ $$( -6 + T )^{3}$$
$79$ $$1024 - 192 T - 2 T^{2} + T^{3}$$
$83$ $$200 - 44 T - 6 T^{2} + T^{3}$$
$89$ $$-48 - 50 T - 10 T^{2} + T^{3}$$
$97$ $$-36 + 258 T - 32 T^{2} + T^{3}$$