# Properties

 Label 4560.2.a.bt 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: 3.3.568.1 Defining polynomial: $$x^{3} - x^{2} - 6 x - 2$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{5} + ( 1 + \beta_{2} ) q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} - q^{5} + ( 1 + \beta_{2} ) q^{7} + q^{9} + ( -1 + \beta_{2} ) q^{11} + ( -2 + \beta_{1} ) q^{13} - q^{15} + 2 \beta_{1} q^{17} - q^{19} + ( 1 + \beta_{2} ) q^{21} + ( 3 + \beta_{1} + \beta_{2} ) q^{23} + q^{25} + q^{27} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{29} + ( 2 + 2 \beta_{1} ) q^{31} + ( -1 + \beta_{2} ) q^{33} + ( -1 - \beta_{2} ) q^{35} + ( -2 - 3 \beta_{1} ) q^{37} + ( -2 + \beta_{1} ) q^{39} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{41} -3 \beta_{1} q^{43} - q^{45} + ( 5 - \beta_{1} - \beta_{2} ) q^{47} + ( 6 + \beta_{1} - \beta_{2} ) q^{49} + 2 \beta_{1} q^{51} + ( 7 - \beta_{1} - \beta_{2} ) q^{53} + ( 1 - \beta_{2} ) q^{55} - q^{57} + ( -7 - 3 \beta_{1} + \beta_{2} ) q^{59} + ( 1 + 3 \beta_{1} + \beta_{2} ) q^{61} + ( 1 + \beta_{2} ) q^{63} + ( 2 - \beta_{1} ) q^{65} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{67} + ( 3 + \beta_{1} + \beta_{2} ) q^{69} + ( 2 - 4 \beta_{1} - 2 \beta_{2} ) q^{71} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{73} + q^{75} + ( 11 + \beta_{1} - 3 \beta_{2} ) q^{77} + ( 3 + \beta_{1} - \beta_{2} ) q^{79} + q^{81} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{83} -2 \beta_{1} q^{85} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{87} + ( 8 + 3 \beta_{1} ) q^{89} + ( -3 + 3 \beta_{1} - \beta_{2} ) q^{91} + ( 2 + 2 \beta_{1} ) q^{93} + q^{95} + ( -2 - 3 \beta_{1} ) q^{97} + ( -1 + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} + O(q^{10})$$ $$3 q + 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 4 q^{11} - 6 q^{13} - 3 q^{15} - 3 q^{19} + 2 q^{21} + 8 q^{23} + 3 q^{25} + 3 q^{27} + 10 q^{29} + 6 q^{31} - 4 q^{33} - 2 q^{35} - 6 q^{37} - 6 q^{39} + 4 q^{41} - 3 q^{45} + 16 q^{47} + 19 q^{49} + 22 q^{53} + 4 q^{55} - 3 q^{57} - 22 q^{59} + 2 q^{61} + 2 q^{63} + 6 q^{65} - 4 q^{67} + 8 q^{69} + 8 q^{71} + 10 q^{73} + 3 q^{75} + 36 q^{77} + 10 q^{79} + 3 q^{81} - 2 q^{83} + 10 q^{87} + 24 q^{89} - 8 q^{91} + 6 q^{93} + 3 q^{95} - 6 q^{97} - 4 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu - 4$$ $$\beta_{2}$$ $$=$$ $$-\nu^{2} + 3 \nu + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + 3 \beta_{1} + 9$$$$)/2$$

## 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
 −1.76156 −0.363328 3.12489
0 1.00000 0 −1.00000 0 −4.38776 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 2.77801 0 1.00000 0
1.3 0 1.00000 0 −1.00000 0 3.60975 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.bt 3
4.b odd 2 1 2280.2.a.r 3
12.b even 2 1 6840.2.a.bk 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2280.2.a.r 3 4.b odd 2 1
4560.2.a.bt 3 1.a even 1 1 trivial
6840.2.a.bk 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4560))$$:

 $$T_{7}^{3} - 2 T_{7}^{2} - 18 T_{7} + 44$$ $$T_{11}^{3} + 4 T_{11}^{2} - 14 T_{11} + 8$$ $$T_{13}^{3} + 6 T_{13}^{2} + 2 T_{13} - 4$$ $$T_{17}^{3} - 40 T_{17} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$( 1 + T )^{3}$$
$7$ $$44 - 18 T - 2 T^{2} + T^{3}$$
$11$ $$8 - 14 T + 4 T^{2} + T^{3}$$
$13$ $$-4 + 2 T + 6 T^{2} + T^{3}$$
$17$ $$64 - 40 T + T^{3}$$
$19$ $$( 1 + T )^{3}$$
$23$ $$16 - 4 T - 8 T^{2} + T^{3}$$
$29$ $$268 - 18 T - 10 T^{2} + T^{3}$$
$31$ $$136 - 28 T - 6 T^{2} + T^{3}$$
$37$ $$-388 - 78 T + 6 T^{2} + T^{3}$$
$41$ $$400 - 90 T - 4 T^{2} + T^{3}$$
$43$ $$-216 - 90 T + T^{3}$$
$47$ $$16 + 60 T - 16 T^{2} + T^{3}$$
$53$ $$-176 + 136 T - 22 T^{2} + T^{3}$$
$59$ $$-976 + 40 T + 22 T^{2} + T^{3}$$
$61$ $$-160 - 96 T - 2 T^{2} + T^{3}$$
$67$ $$128 - 96 T + 4 T^{2} + T^{3}$$
$71$ $$1600 - 184 T - 8 T^{2} + T^{3}$$
$73$ $$488 - 100 T - 10 T^{2} + T^{3}$$
$79$ $$64 - 10 T^{2} + T^{3}$$
$83$ $$-328 - 100 T + 2 T^{2} + T^{3}$$
$89$ $$424 + 102 T - 24 T^{2} + T^{3}$$
$97$ $$-388 - 78 T + 6 T^{2} + T^{3}$$