# Properties

 Label 4560.2.a.bu 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: 3.3.1524.1 Defining polynomial: $$x^{3} - x^{2} - 7 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1140) 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{5} + \beta_{1} q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + q^{5} + \beta_{1} q^{7} + q^{9} + ( 1 + \beta_{2} ) q^{11} + ( 2 - \beta_{1} ) q^{13} + q^{15} + ( 1 + \beta_{1} + \beta_{2} ) q^{17} - q^{19} + \beta_{1} q^{21} + ( -1 - \beta_{1} - \beta_{2} ) q^{23} + q^{25} + q^{27} + ( 1 - \beta_{2} ) q^{29} + ( -3 - \beta_{1} - \beta_{2} ) q^{31} + ( 1 + \beta_{2} ) q^{33} + \beta_{1} q^{35} + ( 2 + \beta_{1} ) q^{37} + ( 2 - \beta_{1} ) q^{39} + ( 5 - \beta_{2} ) q^{41} + ( 2 + \beta_{1} + 2 \beta_{2} ) q^{43} + q^{45} + ( 5 + \beta_{1} + \beta_{2} ) q^{47} + ( 10 + \beta_{1} + 3 \beta_{2} ) q^{49} + ( 1 + \beta_{1} + \beta_{2} ) q^{51} + ( 5 - \beta_{1} - \beta_{2} ) q^{53} + ( 1 + \beta_{2} ) q^{55} - q^{57} + ( -2 - 2 \beta_{2} ) q^{59} + ( 7 + \beta_{1} - \beta_{2} ) q^{61} + \beta_{1} q^{63} + ( 2 - \beta_{1} ) q^{65} + ( -1 - \beta_{1} - \beta_{2} ) q^{69} + ( -6 - 2 \beta_{2} ) q^{71} + ( 4 - 2 \beta_{1} - 2 \beta_{2} ) q^{73} + q^{75} + ( -7 + 3 \beta_{1} - \beta_{2} ) q^{77} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{79} + q^{81} + ( -3 - \beta_{1} - 3 \beta_{2} ) q^{83} + ( 1 + \beta_{1} + \beta_{2} ) q^{85} + ( 1 - \beta_{2} ) q^{87} + ( 1 - 4 \beta_{1} - \beta_{2} ) q^{89} + ( -17 + \beta_{1} - 3 \beta_{2} ) q^{91} + ( -3 - \beta_{1} - \beta_{2} ) q^{93} - q^{95} + ( -2 - \beta_{1} ) q^{97} + ( 1 + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} + 3 q^{5} + 3 q^{9} + O(q^{10})$$ $$3 q + 3 q^{3} + 3 q^{5} + 3 q^{9} + 2 q^{11} + 6 q^{13} + 3 q^{15} + 2 q^{17} - 3 q^{19} - 2 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} - 8 q^{31} + 2 q^{33} + 6 q^{37} + 6 q^{39} + 16 q^{41} + 4 q^{43} + 3 q^{45} + 14 q^{47} + 27 q^{49} + 2 q^{51} + 16 q^{53} + 2 q^{55} - 3 q^{57} - 4 q^{59} + 22 q^{61} + 6 q^{65} - 2 q^{69} - 16 q^{71} + 14 q^{73} + 3 q^{75} - 20 q^{77} - 8 q^{79} + 3 q^{81} - 6 q^{83} + 2 q^{85} + 4 q^{87} + 4 q^{89} - 48 q^{91} - 8 q^{93} - 3 q^{95} - 6 q^{97} + 2 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 7 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} - 5$$ $$\beta_{2}$$ $$=$$ $$-\nu^{2} + 2 \nu + 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{1} + 5$$

## 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
 0.140435 −2.27307 3.13264
0 1.00000 0 1.00000 0 −4.98028 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 0.166860 0 1.00000 0
1.3 0 1.00000 0 1.00000 0 4.81342 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.bu 3
4.b odd 2 1 1140.2.a.f 3
12.b even 2 1 3420.2.a.k 3
20.d odd 2 1 5700.2.a.z 3
20.e even 4 2 5700.2.f.p 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1140.2.a.f 3 4.b odd 2 1
3420.2.a.k 3 12.b even 2 1
4560.2.a.bu 3 1.a even 1 1 trivial
5700.2.a.z 3 20.d odd 2 1
5700.2.f.p 6 20.e even 4 2

## Hecke kernels

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

 $$T_{7}^{3} - 24 T_{7} + 4$$ $$T_{11}^{3} - 2 T_{11}^{2} - 24 T_{11} + 36$$ $$T_{13}^{3} - 6 T_{13}^{2} - 12 T_{13} + 36$$ $$T_{17}^{3} - 2 T_{17}^{2} - 28 T_{17} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$( -1 + T )^{3}$$
$7$ $$4 - 24 T + T^{3}$$
$11$ $$36 - 24 T - 2 T^{2} + T^{3}$$
$13$ $$36 - 12 T - 6 T^{2} + T^{3}$$
$17$ $$8 - 28 T - 2 T^{2} + T^{3}$$
$19$ $$( 1 + T )^{3}$$
$23$ $$-8 - 28 T + 2 T^{2} + T^{3}$$
$29$ $$12 - 20 T - 4 T^{2} + T^{3}$$
$31$ $$-48 - 8 T + 8 T^{2} + T^{3}$$
$37$ $$44 - 12 T - 6 T^{2} + T^{3}$$
$41$ $$-36 + 60 T - 16 T^{2} + T^{3}$$
$43$ $$396 - 80 T - 4 T^{2} + T^{3}$$
$47$ $$24 + 36 T - 14 T^{2} + T^{3}$$
$53$ $$16 + 56 T - 16 T^{2} + T^{3}$$
$59$ $$-288 - 96 T + 4 T^{2} + T^{3}$$
$61$ $$328 + 92 T - 22 T^{2} + T^{3}$$
$67$ $$T^{3}$$
$71$ $$-544 - 16 T + 16 T^{2} + T^{3}$$
$73$ $$536 - 52 T - 14 T^{2} + T^{3}$$
$79$ $$-2432 - 256 T + 8 T^{2} + T^{3}$$
$83$ $$-1384 - 180 T + 6 T^{2} + T^{3}$$
$89$ $$1884 - 324 T - 4 T^{2} + T^{3}$$
$97$ $$-44 - 12 T + 6 T^{2} + T^{3}$$