# Properties

 Label 4560.2.a.br 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: 3.3.564.1 Defining polynomial: $$x^{3} - x^{2} - 5 x + 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{5} + ( -1 + \beta_{2} ) q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + q^{5} + ( -1 + \beta_{2} ) q^{7} + q^{9} + ( -2 - \beta_{1} + \beta_{2} ) q^{11} + ( 1 + \beta_{2} ) q^{13} - q^{15} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{17} + q^{19} + ( 1 - \beta_{2} ) q^{21} + ( -1 + \beta_{1} + 2 \beta_{2} ) q^{23} + q^{25} - q^{27} + ( -\beta_{1} + \beta_{2} ) q^{29} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{31} + ( 2 + \beta_{1} - \beta_{2} ) q^{33} + ( -1 + \beta_{2} ) q^{35} + ( -1 + 3 \beta_{2} ) q^{37} + ( -1 - \beta_{2} ) q^{39} + ( 4 - \beta_{1} + \beta_{2} ) q^{41} + ( 3 + 2 \beta_{1} - \beta_{2} ) q^{43} + q^{45} + ( -3 - \beta_{1} - 2 \beta_{2} ) q^{47} + ( \beta_{1} - 4 \beta_{2} ) q^{49} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{51} + ( 3 + 3 \beta_{1} - 2 \beta_{2} ) q^{53} + ( -2 - \beta_{1} + \beta_{2} ) q^{55} - q^{57} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{59} + ( 5 + \beta_{1} ) q^{61} + ( -1 + \beta_{2} ) q^{63} + ( 1 + \beta_{2} ) q^{65} + ( 4 + 4 \beta_{2} ) q^{67} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{69} + ( -4 + 2 \beta_{1} + 2 \beta_{2} ) q^{71} + ( -4 - 2 \beta_{1} ) q^{73} - q^{75} + ( 5 + \beta_{1} - 6 \beta_{2} ) q^{77} + ( 6 + 2 \beta_{1} ) q^{79} + q^{81} + ( -1 - \beta_{1} ) q^{83} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{85} + ( \beta_{1} - \beta_{2} ) q^{87} + ( -\beta_{1} + \beta_{2} ) q^{89} + ( 5 + \beta_{1} - 2 \beta_{2} ) q^{91} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{93} + q^{95} + ( -3 + \beta_{2} ) q^{97} + ( -2 - \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9} + O(q^{10})$$ $$3 q - 3 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9} - 6 q^{11} + 2 q^{13} - 3 q^{15} + 2 q^{17} + 3 q^{19} + 4 q^{21} - 6 q^{23} + 3 q^{25} - 3 q^{27} + 6 q^{33} - 4 q^{35} - 6 q^{37} - 2 q^{39} + 12 q^{41} + 8 q^{43} + 3 q^{45} - 6 q^{47} + 3 q^{49} - 2 q^{51} + 8 q^{53} - 6 q^{55} - 3 q^{57} + 4 q^{59} + 14 q^{61} - 4 q^{63} + 2 q^{65} + 8 q^{67} + 6 q^{69} - 16 q^{71} - 10 q^{73} - 3 q^{75} + 20 q^{77} + 16 q^{79} + 3 q^{81} - 2 q^{83} + 2 q^{85} + 16 q^{91} + 3 q^{95} - 10 q^{97} - 6 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 5 x + 3$$:

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

## 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
 0.571993 −2.08613 2.51414
0 −1.00000 0 1.00000 0 −4.67282 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −0.648061 0 1.00000 0
1.3 0 −1.00000 0 1.00000 0 1.32088 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.br 3
4.b odd 2 1 1140.2.a.g 3
12.b even 2 1 3420.2.a.m 3
20.d odd 2 1 5700.2.a.w 3
20.e even 4 2 5700.2.f.q 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1140.2.a.g 3 4.b odd 2 1
3420.2.a.m 3 12.b even 2 1
4560.2.a.br 3 1.a even 1 1 trivial
5700.2.a.w 3 20.d odd 2 1
5700.2.f.q 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4560))$$:

 $$T_{7}^{3} + 4 T_{7}^{2} - 4 T_{7} - 4$$ $$T_{11}^{3} + 6 T_{11}^{2} - 12 T_{11} - 76$$ $$T_{13}^{3} - 2 T_{13}^{2} - 8 T_{13} + 12$$ $$T_{17}^{3} - 2 T_{17}^{2} - 44 T_{17} + 72$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$( -1 + T )^{3}$$
$7$ $$-4 - 4 T + 4 T^{2} + T^{3}$$
$11$ $$-76 - 12 T + 6 T^{2} + T^{3}$$
$13$ $$12 - 8 T - 2 T^{2} + T^{3}$$
$17$ $$72 - 44 T - 2 T^{2} + T^{3}$$
$19$ $$( -1 + T )^{3}$$
$23$ $$-344 - 60 T + 6 T^{2} + T^{3}$$
$29$ $$-36 - 24 T + T^{3}$$
$31$ $$208 - 72 T + T^{3}$$
$37$ $$4 - 72 T + 6 T^{2} + T^{3}$$
$41$ $$-4 + 24 T - 12 T^{2} + T^{3}$$
$43$ $$468 - 60 T - 8 T^{2} + T^{3}$$
$47$ $$72 - 60 T + 6 T^{2} + T^{3}$$
$53$ $$1488 - 168 T - 8 T^{2} + T^{3}$$
$59$ $$864 - 144 T - 4 T^{2} + T^{3}$$
$61$ $$8 + 44 T - 14 T^{2} + T^{3}$$
$67$ $$768 - 128 T - 8 T^{2} + T^{3}$$
$71$ $$-1312 - 64 T + 16 T^{2} + T^{3}$$
$73$ $$-328 - 52 T + 10 T^{2} + T^{3}$$
$79$ $$384 - 16 T^{2} + T^{3}$$
$83$ $$-24 - 20 T + 2 T^{2} + T^{3}$$
$89$ $$-36 - 24 T + T^{3}$$
$97$ $$12 + 24 T + 10 T^{2} + T^{3}$$