# Properties

 Label 4560.2.a.bl Level $4560$ Weight $2$ Character orbit 4560.a Self dual yes Analytic conductor $36.412$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ 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 $$\beta = \sqrt{13}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{5} + ( -1 - \beta ) q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} - q^{5} + ( -1 - \beta ) q^{7} + q^{9} + ( 1 - \beta ) q^{11} + ( 1 - \beta ) q^{13} - q^{15} + 2 q^{17} + q^{19} + ( -1 - \beta ) q^{21} + 2 q^{23} + q^{25} + q^{27} + ( -1 - \beta ) q^{29} -4 q^{31} + ( 1 - \beta ) q^{33} + ( 1 + \beta ) q^{35} + ( 7 + \beta ) q^{37} + ( 1 - \beta ) q^{39} + ( 3 - \beta ) q^{41} + ( -7 + \beta ) q^{43} - q^{45} -6 q^{47} + ( 7 + 2 \beta ) q^{49} + 2 q^{51} + ( -1 + \beta ) q^{55} + q^{57} + ( 2 + 2 \beta ) q^{59} + 2 \beta q^{61} + ( -1 - \beta ) q^{63} + ( -1 + \beta ) q^{65} -4 q^{67} + 2 q^{69} + ( 2 - 2 \beta ) q^{71} + 6 q^{73} + q^{75} + 12 q^{77} -8 q^{79} + q^{81} + ( 4 - 2 \beta ) q^{83} -2 q^{85} + ( -1 - \beta ) q^{87} + ( 3 - \beta ) q^{89} + 12 q^{91} -4 q^{93} - q^{95} + ( 13 - \beta ) q^{97} + ( 1 - \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + O(q^{10})$$ $$2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} - 2 q^{15} + 4 q^{17} + 2 q^{19} - 2 q^{21} + 4 q^{23} + 2 q^{25} + 2 q^{27} - 2 q^{29} - 8 q^{31} + 2 q^{33} + 2 q^{35} + 14 q^{37} + 2 q^{39} + 6 q^{41} - 14 q^{43} - 2 q^{45} - 12 q^{47} + 14 q^{49} + 4 q^{51} - 2 q^{55} + 2 q^{57} + 4 q^{59} - 2 q^{63} - 2 q^{65} - 8 q^{67} + 4 q^{69} + 4 q^{71} + 12 q^{73} + 2 q^{75} + 24 q^{77} - 16 q^{79} + 2 q^{81} + 8 q^{83} - 4 q^{85} - 2 q^{87} + 6 q^{89} + 24 q^{91} - 8 q^{93} - 2 q^{95} + 26 q^{97} + 2 q^{99} + O(q^{100})$$

## 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
 2.30278 −1.30278
0 1.00000 0 −1.00000 0 −4.60555 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 2.60555 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.bl 2
4.b odd 2 1 1140.2.a.e 2
12.b even 2 1 3420.2.a.i 2
20.d odd 2 1 5700.2.a.u 2
20.e even 4 2 5700.2.f.n 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1140.2.a.e 2 4.b odd 2 1
3420.2.a.i 2 12.b even 2 1
4560.2.a.bl 2 1.a even 1 1 trivial
5700.2.a.u 2 20.d odd 2 1
5700.2.f.n 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4560))$$:

 $$T_{7}^{2} + 2 T_{7} - 12$$ $$T_{11}^{2} - 2 T_{11} - 12$$ $$T_{13}^{2} - 2 T_{13} - 12$$ $$T_{17} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$-12 + 2 T + T^{2}$$
$11$ $$-12 - 2 T + T^{2}$$
$13$ $$-12 - 2 T + T^{2}$$
$17$ $$( -2 + T )^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$( -2 + T )^{2}$$
$29$ $$-12 + 2 T + T^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$36 - 14 T + T^{2}$$
$41$ $$-4 - 6 T + T^{2}$$
$43$ $$36 + 14 T + T^{2}$$
$47$ $$( 6 + T )^{2}$$
$53$ $$T^{2}$$
$59$ $$-48 - 4 T + T^{2}$$
$61$ $$-52 + T^{2}$$
$67$ $$( 4 + T )^{2}$$
$71$ $$-48 - 4 T + T^{2}$$
$73$ $$( -6 + T )^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$-36 - 8 T + T^{2}$$
$89$ $$-4 - 6 T + T^{2}$$
$97$ $$156 - 26 T + T^{2}$$