# Properties

 Label 2160.2.a.bb Level $2160$ Weight $2$ Character orbit 2160.a Self dual yes Analytic conductor $17.248$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2160 = 2^{4} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2160.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$17.2476868366$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{73})$$ Defining polynomial: $$x^{2} - x - 18$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1080) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{73})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{5} -\beta q^{7} +O(q^{10})$$ $$q + q^{5} -\beta q^{7} + ( 1 - \beta ) q^{11} + 3 q^{13} + ( 3 - \beta ) q^{17} + ( -2 + \beta ) q^{19} + ( 3 + \beta ) q^{23} + q^{25} + ( 1 - \beta ) q^{29} + ( -7 + \beta ) q^{31} -\beta q^{35} + ( 4 - \beta ) q^{37} + 2 \beta q^{41} + ( -3 + \beta ) q^{43} + ( 1 + \beta ) q^{47} + ( 11 + \beta ) q^{49} + 2 \beta q^{53} + ( 1 - \beta ) q^{55} + 12 q^{59} + ( -4 + \beta ) q^{61} + 3 q^{65} + ( -10 - \beta ) q^{67} + ( 4 + 2 \beta ) q^{71} + \beta q^{73} + 18 q^{77} + 5 q^{79} + 6 q^{83} + ( 3 - \beta ) q^{85} -8 q^{89} -3 \beta q^{91} + ( -2 + \beta ) q^{95} + ( -4 - 3 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} - q^{7} + O(q^{10})$$ $$2 q + 2 q^{5} - q^{7} + q^{11} + 6 q^{13} + 5 q^{17} - 3 q^{19} + 7 q^{23} + 2 q^{25} + q^{29} - 13 q^{31} - q^{35} + 7 q^{37} + 2 q^{41} - 5 q^{43} + 3 q^{47} + 23 q^{49} + 2 q^{53} + q^{55} + 24 q^{59} - 7 q^{61} + 6 q^{65} - 21 q^{67} + 10 q^{71} + q^{73} + 36 q^{77} + 10 q^{79} + 12 q^{83} + 5 q^{85} - 16 q^{89} - 3 q^{91} - 3 q^{95} - 11 q^{97} + O(q^{100})$$

## 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
 4.77200 −3.77200
0 0 0 1.00000 0 −4.77200 0 0 0
1.2 0 0 0 1.00000 0 3.77200 0 0 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
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$-1$$

## 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 2160.2.a.bb 2
3.b odd 2 1 2160.2.a.z 2
4.b odd 2 1 1080.2.a.n yes 2
8.b even 2 1 8640.2.a.cn 2
8.d odd 2 1 8640.2.a.cq 2
12.b even 2 1 1080.2.a.m 2
20.d odd 2 1 5400.2.a.ca 2
20.e even 4 2 5400.2.f.bd 4
24.f even 2 1 8640.2.a.de 2
24.h odd 2 1 8640.2.a.db 2
36.f odd 6 2 3240.2.q.z 4
36.h even 6 2 3240.2.q.bc 4
60.h even 2 1 5400.2.a.cb 2
60.l odd 4 2 5400.2.f.be 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.a.m 2 12.b even 2 1
1080.2.a.n yes 2 4.b odd 2 1
2160.2.a.z 2 3.b odd 2 1
2160.2.a.bb 2 1.a even 1 1 trivial
3240.2.q.z 4 36.f odd 6 2
3240.2.q.bc 4 36.h even 6 2
5400.2.a.ca 2 20.d odd 2 1
5400.2.a.cb 2 60.h even 2 1
5400.2.f.bd 4 20.e even 4 2
5400.2.f.be 4 60.l odd 4 2
8640.2.a.cn 2 8.b even 2 1
8640.2.a.cq 2 8.d odd 2 1
8640.2.a.db 2 24.h odd 2 1
8640.2.a.de 2 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2160))$$:

 $$T_{7}^{2} + T_{7} - 18$$ $$T_{11}^{2} - T_{11} - 18$$ $$T_{13} - 3$$ $$T_{17}^{2} - 5 T_{17} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$-18 + T + T^{2}$$
$11$ $$-18 - T + T^{2}$$
$13$ $$( -3 + T )^{2}$$
$17$ $$-12 - 5 T + T^{2}$$
$19$ $$-16 + 3 T + T^{2}$$
$23$ $$-6 - 7 T + T^{2}$$
$29$ $$-18 - T + T^{2}$$
$31$ $$24 + 13 T + T^{2}$$
$37$ $$-6 - 7 T + T^{2}$$
$41$ $$-72 - 2 T + T^{2}$$
$43$ $$-12 + 5 T + T^{2}$$
$47$ $$-16 - 3 T + T^{2}$$
$53$ $$-72 - 2 T + T^{2}$$
$59$ $$( -12 + T )^{2}$$
$61$ $$-6 + 7 T + T^{2}$$
$67$ $$92 + 21 T + T^{2}$$
$71$ $$-48 - 10 T + T^{2}$$
$73$ $$-18 - T + T^{2}$$
$79$ $$( -5 + T )^{2}$$
$83$ $$( -6 + T )^{2}$$
$89$ $$( 8 + T )^{2}$$
$97$ $$-134 + 11 T + T^{2}$$