# Properties

 Label 2160.2.a.t Level $2160$ Weight $2$ Character orbit 2160.a Self dual yes Analytic conductor $17.248$ Analytic rank $0$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1080) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{5} + q^{7} + O(q^{10})$$ $$q + q^{5} + q^{7} + 2 q^{11} - 5 q^{13} - 4 q^{17} + 5 q^{19} + 2 q^{23} + q^{25} + 10 q^{29} + 8 q^{31} + q^{35} - 3 q^{37} + 6 q^{41} - 4 q^{43} + 8 q^{47} - 6 q^{49} + 6 q^{53} + 2 q^{55} + 4 q^{59} - 5 q^{61} - 5 q^{65} + 7 q^{67} - 6 q^{71} - 9 q^{73} + 2 q^{77} - 3 q^{79} - 2 q^{83} - 4 q^{85} - 5 q^{91} + 5 q^{95} + 7 q^{97} + O(q^{100})$$

## 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
 0
0 0 0 1.00000 0 1.00000 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.t 1
3.b odd 2 1 2160.2.a.g 1
4.b odd 2 1 1080.2.a.i yes 1
8.b even 2 1 8640.2.a.q 1
8.d odd 2 1 8640.2.a.n 1
12.b even 2 1 1080.2.a.c 1
20.d odd 2 1 5400.2.a.ba 1
20.e even 4 2 5400.2.f.k 2
24.f even 2 1 8640.2.a.bp 1
24.h odd 2 1 8640.2.a.bw 1
36.f odd 6 2 3240.2.q.h 2
36.h even 6 2 3240.2.q.t 2
60.h even 2 1 5400.2.a.bc 1
60.l odd 4 2 5400.2.f.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.a.c 1 12.b even 2 1
1080.2.a.i yes 1 4.b odd 2 1
2160.2.a.g 1 3.b odd 2 1
2160.2.a.t 1 1.a even 1 1 trivial
3240.2.q.h 2 36.f odd 6 2
3240.2.q.t 2 36.h even 6 2
5400.2.a.ba 1 20.d odd 2 1
5400.2.a.bc 1 60.h even 2 1
5400.2.f.k 2 20.e even 4 2
5400.2.f.t 2 60.l odd 4 2
8640.2.a.n 1 8.d odd 2 1
8640.2.a.q 1 8.b even 2 1
8640.2.a.bp 1 24.f even 2 1
8640.2.a.bw 1 24.h odd 2 1

## Hecke kernels

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

 $$T_{7} - 1$$ $$T_{11} - 2$$ $$T_{13} + 5$$ $$T_{17} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-1 + T$$
$7$ $$-1 + T$$
$11$ $$-2 + T$$
$13$ $$5 + T$$
$17$ $$4 + T$$
$19$ $$-5 + T$$
$23$ $$-2 + T$$
$29$ $$-10 + T$$
$31$ $$-8 + T$$
$37$ $$3 + T$$
$41$ $$-6 + T$$
$43$ $$4 + T$$
$47$ $$-8 + T$$
$53$ $$-6 + T$$
$59$ $$-4 + T$$
$61$ $$5 + T$$
$67$ $$-7 + T$$
$71$ $$6 + T$$
$73$ $$9 + T$$
$79$ $$3 + T$$
$83$ $$2 + T$$
$89$ $$T$$
$97$ $$-7 + T$$