# Properties

 Label 2160.4.a.s Level $2160$ Weight $4$ Character orbit 2160.a Self dual yes Analytic conductor $127.444$ 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: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2160.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$127.444125612$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 540) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 5 q^{5} + 22 q^{7}+O(q^{10})$$ q + 5 * q^5 + 22 * q^7 $$q + 5 q^{5} + 22 q^{7} - 9 q^{11} + 17 q^{13} + 75 q^{17} + 4 q^{19} + 183 q^{23} + 25 q^{25} - 129 q^{29} + 187 q^{31} + 110 q^{35} - 34 q^{37} - 264 q^{41} - 443 q^{43} + 609 q^{47} + 141 q^{49} + 228 q^{53} - 45 q^{55} + 60 q^{59} - 454 q^{61} + 85 q^{65} + 244 q^{67} + 444 q^{71} + 398 q^{73} - 198 q^{77} + 349 q^{79} + 1038 q^{83} + 375 q^{85} - 852 q^{89} + 374 q^{91} + 20 q^{95} + 914 q^{97}+O(q^{100})$$ q + 5 * q^5 + 22 * q^7 - 9 * q^11 + 17 * q^13 + 75 * q^17 + 4 * q^19 + 183 * q^23 + 25 * q^25 - 129 * q^29 + 187 * q^31 + 110 * q^35 - 34 * q^37 - 264 * q^41 - 443 * q^43 + 609 * q^47 + 141 * q^49 + 228 * q^53 - 45 * q^55 + 60 * q^59 - 454 * q^61 + 85 * q^65 + 244 * q^67 + 444 * q^71 + 398 * q^73 - 198 * q^77 + 349 * q^79 + 1038 * q^83 + 375 * q^85 - 852 * q^89 + 374 * q^91 + 20 * q^95 + 914 * q^97

## 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 5.00000 0 22.0000 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 2160.4.a.s 1
3.b odd 2 1 2160.4.a.h 1
4.b odd 2 1 540.4.a.c yes 1
12.b even 2 1 540.4.a.a 1
36.f odd 6 2 1620.4.i.e 2
36.h even 6 2 1620.4.i.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.4.a.a 1 12.b even 2 1
540.4.a.c yes 1 4.b odd 2 1
1620.4.i.e 2 36.f odd 6 2
1620.4.i.k 2 36.h even 6 2
2160.4.a.h 1 3.b odd 2 1
2160.4.a.s 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2160))$$:

 $$T_{7} - 22$$ T7 - 22 $$T_{11} + 9$$ T11 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 5$$
$7$ $$T - 22$$
$11$ $$T + 9$$
$13$ $$T - 17$$
$17$ $$T - 75$$
$19$ $$T - 4$$
$23$ $$T - 183$$
$29$ $$T + 129$$
$31$ $$T - 187$$
$37$ $$T + 34$$
$41$ $$T + 264$$
$43$ $$T + 443$$
$47$ $$T - 609$$
$53$ $$T - 228$$
$59$ $$T - 60$$
$61$ $$T + 454$$
$67$ $$T - 244$$
$71$ $$T - 444$$
$73$ $$T - 398$$
$79$ $$T - 349$$
$83$ $$T - 1038$$
$89$ $$T + 852$$
$97$ $$T - 914$$