# Properties

 Label 2160.4.a.i 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 270) 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 $$q - 5 q^{5} + 22 q^{7} + 12 q^{11} + 38 q^{13} - 105 q^{17} + 157 q^{19} + 117 q^{23} + 25 q^{25} + 66 q^{29} + 25 q^{31} - 110 q^{35} + 314 q^{37} - 504 q^{41} - 380 q^{43} + 252 q^{47} + 141 q^{49} + 3 q^{53} - 60 q^{55} + 318 q^{59} + 293 q^{61} - 190 q^{65} + 322 q^{67} + 120 q^{71} + 44 q^{73} + 264 q^{77} - 917 q^{79} - 309 q^{83} + 525 q^{85} + 1272 q^{89} + 836 q^{91} - 785 q^{95} + 1328 q^{97}+O(q^{100})$$ q - 5 * q^5 + 22 * q^7 + 12 * q^11 + 38 * q^13 - 105 * q^17 + 157 * q^19 + 117 * q^23 + 25 * q^25 + 66 * q^29 + 25 * q^31 - 110 * q^35 + 314 * q^37 - 504 * q^41 - 380 * q^43 + 252 * q^47 + 141 * q^49 + 3 * q^53 - 60 * q^55 + 318 * q^59 + 293 * q^61 - 190 * q^65 + 322 * q^67 + 120 * q^71 + 44 * q^73 + 264 * q^77 - 917 * q^79 - 309 * q^83 + 525 * q^85 + 1272 * q^89 + 836 * q^91 - 785 * q^95 + 1328 * 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.i 1
3.b odd 2 1 2160.4.a.r 1
4.b odd 2 1 270.4.a.g yes 1
12.b even 2 1 270.4.a.c 1
20.d odd 2 1 1350.4.a.l 1
20.e even 4 2 1350.4.c.i 2
36.f odd 6 2 810.4.e.k 2
36.h even 6 2 810.4.e.s 2
60.h even 2 1 1350.4.a.z 1
60.l odd 4 2 1350.4.c.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.4.a.c 1 12.b even 2 1
270.4.a.g yes 1 4.b odd 2 1
810.4.e.k 2 36.f odd 6 2
810.4.e.s 2 36.h even 6 2
1350.4.a.l 1 20.d odd 2 1
1350.4.a.z 1 60.h even 2 1
1350.4.c.i 2 20.e even 4 2
1350.4.c.l 2 60.l odd 4 2
2160.4.a.i 1 1.a even 1 1 trivial
2160.4.a.r 1 3.b odd 2 1

## Hecke kernels

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

 $$T_{7} - 22$$ T7 - 22 $$T_{11} - 12$$ T11 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 5$$
$7$ $$T - 22$$
$11$ $$T - 12$$
$13$ $$T - 38$$
$17$ $$T + 105$$
$19$ $$T - 157$$
$23$ $$T - 117$$
$29$ $$T - 66$$
$31$ $$T - 25$$
$37$ $$T - 314$$
$41$ $$T + 504$$
$43$ $$T + 380$$
$47$ $$T - 252$$
$53$ $$T - 3$$
$59$ $$T - 318$$
$61$ $$T - 293$$
$67$ $$T - 322$$
$71$ $$T - 120$$
$73$ $$T - 44$$
$79$ $$T + 917$$
$83$ $$T + 309$$
$89$ $$T - 1272$$
$97$ $$T - 1328$$