# Properties

 Label 2160.4.a.m Level $2160$ Weight $4$ Character orbit 2160.a Self dual yes Analytic conductor $127.444$ Analytic rank $1$ 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q + 5 q^{5} - 8 q^{7}+O(q^{10})$$ q + 5 * q^5 - 8 * q^7 $$q + 5 q^{5} - 8 q^{7} - 18 q^{11} + 8 q^{13} + 15 q^{17} - 23 q^{19} - 63 q^{23} + 25 q^{25} + 156 q^{29} + 85 q^{31} - 40 q^{35} + 74 q^{37} + 246 q^{41} + 190 q^{43} - 288 q^{47} - 279 q^{49} - 177 q^{53} - 90 q^{55} - 792 q^{59} - 907 q^{61} + 40 q^{65} + 322 q^{67} + 270 q^{71} + 254 q^{73} + 144 q^{77} + 1123 q^{79} + 771 q^{83} + 75 q^{85} - 198 q^{89} - 64 q^{91} - 115 q^{95} - 1192 q^{97}+O(q^{100})$$ q + 5 * q^5 - 8 * q^7 - 18 * q^11 + 8 * q^13 + 15 * q^17 - 23 * q^19 - 63 * q^23 + 25 * q^25 + 156 * q^29 + 85 * q^31 - 40 * q^35 + 74 * q^37 + 246 * q^41 + 190 * q^43 - 288 * q^47 - 279 * q^49 - 177 * q^53 - 90 * q^55 - 792 * q^59 - 907 * q^61 + 40 * q^65 + 322 * q^67 + 270 * q^71 + 254 * q^73 + 144 * q^77 + 1123 * q^79 + 771 * q^83 + 75 * q^85 - 198 * q^89 - 64 * q^91 - 115 * q^95 - 1192 * 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 −8.00000 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 2160.4.a.m 1
3.b odd 2 1 2160.4.a.c 1
4.b odd 2 1 270.4.a.l yes 1
12.b even 2 1 270.4.a.b 1
20.d odd 2 1 1350.4.a.f 1
20.e even 4 2 1350.4.c.n 2
36.f odd 6 2 810.4.e.b 2
36.h even 6 2 810.4.e.v 2
60.h even 2 1 1350.4.a.t 1
60.l odd 4 2 1350.4.c.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.4.a.b 1 12.b even 2 1
270.4.a.l yes 1 4.b odd 2 1
810.4.e.b 2 36.f odd 6 2
810.4.e.v 2 36.h even 6 2
1350.4.a.f 1 20.d odd 2 1
1350.4.a.t 1 60.h even 2 1
1350.4.c.g 2 60.l odd 4 2
1350.4.c.n 2 20.e even 4 2
2160.4.a.c 1 3.b odd 2 1
2160.4.a.m 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2160))$$:

 $$T_{7} + 8$$ T7 + 8 $$T_{11} + 18$$ T11 + 18

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 5$$
$7$ $$T + 8$$
$11$ $$T + 18$$
$13$ $$T - 8$$
$17$ $$T - 15$$
$19$ $$T + 23$$
$23$ $$T + 63$$
$29$ $$T - 156$$
$31$ $$T - 85$$
$37$ $$T - 74$$
$41$ $$T - 246$$
$43$ $$T - 190$$
$47$ $$T + 288$$
$53$ $$T + 177$$
$59$ $$T + 792$$
$61$ $$T + 907$$
$67$ $$T - 322$$
$71$ $$T - 270$$
$73$ $$T - 254$$
$79$ $$T - 1123$$
$83$ $$T - 771$$
$89$ $$T + 198$$
$97$ $$T + 1192$$