# Properties

 Label 2160.4.a.g 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

 $$f(q)$$ $$=$$ $$q - 5 q^{5} + 13 q^{7}+O(q^{10})$$ q - 5 * q^5 + 13 * q^7 $$q - 5 q^{5} + 13 q^{7} + 30 q^{11} - 61 q^{13} + 12 q^{17} + 49 q^{19} - 18 q^{23} + 25 q^{25} - 186 q^{29} + 160 q^{31} - 65 q^{35} - 91 q^{37} + 378 q^{41} + 268 q^{43} - 144 q^{47} - 174 q^{49} + 570 q^{53} - 150 q^{55} - 204 q^{59} - 877 q^{61} + 305 q^{65} + 187 q^{67} + 606 q^{71} + 431 q^{73} + 390 q^{77} - 1151 q^{79} - 102 q^{83} - 60 q^{85} + 984 q^{89} - 793 q^{91} - 245 q^{95} - 265 q^{97}+O(q^{100})$$ q - 5 * q^5 + 13 * q^7 + 30 * q^11 - 61 * q^13 + 12 * q^17 + 49 * q^19 - 18 * q^23 + 25 * q^25 - 186 * q^29 + 160 * q^31 - 65 * q^35 - 91 * q^37 + 378 * q^41 + 268 * q^43 - 144 * q^47 - 174 * q^49 + 570 * q^53 - 150 * q^55 - 204 * q^59 - 877 * q^61 + 305 * q^65 + 187 * q^67 + 606 * q^71 + 431 * q^73 + 390 * q^77 - 1151 * q^79 - 102 * q^83 - 60 * q^85 + 984 * q^89 - 793 * q^91 - 245 * q^95 - 265 * 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 13.0000 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.g 1
3.b odd 2 1 2160.4.a.q 1
4.b odd 2 1 270.4.a.h yes 1
12.b even 2 1 270.4.a.d 1
20.d odd 2 1 1350.4.a.i 1
20.e even 4 2 1350.4.c.f 2
36.f odd 6 2 810.4.e.j 2
36.h even 6 2 810.4.e.r 2
60.h even 2 1 1350.4.a.w 1
60.l odd 4 2 1350.4.c.o 2

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

 $$T_{7} - 13$$ T7 - 13 $$T_{11} - 30$$ T11 - 30

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 5$$
$7$ $$T - 13$$
$11$ $$T - 30$$
$13$ $$T + 61$$
$17$ $$T - 12$$
$19$ $$T - 49$$
$23$ $$T + 18$$
$29$ $$T + 186$$
$31$ $$T - 160$$
$37$ $$T + 91$$
$41$ $$T - 378$$
$43$ $$T - 268$$
$47$ $$T + 144$$
$53$ $$T - 570$$
$59$ $$T + 204$$
$61$ $$T + 877$$
$67$ $$T - 187$$
$71$ $$T - 606$$
$73$ $$T - 431$$
$79$ $$T + 1151$$
$83$ $$T + 102$$
$89$ $$T - 984$$
$97$ $$T + 265$$