# Properties

 Label 2160.4.a.j 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

 $$f(q)$$ $$=$$ $$q - 5 q^{5} + 34 q^{7}+O(q^{10})$$ q - 5 * q^5 + 34 * q^7 $$q - 5 q^{5} + 34 q^{7} - 48 q^{11} - 70 q^{13} + 27 q^{17} - 119 q^{19} - 51 q^{23} + 25 q^{25} + 30 q^{29} + 133 q^{31} - 170 q^{35} + 218 q^{37} - 156 q^{41} + 88 q^{43} - 516 q^{47} + 813 q^{49} + 639 q^{53} + 240 q^{55} - 654 q^{59} + 461 q^{61} + 350 q^{65} - 182 q^{67} + 900 q^{71} + 704 q^{73} - 1632 q^{77} + 1375 q^{79} + 915 q^{83} - 135 q^{85} + 1116 q^{89} - 2380 q^{91} + 595 q^{95} - 16 q^{97}+O(q^{100})$$ q - 5 * q^5 + 34 * q^7 - 48 * q^11 - 70 * q^13 + 27 * q^17 - 119 * q^19 - 51 * q^23 + 25 * q^25 + 30 * q^29 + 133 * q^31 - 170 * q^35 + 218 * q^37 - 156 * q^41 + 88 * q^43 - 516 * q^47 + 813 * q^49 + 639 * q^53 + 240 * q^55 - 654 * q^59 + 461 * q^61 + 350 * q^65 - 182 * q^67 + 900 * q^71 + 704 * q^73 - 1632 * q^77 + 1375 * q^79 + 915 * q^83 - 135 * q^85 + 1116 * q^89 - 2380 * q^91 + 595 * q^95 - 16 * 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 34.0000 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.j 1
3.b odd 2 1 2160.4.a.t 1
4.b odd 2 1 270.4.a.a 1
12.b even 2 1 270.4.a.k yes 1
20.d odd 2 1 1350.4.a.bb 1
20.e even 4 2 1350.4.c.r 2
36.f odd 6 2 810.4.e.x 2
36.h even 6 2 810.4.e.d 2
60.h even 2 1 1350.4.a.n 1
60.l odd 4 2 1350.4.c.c 2

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

 $$T_{7} - 34$$ T7 - 34 $$T_{11} + 48$$ T11 + 48

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 5$$
$7$ $$T - 34$$
$11$ $$T + 48$$
$13$ $$T + 70$$
$17$ $$T - 27$$
$19$ $$T + 119$$
$23$ $$T + 51$$
$29$ $$T - 30$$
$31$ $$T - 133$$
$37$ $$T - 218$$
$41$ $$T + 156$$
$43$ $$T - 88$$
$47$ $$T + 516$$
$53$ $$T - 639$$
$59$ $$T + 654$$
$61$ $$T - 461$$
$67$ $$T + 182$$
$71$ $$T - 900$$
$73$ $$T - 704$$
$79$ $$T - 1375$$
$83$ $$T - 915$$
$89$ $$T - 1116$$
$97$ $$T + 16$$