# Properties

 Label 2160.4.a.a 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 540) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 5 q^{5} - 17 q^{7}+O(q^{10})$$ q - 5 * q^5 - 17 * q^7 $$q - 5 q^{5} - 17 q^{7} - 30 q^{11} - 61 q^{13} + 120 q^{17} + 43 q^{19} + 90 q^{23} + 25 q^{25} + 90 q^{29} - 8 q^{31} + 85 q^{35} + 317 q^{37} + 30 q^{41} + 220 q^{43} - 180 q^{47} - 54 q^{49} + 630 q^{53} + 150 q^{55} - 840 q^{59} + 599 q^{61} + 305 q^{65} - 107 q^{67} - 210 q^{71} - 421 q^{73} + 510 q^{77} - 353 q^{79} - 1350 q^{83} - 600 q^{85} - 1020 q^{89} + 1037 q^{91} - 215 q^{95} - 997 q^{97}+O(q^{100})$$ q - 5 * q^5 - 17 * q^7 - 30 * q^11 - 61 * q^13 + 120 * q^17 + 43 * q^19 + 90 * q^23 + 25 * q^25 + 90 * q^29 - 8 * q^31 + 85 * q^35 + 317 * q^37 + 30 * q^41 + 220 * q^43 - 180 * q^47 - 54 * q^49 + 630 * q^53 + 150 * q^55 - 840 * q^59 + 599 * q^61 + 305 * q^65 - 107 * q^67 - 210 * q^71 - 421 * q^73 + 510 * q^77 - 353 * q^79 - 1350 * q^83 - 600 * q^85 - 1020 * q^89 + 1037 * q^91 - 215 * q^95 - 997 * 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 −17.0000 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.a 1
3.b odd 2 1 2160.4.a.k 1
4.b odd 2 1 540.4.a.b 1
12.b even 2 1 540.4.a.d yes 1
36.f odd 6 2 1620.4.i.h 2
36.h even 6 2 1620.4.i.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.4.a.b 1 4.b odd 2 1
540.4.a.d yes 1 12.b even 2 1
1620.4.i.b 2 36.h even 6 2
1620.4.i.h 2 36.f odd 6 2
2160.4.a.a 1 1.a even 1 1 trivial
2160.4.a.k 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2160))$$:

 $$T_{7} + 17$$ T7 + 17 $$T_{11} + 30$$ T11 + 30

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 5$$
$7$ $$T + 17$$
$11$ $$T + 30$$
$13$ $$T + 61$$
$17$ $$T - 120$$
$19$ $$T - 43$$
$23$ $$T - 90$$
$29$ $$T - 90$$
$31$ $$T + 8$$
$37$ $$T - 317$$
$41$ $$T - 30$$
$43$ $$T - 220$$
$47$ $$T + 180$$
$53$ $$T - 630$$
$59$ $$T + 840$$
$61$ $$T - 599$$
$67$ $$T + 107$$
$71$ $$T + 210$$
$73$ $$T + 421$$
$79$ $$T + 353$$
$83$ $$T + 1350$$
$89$ $$T + 1020$$
$97$ $$T + 997$$