# Properties

 Label 2160.4.a.b 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

 $$f(q)$$ $$=$$ $$q - 5 q^{5} - 14 q^{7}+O(q^{10})$$ q - 5 * q^5 - 14 * q^7 $$q - 5 q^{5} - 14 q^{7} + 3 q^{11} + 47 q^{13} + 39 q^{17} - 32 q^{19} - 99 q^{23} + 25 q^{25} - 51 q^{29} - 83 q^{31} + 70 q^{35} + 314 q^{37} + 108 q^{41} - 299 q^{43} + 531 q^{47} - 147 q^{49} - 564 q^{53} - 15 q^{55} + 12 q^{59} + 230 q^{61} - 235 q^{65} + 268 q^{67} + 120 q^{71} + 1106 q^{73} - 42 q^{77} + 739 q^{79} + 1086 q^{83} - 195 q^{85} + 120 q^{89} - 658 q^{91} + 160 q^{95} - 1642 q^{97}+O(q^{100})$$ q - 5 * q^5 - 14 * q^7 + 3 * q^11 + 47 * q^13 + 39 * q^17 - 32 * q^19 - 99 * q^23 + 25 * q^25 - 51 * q^29 - 83 * q^31 + 70 * q^35 + 314 * q^37 + 108 * q^41 - 299 * q^43 + 531 * q^47 - 147 * q^49 - 564 * q^53 - 15 * q^55 + 12 * q^59 + 230 * q^61 - 235 * q^65 + 268 * q^67 + 120 * q^71 + 1106 * q^73 - 42 * q^77 + 739 * q^79 + 1086 * q^83 - 195 * q^85 + 120 * q^89 - 658 * q^91 + 160 * q^95 - 1642 * 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 −14.0000 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.b 1
3.b odd 2 1 2160.4.a.l 1
4.b odd 2 1 270.4.a.j yes 1
12.b even 2 1 270.4.a.f 1
20.d odd 2 1 1350.4.a.e 1
20.e even 4 2 1350.4.c.j 2
36.f odd 6 2 810.4.e.f 2
36.h even 6 2 810.4.e.n 2
60.h even 2 1 1350.4.a.r 1
60.l odd 4 2 1350.4.c.k 2

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

 $$T_{7} + 14$$ T7 + 14 $$T_{11} - 3$$ T11 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 5$$
$7$ $$T + 14$$
$11$ $$T - 3$$
$13$ $$T - 47$$
$17$ $$T - 39$$
$19$ $$T + 32$$
$23$ $$T + 99$$
$29$ $$T + 51$$
$31$ $$T + 83$$
$37$ $$T - 314$$
$41$ $$T - 108$$
$43$ $$T + 299$$
$47$ $$T - 531$$
$53$ $$T + 564$$
$59$ $$T - 12$$
$61$ $$T - 230$$
$67$ $$T - 268$$
$71$ $$T - 120$$
$73$ $$T - 1106$$
$79$ $$T - 739$$
$83$ $$T - 1086$$
$89$ $$T - 120$$
$97$ $$T + 1642$$