# Properties

 Label 2160.4.a.f 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 135) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 5 q^{5} + 6 q^{7}+O(q^{10})$$ q - 5 * q^5 + 6 * q^7 $$q - 5 q^{5} + 6 q^{7} - 47 q^{11} - 5 q^{13} + 131 q^{17} + 56 q^{19} + 3 q^{23} + 25 q^{25} + 157 q^{29} - 225 q^{31} - 30 q^{35} - 70 q^{37} - 140 q^{41} - 397 q^{43} - 347 q^{47} - 307 q^{49} - 4 q^{53} + 235 q^{55} + 748 q^{59} - 338 q^{61} + 25 q^{65} - 492 q^{67} + 32 q^{71} + 970 q^{73} - 282 q^{77} + 1257 q^{79} - 102 q^{83} - 655 q^{85} + 1488 q^{89} - 30 q^{91} - 280 q^{95} + 974 q^{97}+O(q^{100})$$ q - 5 * q^5 + 6 * q^7 - 47 * q^11 - 5 * q^13 + 131 * q^17 + 56 * q^19 + 3 * q^23 + 25 * q^25 + 157 * q^29 - 225 * q^31 - 30 * q^35 - 70 * q^37 - 140 * q^41 - 397 * q^43 - 347 * q^47 - 307 * q^49 - 4 * q^53 + 235 * q^55 + 748 * q^59 - 338 * q^61 + 25 * q^65 - 492 * q^67 + 32 * q^71 + 970 * q^73 - 282 * q^77 + 1257 * q^79 - 102 * q^83 - 655 * q^85 + 1488 * q^89 - 30 * q^91 - 280 * q^95 + 974 * 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 6.00000 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.f 1
3.b odd 2 1 2160.4.a.p 1
4.b odd 2 1 135.4.a.b 1
12.b even 2 1 135.4.a.c yes 1
20.d odd 2 1 675.4.a.h 1
20.e even 4 2 675.4.b.f 2
36.f odd 6 2 405.4.e.h 2
36.h even 6 2 405.4.e.f 2
60.h even 2 1 675.4.a.c 1
60.l odd 4 2 675.4.b.e 2

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

 $$T_{7} - 6$$ T7 - 6 $$T_{11} + 47$$ T11 + 47

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 5$$
$7$ $$T - 6$$
$11$ $$T + 47$$
$13$ $$T + 5$$
$17$ $$T - 131$$
$19$ $$T - 56$$
$23$ $$T - 3$$
$29$ $$T - 157$$
$31$ $$T + 225$$
$37$ $$T + 70$$
$41$ $$T + 140$$
$43$ $$T + 397$$
$47$ $$T + 347$$
$53$ $$T + 4$$
$59$ $$T - 748$$
$61$ $$T + 338$$
$67$ $$T + 492$$
$71$ $$T - 32$$
$73$ $$T - 970$$
$79$ $$T - 1257$$
$83$ $$T + 102$$
$89$ $$T - 1488$$
$97$ $$T - 974$$