# Properties

 Label 2160.4.a.d 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,4,Mod(1,2160)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2160.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2160 = 2^{4} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2160.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 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 135) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 5 q^{5}+O(q^{10})$$ q - 5 * q^5 $$q - 5 q^{5} + 10 q^{11} - 80 q^{13} - 7 q^{17} + 113 q^{19} - 81 q^{23} + 25 q^{25} + 220 q^{29} + 189 q^{31} + 170 q^{37} + 130 q^{41} - 10 q^{43} + 160 q^{47} - 343 q^{49} - 631 q^{53} - 50 q^{55} - 560 q^{59} + 229 q^{61} + 400 q^{65} - 750 q^{67} + 890 q^{71} - 890 q^{73} + 27 q^{79} + 429 q^{83} + 35 q^{85} + 750 q^{89} - 565 q^{95} - 1480 q^{97}+O(q^{100})$$ q - 5 * q^5 + 10 * q^11 - 80 * q^13 - 7 * q^17 + 113 * q^19 - 81 * q^23 + 25 * q^25 + 220 * q^29 + 189 * q^31 + 170 * q^37 + 130 * q^41 - 10 * q^43 + 160 * q^47 - 343 * q^49 - 631 * q^53 - 50 * q^55 - 560 * q^59 + 229 * q^61 + 400 * q^65 - 750 * q^67 + 890 * q^71 - 890 * q^73 + 27 * q^79 + 429 * q^83 + 35 * q^85 + 750 * q^89 - 565 * q^95 - 1480 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 0 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.d 1
3.b odd 2 1 2160.4.a.n 1
4.b odd 2 1 135.4.a.d yes 1
12.b even 2 1 135.4.a.a 1
20.d odd 2 1 675.4.a.b 1
20.e even 4 2 675.4.b.c 2
36.f odd 6 2 405.4.e.e 2
36.h even 6 2 405.4.e.j 2
60.h even 2 1 675.4.a.i 1
60.l odd 4 2 675.4.b.d 2

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

 $$T_{7}$$ T7 $$T_{11} - 10$$ T11 - 10

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 5$$
$7$ $$T$$
$11$ $$T - 10$$
$13$ $$T + 80$$
$17$ $$T + 7$$
$19$ $$T - 113$$
$23$ $$T + 81$$
$29$ $$T - 220$$
$31$ $$T - 189$$
$37$ $$T - 170$$
$41$ $$T - 130$$
$43$ $$T + 10$$
$47$ $$T - 160$$
$53$ $$T + 631$$
$59$ $$T + 560$$
$61$ $$T - 229$$
$67$ $$T + 750$$
$71$ $$T - 890$$
$73$ $$T + 890$$
$79$ $$T - 27$$
$83$ $$T - 429$$
$89$ $$T - 750$$
$97$ $$T + 1480$$