# Properties

 Label 1080.6.a.b Level $1080$ Weight $6$ Character orbit 1080.a Self dual yes Analytic conductor $173.215$ 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] = [1080,6,Mod(1,1080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1080, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("1080.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1080 = 2^{3} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 1080.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: $$173.214525398$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q + 25 q^{5} - 234 q^{7}+O(q^{10})$$ q + 25 * q^5 - 234 * q^7 $$q + 25 q^{5} - 234 q^{7} + 347 q^{11} - 33 q^{13} + 237 q^{17} + 1496 q^{19} - 2811 q^{23} + 625 q^{25} - 5513 q^{29} + 2911 q^{31} - 5850 q^{35} + 5602 q^{37} + 4716 q^{41} + 10479 q^{43} + 5963 q^{47} + 37949 q^{49} - 17964 q^{53} + 8675 q^{55} + 30372 q^{59} - 35530 q^{61} - 825 q^{65} - 12476 q^{67} + 7520 q^{71} + 36378 q^{73} - 81198 q^{77} - 22727 q^{79} + 46254 q^{83} + 5925 q^{85} - 58832 q^{89} + 7722 q^{91} + 37400 q^{95} - 145906 q^{97}+O(q^{100})$$ q + 25 * q^5 - 234 * q^7 + 347 * q^11 - 33 * q^13 + 237 * q^17 + 1496 * q^19 - 2811 * q^23 + 625 * q^25 - 5513 * q^29 + 2911 * q^31 - 5850 * q^35 + 5602 * q^37 + 4716 * q^41 + 10479 * q^43 + 5963 * q^47 + 37949 * q^49 - 17964 * q^53 + 8675 * q^55 + 30372 * q^59 - 35530 * q^61 - 825 * q^65 - 12476 * q^67 + 7520 * q^71 + 36378 * q^73 - 81198 * q^77 - 22727 * q^79 + 46254 * q^83 + 5925 * q^85 - 58832 * q^89 + 7722 * q^91 + 37400 * q^95 - 145906 * 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 25.0000 0 −234.000 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 1080.6.a.b yes 1
3.b odd 2 1 1080.6.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.6.a.a 1 3.b odd 2 1
1080.6.a.b yes 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(1080))$$:

 $$T_{7} + 234$$ T7 + 234 $$T_{11} - 347$$ T11 - 347

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 25$$
$7$ $$T + 234$$
$11$ $$T - 347$$
$13$ $$T + 33$$
$17$ $$T - 237$$
$19$ $$T - 1496$$
$23$ $$T + 2811$$
$29$ $$T + 5513$$
$31$ $$T - 2911$$
$37$ $$T - 5602$$
$41$ $$T - 4716$$
$43$ $$T - 10479$$
$47$ $$T - 5963$$
$53$ $$T + 17964$$
$59$ $$T - 30372$$
$61$ $$T + 35530$$
$67$ $$T + 12476$$
$71$ $$T - 7520$$
$73$ $$T - 36378$$
$79$ $$T + 22727$$
$83$ $$T - 46254$$
$89$ $$T + 58832$$
$97$ $$T + 145906$$