# Properties

 Label 1080.1.i.b Level $1080$ Weight $1$ Character orbit 1080.i Self dual yes Analytic conductor $0.539$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -120 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1080,1,Mod(269,1080)]

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, 1, 1, 1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("1080.269");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1080 = 2^{3} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1080.i (of order $$2$$, degree $$1$$, minimal)

## 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: $$0.538990213644$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.1080.1 Artin image: $D_6$ Artin field: Galois closure of 6.2.5832000.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + q^{5} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + q^5 - q^8 $$q - q^{2} + q^{4} + q^{5} - q^{8} - q^{10} - q^{11} + q^{13} + q^{16} + q^{17} + q^{20} + q^{22} + q^{23} + q^{25} - q^{26} - q^{29} - q^{31} - q^{32} - q^{34} - 2 q^{37} - q^{40} + q^{43} - q^{44} - q^{46} + q^{47} + q^{49} - q^{50} + q^{52} - q^{55} + q^{58} + 2 q^{59} + q^{62} + q^{64} + q^{65} - 2 q^{67} + q^{68} + 2 q^{74} - q^{79} + q^{80} + q^{85} - q^{86} + q^{88} + q^{92} - q^{94} - q^{98}+O(q^{100})$$ q - q^2 + q^4 + q^5 - q^8 - q^10 - q^11 + q^13 + q^16 + q^17 + q^20 + q^22 + q^23 + q^25 - q^26 - q^29 - q^31 - q^32 - q^34 - 2 * q^37 - q^40 + q^43 - q^44 - q^46 + q^47 + q^49 - q^50 + q^52 - q^55 + q^58 + 2 * q^59 + q^62 + q^64 + q^65 - 2 * q^67 + q^68 + 2 * q^74 - q^79 + q^80 + q^85 - q^86 + q^88 + q^92 - q^94 - q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times$$.

 $$n$$ $$217$$ $$271$$ $$541$$ $$1001$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
269.1
 0
−1.00000 0 1.00000 1.00000 0 0 −1.00000 0 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
120.i odd 2 1 CM by $$\Q(\sqrt{-30})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1080.1.i.b yes 1
3.b odd 2 1 1080.1.i.c yes 1
5.b even 2 1 1080.1.i.d yes 1
8.b even 2 1 1080.1.i.a 1
9.c even 3 2 3240.1.bh.c 2
9.d odd 6 2 3240.1.bh.b 2
15.d odd 2 1 1080.1.i.a 1
24.h odd 2 1 1080.1.i.d yes 1
40.f even 2 1 1080.1.i.c yes 1
45.h odd 6 2 3240.1.bh.d 2
45.j even 6 2 3240.1.bh.a 2
72.j odd 6 2 3240.1.bh.a 2
72.n even 6 2 3240.1.bh.d 2
120.i odd 2 1 CM 1080.1.i.b yes 1
360.bh odd 6 2 3240.1.bh.c 2
360.bk even 6 2 3240.1.bh.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.1.i.a 1 8.b even 2 1
1080.1.i.a 1 15.d odd 2 1
1080.1.i.b yes 1 1.a even 1 1 trivial
1080.1.i.b yes 1 120.i odd 2 1 CM
1080.1.i.c yes 1 3.b odd 2 1
1080.1.i.c yes 1 40.f even 2 1
1080.1.i.d yes 1 5.b even 2 1
1080.1.i.d yes 1 24.h odd 2 1
3240.1.bh.a 2 45.j even 6 2
3240.1.bh.a 2 72.j odd 6 2
3240.1.bh.b 2 9.d odd 6 2
3240.1.bh.b 2 360.bk even 6 2
3240.1.bh.c 2 9.c even 3 2
3240.1.bh.c 2 360.bh odd 6 2
3240.1.bh.d 2 45.h odd 6 2
3240.1.bh.d 2 72.n even 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1080, [\chi])$$:

 $$T_{7}$$ T7 $$T_{11} + 1$$ T11 + 1 $$T_{13} - 1$$ T13 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T$$
$11$ $$T + 1$$
$13$ $$T - 1$$
$17$ $$T - 1$$
$19$ $$T$$
$23$ $$T - 1$$
$29$ $$T + 1$$
$31$ $$T + 1$$
$37$ $$T + 2$$
$41$ $$T$$
$43$ $$T - 1$$
$47$ $$T - 1$$
$53$ $$T$$
$59$ $$T - 2$$
$61$ $$T$$
$67$ $$T + 2$$
$71$ $$T$$
$73$ $$T$$
$79$ $$T + 1$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T$$