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 $2$

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

Character values

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

\(n\) \(217\) \(271\) \(541\) \(1001\)
\(\chi(n)\) \(1\) \(0\) \(1\) \(1\)

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:

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 \(S_{1}^{\mathrm{new}}(1080, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{11} + 1 \) Copy content Toggle raw display
\( T_{13} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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