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Properties

Label 1080.1.i.e
Level $1080$
Weight $1$
Character orbit 1080.i
Analytic conductor $0.539$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -24
Inner twists $8$

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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: no
Analytic conductor: \(0.538990213644\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.46656000.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{12}^{3} q^{2} - q^{4} - \zeta_{12} q^{5} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12}^{3} q^{2} - q^{4} - \zeta_{12} q^{5} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{7} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{4} q^{10} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{11} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{14} + q^{16} + \zeta_{12} q^{20} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{22} + \zeta_{12}^{2} q^{25} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{28} + q^{31} - \zeta_{12}^{3} q^{32} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{35} - \zeta_{12}^{4} q^{40} + (\zeta_{12}^{5} - \zeta_{12}) q^{44} + (\zeta_{12}^{4} - \zeta_{12}^{2} - 1) q^{49} - \zeta_{12}^{5} q^{50} + \zeta_{12}^{3} q^{53} + ( - \zeta_{12}^{2} - 1) q^{55} + (\zeta_{12}^{5} - \zeta_{12}) q^{56} - \zeta_{12}^{3} q^{62} - q^{64} + ( - \zeta_{12}^{2} - 1) q^{70} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{73} + (\zeta_{12}^{5} + \zeta_{12}^{3} + \zeta_{12}) q^{77} + q^{79} - \zeta_{12} q^{80} - \zeta_{12}^{3} q^{83} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{88} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{97} + (\zeta_{12}^{5} + \zeta_{12}^{3} + \zeta_{12}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 2 q^{10} + 4 q^{16} + 2 q^{25} + 4 q^{31} + 2 q^{40} - 8 q^{49} - 6 q^{55} - 4 q^{64} - 6 q^{70} + 8 q^{79}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i
− 1.00000i 0 −1.00000 −0.866025 − 0.500000i 0 1.73205i 1.00000i 0 −0.500000 + 0.866025i
269.2 − 1.00000i 0 −1.00000 0.866025 − 0.500000i 0 − 1.73205i 1.00000i 0 −0.500000 − 0.866025i
269.3 1.00000i 0 −1.00000 −0.866025 + 0.500000i 0 − 1.73205i − 1.00000i 0 −0.500000 − 0.866025i
269.4 1.00000i 0 −1.00000 0.866025 + 0.500000i 0 1.73205i − 1.00000i 0 −0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
15.d odd 2 1 inner
40.f even 2 1 inner
120.i odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1080.1.i.e ✓ 4
3.b odd 2 1 inner 1080.1.i.e ✓ 4
5.b even 2 1 inner 1080.1.i.e ✓ 4
8.b even 2 1 inner 1080.1.i.e ✓ 4
9.c even 3 1 3240.1.bh.f 4
9.c even 3 1 3240.1.bh.i 4
9.d odd 6 1 3240.1.bh.f 4
9.d odd 6 1 3240.1.bh.i 4
15.d odd 2 1 inner 1080.1.i.e ✓ 4
24.h odd 2 1 CM 1080.1.i.e ✓ 4
40.f even 2 1 inner 1080.1.i.e ✓ 4
45.h odd 6 1 3240.1.bh.f 4
45.h odd 6 1 3240.1.bh.i 4
45.j even 6 1 3240.1.bh.f 4
45.j even 6 1 3240.1.bh.i 4
72.j odd 6 1 3240.1.bh.f 4
72.j odd 6 1 3240.1.bh.i 4
72.n even 6 1 3240.1.bh.f 4
72.n even 6 1 3240.1.bh.i 4
120.i odd 2 1 inner 1080.1.i.e ✓ 4
360.bh odd 6 1 3240.1.bh.f 4
360.bh odd 6 1 3240.1.bh.i 4
360.bk even 6 1 3240.1.bh.f 4
360.bk even 6 1 3240.1.bh.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.1.i.e ✓ 4 1.a even 1 1 trivial
1080.1.i.e ✓ 4 3.b odd 2 1 inner
1080.1.i.e ✓ 4 5.b even 2 1 inner
1080.1.i.e ✓ 4 8.b even 2 1 inner
1080.1.i.e ✓ 4 15.d odd 2 1 inner
1080.1.i.e ✓ 4 24.h odd 2 1 CM
1080.1.i.e ✓ 4 40.f even 2 1 inner
1080.1.i.e ✓ 4 120.i odd 2 1 inner
3240.1.bh.f 4 9.c even 3 1
3240.1.bh.f 4 9.d odd 6 1
3240.1.bh.f 4 45.h odd 6 1
3240.1.bh.f 4 45.j even 6 1
3240.1.bh.f 4 72.j odd 6 1
3240.1.bh.f 4 72.n even 6 1
3240.1.bh.f 4 360.bh odd 6 1
3240.1.bh.f 4 360.bk even 6 1
3240.1.bh.i 4 9.c even 3 1
3240.1.bh.i 4 9.d odd 6 1
3240.1.bh.i 4 45.h odd 6 1
3240.1.bh.i 4 45.j even 6 1
3240.1.bh.i 4 72.j odd 6 1
3240.1.bh.i 4 72.n even 6 1
3240.1.bh.i 4 360.bh odd 6 1
3240.1.bh.i 4 360.bk even 6 1

Hecke kernels

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

\( T_{7}^{2} + 3 \) Copy content Toggle raw display
\( T_{11}^{2} - 3 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$7$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T - 1)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$79$ \( (T - 2)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
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