Properties

Label 2340.1.em.c
Level $2340$
Weight $1$
Character orbit 2340.em
Analytic conductor $1.168$
Analytic rank $0$
Dimension $4$
Projective image $D_{12}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2340,1,Mod(163,2340)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2340, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 0, 9, 11]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2340.163");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2340.em (of order \(12\), degree \(4\), 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: \(1.16781212956\)
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: \(D_{12}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{12}^{2} q^{2} + \zeta_{12}^{4} q^{4} + \zeta_{12} q^{5} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12}^{2} q^{2} + \zeta_{12}^{4} q^{4} + \zeta_{12} q^{5} - q^{8} + \zeta_{12}^{3} q^{10} + \zeta_{12}^{2} q^{13} - \zeta_{12}^{2} q^{16} + ( - \zeta_{12}^{3} + \zeta_{12}^{2}) q^{17} + \zeta_{12}^{5} q^{20} + \zeta_{12}^{2} q^{25} + \zeta_{12}^{4} q^{26} + (\zeta_{12}^{4} - 1) q^{29} - \zeta_{12}^{4} q^{32} + ( - \zeta_{12}^{5} + \zeta_{12}^{4}) q^{34} - \zeta_{12}^{5} q^{37} - \zeta_{12} q^{40} + (\zeta_{12} - 1) q^{41} + \zeta_{12}^{2} q^{49} + \zeta_{12}^{4} q^{50} - q^{52} + ( - \zeta_{12}^{2} - \zeta_{12}) q^{53} + ( - \zeta_{12}^{2} - 1) q^{58} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{61} + q^{64} + \zeta_{12}^{3} q^{65} + (\zeta_{12} - 1) q^{68} + (\zeta_{12}^{5} - \zeta_{12}) q^{73} + \zeta_{12} q^{74} - \zeta_{12}^{3} q^{80} + (\zeta_{12}^{3} - \zeta_{12}^{2}) q^{82} + ( - \zeta_{12}^{4} + \zeta_{12}^{3}) q^{85} + (\zeta_{12}^{5} + \zeta_{12}^{2}) q^{89} - \zeta_{12}^{4} q^{97} + \zeta_{12}^{4} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{4} - 4 q^{8} + 2 q^{13} - 2 q^{16} + 2 q^{17} + 2 q^{25} - 2 q^{26} - 6 q^{29} + 2 q^{32} - 2 q^{34} - 4 q^{41} + 2 q^{49} - 2 q^{50} - 4 q^{52} - 2 q^{53} - 6 q^{58} + 4 q^{64} - 4 q^{68} - 2 q^{82} + 2 q^{85} + 2 q^{89} + 4 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times\).

\(n\) \(937\) \(1081\) \(1171\) \(2081\)
\(\chi(n)\) \(-\zeta_{12}^{3}\) \(-\zeta_{12}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0.500000 0.866025i 0 −0.500000 0.866025i −0.866025 + 0.500000i 0 0 −1.00000 0 1.00000i
487.1 0.500000 0.866025i 0 −0.500000 0.866025i 0.866025 0.500000i 0 0 −1.00000 0 1.00000i
847.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.866025 0.500000i 0 0 −1.00000 0 1.00000i
2143.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.866025 + 0.500000i 0 0 −1.00000 0 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
65.o even 12 1 inner
260.be odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2340.1.em.c yes 4
3.b odd 2 1 2340.1.em.b 4
4.b odd 2 1 CM 2340.1.em.c yes 4
5.c odd 4 1 2340.1.hj.b yes 4
12.b even 2 1 2340.1.em.b 4
13.f odd 12 1 2340.1.hj.b yes 4
15.e even 4 1 2340.1.hj.a yes 4
20.e even 4 1 2340.1.hj.b yes 4
39.k even 12 1 2340.1.hj.a yes 4
52.l even 12 1 2340.1.hj.b yes 4
60.l odd 4 1 2340.1.hj.a yes 4
65.o even 12 1 inner 2340.1.em.c yes 4
156.v odd 12 1 2340.1.hj.a yes 4
195.bn odd 12 1 2340.1.em.b 4
260.be odd 12 1 inner 2340.1.em.c yes 4
780.cf even 12 1 2340.1.em.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2340.1.em.b 4 3.b odd 2 1
2340.1.em.b 4 12.b even 2 1
2340.1.em.b 4 195.bn odd 12 1
2340.1.em.b 4 780.cf even 12 1
2340.1.em.c yes 4 1.a even 1 1 trivial
2340.1.em.c yes 4 4.b odd 2 1 CM
2340.1.em.c yes 4 65.o even 12 1 inner
2340.1.em.c yes 4 260.be odd 12 1 inner
2340.1.hj.a yes 4 15.e even 4 1
2340.1.hj.a yes 4 39.k even 12 1
2340.1.hj.a yes 4 60.l odd 4 1
2340.1.hj.a yes 4 156.v odd 12 1
2340.1.hj.b yes 4 5.c odd 4 1
2340.1.hj.b yes 4 13.f odd 12 1
2340.1.hj.b yes 4 20.e even 4 1
2340.1.hj.b yes 4 52.l even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2340, [\chi])\):

\( T_{17}^{4} - 2T_{17}^{3} + 5T_{17}^{2} - 4T_{17} + 1 \) Copy content Toggle raw display
\( T_{29}^{2} + 3T_{29} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} + 5 T^{2} - 4 T + 1 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$41$ \( T^{4} + 4 T^{3} + 5 T^{2} + 2 T + 1 \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} + 2 T^{3} + 2 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} + 3T^{2} + 9 \) 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
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$97$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
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