Properties

Label 2997.1.u.a
Level $2997$
Weight $1$
Character orbit 2997.u
Analytic conductor $1.496$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2997,1,Mod(269,2997)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2997, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2997.269");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2997 = 3^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2997.u (of order \(6\), degree \(2\), not 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: \(1.49569784286\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 111)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.4107.1
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.26946027.2

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{4} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6}^{2} q^{4} - q^{7} - \zeta_{6}^{2} q^{13} - \zeta_{6} q^{16} - \zeta_{6} q^{19} - \zeta_{6} q^{25} - \zeta_{6}^{2} q^{28} + \zeta_{6} q^{31} + q^{37} - \zeta_{6}^{2} q^{43} + \zeta_{6} q^{52} + q^{61} + q^{64} - \zeta_{6}^{2} q^{67} - q^{73} + 2 q^{76} - q^{79} + \zeta_{6}^{2} q^{91} - \zeta_{6}^{2} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{4} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{4} - 2 q^{7} + q^{13} - q^{16} - 2 q^{19} - q^{25} + q^{28} + q^{31} + 2 q^{37} + q^{43} + q^{52} + 4 q^{61} + 2 q^{64} + q^{67} - 2 q^{73} + 4 q^{76} - 2 q^{79} - q^{91} + q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2997\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1703\)
\(\chi(n)\) \(-\zeta_{6}\) \(\zeta_{6}\)

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
0.500000 + 0.866025i
0.500000 0.866025i
0 0 −0.500000 + 0.866025i 0 0 −1.00000 0 0 0
1025.1 0 0 −0.500000 0.866025i 0 0 −1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
333.g even 3 1 inner
333.u odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2997.1.u.a 2
3.b odd 2 1 CM 2997.1.u.a 2
9.c even 3 1 111.1.i.a 2
9.c even 3 1 2997.1.l.a 2
9.d odd 6 1 111.1.i.a 2
9.d odd 6 1 2997.1.l.a 2
36.f odd 6 1 1776.1.bw.a 2
36.h even 6 1 1776.1.bw.a 2
37.c even 3 1 2997.1.l.a 2
45.h odd 6 1 2775.1.z.a 2
45.j even 6 1 2775.1.z.a 2
45.k odd 12 2 2775.1.x.a 4
45.l even 12 2 2775.1.x.a 4
111.i odd 6 1 2997.1.l.a 2
333.g even 3 1 inner 2997.1.u.a 2
333.h even 3 1 111.1.i.a 2
333.l odd 6 1 111.1.i.a 2
333.u odd 6 1 inner 2997.1.u.a 2
1332.s even 6 1 1776.1.bw.a 2
1332.br odd 6 1 1776.1.bw.a 2
1665.ba even 6 1 2775.1.z.a 2
1665.bz odd 6 1 2775.1.z.a 2
1665.co odd 12 2 2775.1.x.a 4
1665.cp even 12 2 2775.1.x.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.1.i.a 2 9.c even 3 1
111.1.i.a 2 9.d odd 6 1
111.1.i.a 2 333.h even 3 1
111.1.i.a 2 333.l odd 6 1
1776.1.bw.a 2 36.f odd 6 1
1776.1.bw.a 2 36.h even 6 1
1776.1.bw.a 2 1332.s even 6 1
1776.1.bw.a 2 1332.br odd 6 1
2775.1.x.a 4 45.k odd 12 2
2775.1.x.a 4 45.l even 12 2
2775.1.x.a 4 1665.co odd 12 2
2775.1.x.a 4 1665.cp even 12 2
2775.1.z.a 2 45.h odd 6 1
2775.1.z.a 2 45.j even 6 1
2775.1.z.a 2 1665.ba even 6 1
2775.1.z.a 2 1665.bz odd 6 1
2997.1.l.a 2 9.c even 3 1
2997.1.l.a 2 9.d odd 6 1
2997.1.l.a 2 37.c even 3 1
2997.1.l.a 2 111.i odd 6 1
2997.1.u.a 2 1.a even 1 1 trivial
2997.1.u.a 2 3.b odd 2 1 CM
2997.1.u.a 2 333.g even 3 1 inner
2997.1.u.a 2 333.u odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{1}^{\mathrm{new}}(2997, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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