Properties

Label 2156.1.k.b
Level $2156$
Weight $1$
Character orbit 2156.k
Analytic conductor $1.076$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -11
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2156,1,Mod(373,2156)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2156, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2156.373");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2156 = 2^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2156.k (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: \(1.07598416724\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 44)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.44.1
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.51131696.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{6}^{2} q^{3} + \zeta_{6} q^{5} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6}^{2} q^{3} + \zeta_{6} q^{5} + \zeta_{6}^{2} q^{11} + q^{15} + \zeta_{6} q^{23} + q^{27} - \zeta_{6}^{2} q^{31} + \zeta_{6} q^{33} + \zeta_{6} q^{37} - 2 \zeta_{6} q^{47} + 2 \zeta_{6}^{2} q^{53} - q^{55} - \zeta_{6}^{2} q^{59} - \zeta_{6}^{2} q^{67} + q^{69} - q^{71} - \zeta_{6}^{2} q^{81} + \zeta_{6} q^{89} - \zeta_{6} q^{93} - q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + q^{5} - q^{11} + 2 q^{15} + q^{23} + 2 q^{27} + q^{31} + q^{33} + q^{37} - 2 q^{47} - 2 q^{53} - 2 q^{55} + q^{59} + q^{67} + 2 q^{69} - 2 q^{71} + q^{81} + q^{89} - q^{93} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2156\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1079\) \(1277\)
\(\chi(n)\) \(-1\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
373.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0.500000 + 0.866025i 0 0.500000 0.866025i 0 0 0 0 0
1341.1 0 0.500000 0.866025i 0 0.500000 + 0.866025i 0 0 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
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
7.c even 3 1 inner
77.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2156.1.k.b 2
7.b odd 2 1 2156.1.k.a 2
7.c even 3 1 44.1.d.a 1
7.c even 3 1 inner 2156.1.k.b 2
7.d odd 6 1 2156.1.h.a 1
7.d odd 6 1 2156.1.k.a 2
11.b odd 2 1 CM 2156.1.k.b 2
21.h odd 6 1 396.1.f.a 1
28.g odd 6 1 176.1.h.a 1
35.j even 6 1 1100.1.f.a 1
35.l odd 12 2 1100.1.e.a 2
56.k odd 6 1 704.1.h.a 1
56.p even 6 1 704.1.h.b 1
63.g even 3 1 3564.1.m.b 2
63.h even 3 1 3564.1.m.b 2
63.j odd 6 1 3564.1.m.a 2
63.n odd 6 1 3564.1.m.a 2
77.b even 2 1 2156.1.k.a 2
77.h odd 6 1 44.1.d.a 1
77.h odd 6 1 inner 2156.1.k.b 2
77.i even 6 1 2156.1.h.a 1
77.i even 6 1 2156.1.k.a 2
77.m even 15 4 484.1.f.a 4
77.o odd 30 4 484.1.f.a 4
84.n even 6 1 1584.1.j.a 1
112.u odd 12 2 2816.1.b.a 2
112.w even 12 2 2816.1.b.b 2
231.l even 6 1 396.1.f.a 1
308.n even 6 1 176.1.h.a 1
308.bb odd 30 4 1936.1.n.a 4
308.bc even 30 4 1936.1.n.a 4
385.q odd 6 1 1100.1.f.a 1
385.bc even 12 2 1100.1.e.a 2
616.y even 6 1 704.1.h.a 1
616.bg odd 6 1 704.1.h.b 1
693.o odd 6 1 3564.1.m.b 2
693.r even 6 1 3564.1.m.a 2
693.bl odd 6 1 3564.1.m.b 2
693.bn even 6 1 3564.1.m.a 2
924.z odd 6 1 1584.1.j.a 1
1232.ch even 12 2 2816.1.b.a 2
1232.ck odd 12 2 2816.1.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.1.d.a 1 7.c even 3 1
44.1.d.a 1 77.h odd 6 1
176.1.h.a 1 28.g odd 6 1
176.1.h.a 1 308.n even 6 1
396.1.f.a 1 21.h odd 6 1
396.1.f.a 1 231.l even 6 1
484.1.f.a 4 77.m even 15 4
484.1.f.a 4 77.o odd 30 4
704.1.h.a 1 56.k odd 6 1
704.1.h.a 1 616.y even 6 1
704.1.h.b 1 56.p even 6 1
704.1.h.b 1 616.bg odd 6 1
1100.1.e.a 2 35.l odd 12 2
1100.1.e.a 2 385.bc even 12 2
1100.1.f.a 1 35.j even 6 1
1100.1.f.a 1 385.q odd 6 1
1584.1.j.a 1 84.n even 6 1
1584.1.j.a 1 924.z odd 6 1
1936.1.n.a 4 308.bb odd 30 4
1936.1.n.a 4 308.bc even 30 4
2156.1.h.a 1 7.d odd 6 1
2156.1.h.a 1 77.i even 6 1
2156.1.k.a 2 7.b odd 2 1
2156.1.k.a 2 7.d odd 6 1
2156.1.k.a 2 77.b even 2 1
2156.1.k.a 2 77.i even 6 1
2156.1.k.b 2 1.a even 1 1 trivial
2156.1.k.b 2 7.c even 3 1 inner
2156.1.k.b 2 11.b odd 2 1 CM
2156.1.k.b 2 77.h odd 6 1 inner
2816.1.b.a 2 112.u odd 12 2
2816.1.b.a 2 1232.ch even 12 2
2816.1.b.b 2 112.w even 12 2
2816.1.b.b 2 1232.ck odd 12 2
3564.1.m.a 2 63.j odd 6 1
3564.1.m.a 2 63.n odd 6 1
3564.1.m.a 2 693.r even 6 1
3564.1.m.a 2 693.bn even 6 1
3564.1.m.b 2 63.g even 3 1
3564.1.m.b 2 63.h even 3 1
3564.1.m.b 2 693.o odd 6 1
3564.1.m.b 2 693.bl odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2156, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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