Properties

Label 2156.1.k.d
Level $2156$
Weight $1$
Character orbit 2156.k
Analytic conductor $1.076$
Analytic rank $0$
Dimension $8$
Projective image $D_{12}$
CM discriminant -11
Inner twists $8$

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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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 3^{2} \)
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.

\(f(q)\) \(=\) \( q + (\zeta_{24}^{11} - \zeta_{24}^{9}) q^{3} + (\zeta_{24}^{9} + \zeta_{24}^{7}) q^{5} + ( - \zeta_{24}^{10} + \cdots - \zeta_{24}^{6}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{24}^{11} - \zeta_{24}^{9}) q^{3} + (\zeta_{24}^{9} + \zeta_{24}^{7}) q^{5} + ( - \zeta_{24}^{10} + \cdots - \zeta_{24}^{6}) q^{9}+ \cdots + ( - \zeta_{24}^{10} + \zeta_{24}^{2} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{9} + 4 q^{11} + 8 q^{15} - 4 q^{25} - 4 q^{37} - 8 q^{81} + 8 q^{93} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(981\) \(1079\) \(1277\)
\(\chi(n)\) \(-1\) \(1\) \(\zeta_{24}^{8}\)

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
0.258819 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
−0.258819 + 0.965926i
0.258819 + 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
−0.258819 0.965926i
0 −0.965926 1.67303i 0 −0.258819 + 0.448288i 0 0 0 −1.36603 + 2.36603i 0
373.2 0 −0.258819 0.448288i 0 −0.965926 + 1.67303i 0 0 0 0.366025 0.633975i 0
373.3 0 0.258819 + 0.448288i 0 0.965926 1.67303i 0 0 0 0.366025 0.633975i 0
373.4 0 0.965926 + 1.67303i 0 0.258819 0.448288i 0 0 0 −1.36603 + 2.36603i 0
1341.1 0 −0.965926 + 1.67303i 0 −0.258819 0.448288i 0 0 0 −1.36603 2.36603i 0
1341.2 0 −0.258819 + 0.448288i 0 −0.965926 1.67303i 0 0 0 0.366025 + 0.633975i 0
1341.3 0 0.258819 0.448288i 0 0.965926 + 1.67303i 0 0 0 0.366025 + 0.633975i 0
1341.4 0 0.965926 1.67303i 0 0.258819 + 0.448288i 0 0 0 −1.36603 2.36603i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 373.4
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.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
77.b even 2 1 inner
77.h odd 6 1 inner
77.i even 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.d 8
7.b odd 2 1 inner 2156.1.k.d 8
7.c even 3 1 2156.1.h.c 4
7.c even 3 1 inner 2156.1.k.d 8
7.d odd 6 1 2156.1.h.c 4
7.d odd 6 1 inner 2156.1.k.d 8
11.b odd 2 1 CM 2156.1.k.d 8
77.b even 2 1 inner 2156.1.k.d 8
77.h odd 6 1 2156.1.h.c 4
77.h odd 6 1 inner 2156.1.k.d 8
77.i even 6 1 2156.1.h.c 4
77.i even 6 1 inner 2156.1.k.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2156.1.h.c 4 7.c even 3 1
2156.1.h.c 4 7.d odd 6 1
2156.1.h.c 4 77.h odd 6 1
2156.1.h.c 4 77.i even 6 1
2156.1.k.d 8 1.a even 1 1 trivial
2156.1.k.d 8 7.b odd 2 1 inner
2156.1.k.d 8 7.c even 3 1 inner
2156.1.k.d 8 7.d odd 6 1 inner
2156.1.k.d 8 11.b odd 2 1 CM
2156.1.k.d 8 77.b even 2 1 inner
2156.1.k.d 8 77.h odd 6 1 inner
2156.1.k.d 8 77.i even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 4T_{3}^{6} + 15T_{3}^{4} + 4T_{3}^{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2156, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} + 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} + 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} + 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 3)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} + 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( (T^{4} - 4 T^{2} + 1)^{2} \) Copy content Toggle raw display
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