Properties

Label 2736.1.bk.a
Level $2736$
Weight $1$
Character orbit 2736.bk
Analytic conductor $1.365$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2736,1,Mod(847,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 0, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.847");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2736.bk (of order \(6\), degree \(2\), 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: \(1.36544187456\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.225194688.1

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6}^{2} - \zeta_{6}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6}^{2} - \zeta_{6}) q^{7} - \zeta_{6} q^{13} - q^{19} + \zeta_{6} q^{25} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{31} - q^{37} + ( - \zeta_{6} - 1) q^{43} + (\zeta_{6}^{2} - \zeta_{6} - 1) q^{49} + \zeta_{6} q^{61} + ( - \zeta_{6}^{2} + 1) q^{67} + \zeta_{6}^{2} q^{73} + (\zeta_{6} + 1) q^{79} + (\zeta_{6}^{2} - 1) q^{91} - 2 \zeta_{6}^{2} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{13} - 2 q^{19} + q^{25} - 2 q^{37} - 3 q^{43} - 4 q^{49} + q^{61} + 3 q^{67} - q^{73} + 3 q^{79} - 3 q^{91} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
847.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 0 0 1.73205i 0 0 0
2287.1 0 0 0 0 0 1.73205i 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 \(\Q(\sqrt{-3}) \)
76.g odd 6 1 inner
228.m even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.1.bk.a 2
3.b odd 2 1 CM 2736.1.bk.a 2
4.b odd 2 1 2736.1.bk.b yes 2
12.b even 2 1 2736.1.bk.b yes 2
19.c even 3 1 2736.1.bk.b yes 2
57.h odd 6 1 2736.1.bk.b yes 2
76.g odd 6 1 inner 2736.1.bk.a 2
228.m even 6 1 inner 2736.1.bk.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2736.1.bk.a 2 1.a even 1 1 trivial
2736.1.bk.a 2 3.b odd 2 1 CM
2736.1.bk.a 2 76.g odd 6 1 inner
2736.1.bk.a 2 228.m even 6 1 inner
2736.1.bk.b yes 2 4.b odd 2 1
2736.1.bk.b yes 2 12.b even 2 1
2736.1.bk.b yes 2 19.c even 3 1
2736.1.bk.b yes 2 57.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{43}^{2} + 3T_{43} + 3 \) acting on \(S_{1}^{\mathrm{new}}(2736, [\chi])\). 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
$7$ \( T^{2} + 3 \) 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
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 3 \) Copy content Toggle raw display
$37$ \( (T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 3T + 3 \) 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
$61$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$79$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
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