Properties

Label 2667.1.cj.a
Level $2667$
Weight $1$
Character orbit 2667.cj
Analytic conductor $1.331$
Analytic rank $0$
Dimension $6$
Projective image $D_{9}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2667,1,Mod(611,2667)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2667, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 6, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2667.611");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2667 = 3 \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2667.cj (of order \(18\), degree \(6\), 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: \(1.33100638869\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{9}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{9} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{18}^{7} q^{3} + \zeta_{18}^{6} q^{4} - \zeta_{18} q^{7} - \zeta_{18}^{5} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{18}^{7} q^{3} + \zeta_{18}^{6} q^{4} - \zeta_{18} q^{7} - \zeta_{18}^{5} q^{9} + \zeta_{18}^{4} q^{12} + (\zeta_{18}^{8} + 1) q^{13} - \zeta_{18}^{3} q^{16} + ( - \zeta_{18}^{7} - \zeta_{18}^{5}) q^{19} + \zeta_{18}^{8} q^{21} + q^{25} - \zeta_{18}^{3} q^{27} - \zeta_{18}^{7} q^{28} + ( - \zeta_{18}^{5} + 1) q^{31} + \zeta_{18}^{2} q^{36} + ( - \zeta_{18}^{7} + \zeta_{18}^{6}) q^{37} + ( - \zeta_{18}^{7} + \zeta_{18}^{6}) q^{39} - \zeta_{18}^{8} q^{43} - \zeta_{18} q^{48} + \zeta_{18}^{2} q^{49} + (\zeta_{18}^{6} - \zeta_{18}^{5}) q^{52} + ( - \zeta_{18}^{5} - \zeta_{18}^{3}) q^{57} + (\zeta_{18}^{8} - \zeta_{18}) q^{61} + \zeta_{18}^{6} q^{63} + q^{64} + (\zeta_{18}^{6} + \zeta_{18}^{2}) q^{67} + ( - \zeta_{18}^{5} + \zeta_{18}^{4}) q^{73} - \zeta_{18}^{7} q^{75} + (\zeta_{18}^{4} + \zeta_{18}^{2}) q^{76} + ( - \zeta_{18}^{5} - \zeta_{18}^{3}) q^{79} - \zeta_{18} q^{81} - \zeta_{18}^{5} q^{84} + ( - \zeta_{18} + 1) q^{91} + ( - \zeta_{18}^{7} - \zeta_{18}^{3}) q^{93} + \zeta_{18}^{7} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{4} + 6 q^{13} - 3 q^{16} + 6 q^{25} - 3 q^{27} + 6 q^{31} - 3 q^{37} - 3 q^{39} - 3 q^{52} - 3 q^{57} - 3 q^{63} + 6 q^{64} - 3 q^{67} - 3 q^{79} + 6 q^{91} - 3 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2667\mathbb{Z}\right)^\times\).

\(n\) \(890\) \(1144\) \(2416\)
\(\chi(n)\) \(-1\) \(-\zeta_{18}^{3}\) \(\zeta_{18}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
611.1
−0.766044 0.642788i
−0.173648 0.984808i
−0.173648 + 0.984808i
0.939693 + 0.342020i
0.939693 0.342020i
−0.766044 + 0.642788i
0 0.173648 0.984808i −0.500000 0.866025i 0 0 0.766044 + 0.642788i 0 −0.939693 0.342020i 0
830.1 0 −0.939693 0.342020i −0.500000 + 0.866025i 0 0 0.173648 + 0.984808i 0 0.766044 + 0.642788i 0
1115.1 0 −0.939693 + 0.342020i −0.500000 0.866025i 0 0 0.173648 0.984808i 0 0.766044 0.642788i 0
1292.1 0 0.766044 0.642788i −0.500000 + 0.866025i 0 0 −0.939693 0.342020i 0 0.173648 0.984808i 0
1703.1 0 0.766044 + 0.642788i −0.500000 0.866025i 0 0 −0.939693 + 0.342020i 0 0.173648 + 0.984808i 0
2069.1 0 0.173648 + 0.984808i −0.500000 + 0.866025i 0 0 0.766044 0.642788i 0 −0.939693 + 0.342020i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 611.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
889.w even 9 1 inner
2667.cj odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2667.1.cj.a 6
3.b odd 2 1 CM 2667.1.cj.a 6
7.c even 3 1 2667.1.co.a yes 6
21.h odd 6 1 2667.1.co.a yes 6
127.f even 9 1 2667.1.co.a yes 6
381.o odd 18 1 2667.1.co.a yes 6
889.w even 9 1 inner 2667.1.cj.a 6
2667.cj odd 18 1 inner 2667.1.cj.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2667.1.cj.a 6 1.a even 1 1 trivial
2667.1.cj.a 6 3.b odd 2 1 CM
2667.1.cj.a 6 889.w even 9 1 inner
2667.1.cj.a 6 2667.cj odd 18 1 inner
2667.1.co.a yes 6 7.c even 3 1
2667.1.co.a yes 6 21.h odd 6 1
2667.1.co.a yes 6 127.f even 9 1
2667.1.co.a yes 6 381.o odd 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2667, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - 6 T^{5} + 15 T^{4} - 19 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( T^{6} - 6 T^{5} + 15 T^{4} - 19 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{6} \) Copy content Toggle raw display
$43$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$47$ \( T^{6} \) Copy content Toggle raw display
$53$ \( T^{6} \) Copy content Toggle raw display
$59$ \( T^{6} \) Copy content Toggle raw display
$61$ \( (T^{3} - 3 T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( (T^{3} - 3 T + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{6} \) Copy content Toggle raw display
$89$ \( T^{6} \) Copy content Toggle raw display
$97$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
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