Properties

Label 2667.1.bi.a
Level $2667$
Weight $1$
Character orbit 2667.bi
Analytic conductor $1.331$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
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(527,2667)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2667, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2667.527");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2667 = 3 \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2667.bi (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: \(1.33100638869\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.7112889.1
Artin image: $\SL(2,3):C_2$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{12}^{5} q^{2} + q^{3} - \zeta_{12}^{5} q^{5} + \zeta_{12}^{5} q^{6} + q^{7} - \zeta_{12}^{3} q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12}^{5} q^{2} + q^{3} - \zeta_{12}^{5} q^{5} + \zeta_{12}^{5} q^{6} + q^{7} - \zeta_{12}^{3} q^{8} + q^{9} + \zeta_{12}^{4} q^{10} - \zeta_{12}^{5} q^{11} - \zeta_{12}^{2} q^{13} + \zeta_{12}^{5} q^{14} - \zeta_{12}^{5} q^{15} + \zeta_{12}^{2} q^{16} + \zeta_{12}^{5} q^{17} + \zeta_{12}^{5} q^{18} + \zeta_{12}^{2} q^{19} + q^{21} + \zeta_{12}^{4} q^{22} - \zeta_{12} q^{23} - \zeta_{12}^{3} q^{24} + \zeta_{12} q^{26} + q^{27} + \zeta_{12} q^{29} + \zeta_{12}^{4} q^{30} - q^{31} + \zeta_{12} q^{32} - \zeta_{12}^{5} q^{33} - \zeta_{12}^{4} q^{34} - \zeta_{12}^{5} q^{35} - \zeta_{12} q^{38} - \zeta_{12}^{2} q^{39} - \zeta_{12}^{2} q^{40} - \zeta_{12} q^{41} + \zeta_{12}^{5} q^{42} - \zeta_{12}^{4} q^{43} - \zeta_{12}^{5} q^{45} + q^{46} - \zeta_{12}^{5} q^{47} + \zeta_{12}^{2} q^{48} + q^{49} + \zeta_{12}^{5} q^{51} + \zeta_{12}^{5} q^{53} + \zeta_{12}^{5} q^{54} - \zeta_{12}^{4} q^{55} - \zeta_{12}^{3} q^{56} + \zeta_{12}^{2} q^{57} - q^{58} + \zeta_{12}^{5} q^{59} - \zeta_{12}^{2} q^{61} - 2 \zeta_{12}^{5} q^{62} + q^{63} - q^{64} - \zeta_{12} q^{65} + \zeta_{12}^{4} q^{66} - \zeta_{12} q^{69} + \zeta_{12}^{4} q^{70} + \zeta_{12}^{5} q^{71} - \zeta_{12}^{3} q^{72} - \zeta_{12}^{4} q^{73} - \zeta_{12}^{5} q^{77} + \zeta_{12} q^{78} + \zeta_{12}^{4} q^{79} + \zeta_{12} q^{80} + q^{81} + q^{82} + \zeta_{12} q^{83} + \zeta_{12}^{4} q^{85} + \zeta_{12}^{3} q^{86} + \zeta_{12} q^{87} - \zeta_{12}^{2} q^{88} + \zeta_{12}^{5} q^{89} + \zeta_{12}^{4} q^{90} - \zeta_{12}^{2} q^{91} - 2 q^{93} + \zeta_{12}^{4} q^{94} + \zeta_{12} q^{95} - \zeta_{12}^{2} q^{97} + \zeta_{12}^{5} q^{98} - \zeta_{12}^{5} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 4 q^{7} + 4 q^{9} - 2 q^{10} - 2 q^{13} + 2 q^{16} + 2 q^{19} + 4 q^{21} - 2 q^{22} + 4 q^{27} - 2 q^{30} - 8 q^{31} + 2 q^{34} - 2 q^{39} - 2 q^{40} + 2 q^{43} + 4 q^{46} + 2 q^{48} + 4 q^{49} + 2 q^{55} + 2 q^{57} - 4 q^{58} - 2 q^{61} + 4 q^{63} - 4 q^{64} - 2 q^{66} - 2 q^{70} + 2 q^{73} - 2 q^{79} + 4 q^{81} + 4 q^{82} - 2 q^{85} - 2 q^{88} - 2 q^{90} - 2 q^{91} - 8 q^{93} - 2 q^{94} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(890\) \(1144\) \(2416\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{2}\) \(\zeta_{12}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
527.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 0.500000i 1.00000 0 0.866025 + 0.500000i −0.866025 0.500000i 1.00000 1.00000i 1.00000 −0.500000 0.866025i
527.2 0.866025 + 0.500000i 1.00000 0 −0.866025 0.500000i 0.866025 + 0.500000i 1.00000 1.00000i 1.00000 −0.500000 0.866025i
1250.1 −0.866025 + 0.500000i 1.00000 0 0.866025 0.500000i −0.866025 + 0.500000i 1.00000 1.00000i 1.00000 −0.500000 + 0.866025i
1250.2 0.866025 0.500000i 1.00000 0 −0.866025 + 0.500000i 0.866025 0.500000i 1.00000 1.00000i 1.00000 −0.500000 + 0.866025i
\(n\): 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 inner
889.g even 3 1 inner
2667.bi odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2667.1.bi.a yes 4
3.b odd 2 1 inner 2667.1.bi.a yes 4
7.c even 3 1 2667.1.bf.a 4
21.h odd 6 1 2667.1.bf.a 4
127.c even 3 1 2667.1.bf.a 4
381.f odd 6 1 2667.1.bf.a 4
889.g even 3 1 inner 2667.1.bi.a yes 4
2667.bi odd 6 1 inner 2667.1.bi.a yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2667.1.bf.a 4 7.c even 3 1
2667.1.bf.a 4 21.h odd 6 1
2667.1.bf.a 4 127.c even 3 1
2667.1.bf.a 4 381.f odd 6 1
2667.1.bi.a yes 4 1.a even 1 1 trivial
2667.1.bi.a yes 4 3.b odd 2 1 inner
2667.1.bi.a yes 4 889.g even 3 1 inner
2667.1.bi.a yes 4 2667.bi odd 6 1 inner

Hecke kernels

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

Hecke characteristic polynomials

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