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Properties

Label 2163.1.bu.b
Level $2163$
Weight $1$
Character orbit 2163.bu
Analytic conductor $1.079$
Analytic rank $0$
Dimension $16$
Projective image $D_{34}$
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] = [2163,1,Mod(125,2163)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2163, base_ring=CyclotomicField(34))
 
chi = DirichletCharacter(H, H._module([17, 17, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2163.125");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2163 = 3 \cdot 7 \cdot 103 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2163.bu (of order \(34\), degree \(16\), 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: \(1.07947762233\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{34})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{34}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{34} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{34}^{8} q^{3} - \zeta_{34}^{15} q^{4} - \zeta_{34}^{12} q^{7} + \zeta_{34}^{16} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{34}^{8} q^{3} - \zeta_{34}^{15} q^{4} - \zeta_{34}^{12} q^{7} + \zeta_{34}^{16} q^{9} - \zeta_{34}^{6} q^{12} + ( - \zeta_{34}^{5} + \zeta_{34}) q^{13} - \zeta_{34}^{13} q^{16} + (\zeta_{34}^{14} - \zeta_{34}^{4}) q^{19} - \zeta_{34}^{3} q^{21} + \zeta_{34}^{10} q^{25} + \zeta_{34}^{7} q^{27} - \zeta_{34}^{10} q^{28} + ( - \zeta_{34}^{7} + \zeta_{34}^{2}) q^{31} + \zeta_{34}^{14} q^{36} + (\zeta_{34}^{9} - \zeta_{34}^{3}) q^{37} + (\zeta_{34}^{13} - \zeta_{34}^{9}) q^{39} + (\zeta_{34}^{4} + \zeta_{34}) q^{43} - \zeta_{34}^{4} q^{48} - \zeta_{34}^{7} q^{49} + ( - \zeta_{34}^{16} - \zeta_{34}^{3}) q^{52} + (\zeta_{34}^{12} + \zeta_{34}^{5}) q^{57} + ( - \zeta_{34}^{14} + \zeta_{34}^{6}) q^{61} + \zeta_{34}^{11} q^{63} - \zeta_{34}^{11} q^{64} + ( - \zeta_{34}^{15} + \zeta_{34}^{13}) q^{67} + (\zeta_{34}^{16} + \zeta_{34}^{8}) q^{73} + \zeta_{34} q^{75} + (\zeta_{34}^{12} - \zeta_{34}^{2}) q^{76} + ( - \zeta_{34}^{8} - \zeta_{34}^{2}) q^{79} - \zeta_{34}^{15} q^{81} - \zeta_{34} q^{84} + ( - \zeta_{34}^{13} - 1) q^{91} + (\zeta_{34}^{15} - \zeta_{34}^{10}) q^{93} + (\zeta_{34}^{14} - 1) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + q^{3} - q^{4} + q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + q^{3} - q^{4} + q^{7} - q^{9} + q^{12} - q^{16} - q^{21} - q^{25} + q^{27} + q^{28} - 2 q^{31} - q^{36} + q^{48} - q^{49} + q^{63} - q^{64} - 2 q^{73} + q^{75} + 2 q^{79} - q^{81} - q^{84} - 17 q^{91} + 2 q^{93} - 17 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2163\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(619\) \(722\)
\(\chi(n)\) \(-\zeta_{34}^{10}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
125.1
0.602635 − 0.798017i
0.850217 + 0.526432i
−0.739009 − 0.673696i
0.602635 + 0.798017i
0.982973 + 0.183750i
−0.445738 + 0.895163i
−0.0922684 − 0.995734i
−0.0922684 + 0.995734i
0.850217 − 0.526432i
0.982973 − 0.183750i
0.273663 − 0.961826i
0.273663 + 0.961826i
−0.739009 + 0.673696i
−0.445738 − 0.895163i
−0.932472 + 0.361242i
−0.932472 − 0.361242i
0 −0.445738 + 0.895163i −0.273663 + 0.961826i 0 0 −0.0922684 − 0.995734i 0 −0.602635 − 0.798017i 0
209.1 0 0.273663 + 0.961826i 0.445738 − 0.895163i 0 0 −0.932472 − 0.361242i 0 −0.850217 + 0.526432i 0
230.1 0 −0.932472 + 0.361242i 0.0922684 − 0.995734i 0 0 0.850217 − 0.526432i 0 0.739009 − 0.673696i 0
398.1 0 −0.445738 − 0.895163i −0.273663 − 0.961826i 0 0 −0.0922684 + 0.995734i 0 −0.602635 + 0.798017i 0
713.1 0 −0.0922684 − 0.995734i 0.932472 − 0.361242i 0 0 0.602635 − 0.798017i 0 −0.982973 + 0.183750i 0
1007.1 0 0.850217 + 0.526432i −0.602635 + 0.798017i 0 0 −0.739009 + 0.673696i 0 0.445738 + 0.895163i 0
1175.1 0 −0.739009 + 0.673696i −0.982973 − 0.183750i 0 0 −0.445738 + 0.895163i 0 0.0922684 − 0.995734i 0
1469.1 0 −0.739009 − 0.673696i −0.982973 + 0.183750i 0 0 −0.445738 − 0.895163i 0 0.0922684 + 0.995734i 0
1511.1 0 0.273663 − 0.961826i 0.445738 + 0.895163i 0 0 −0.932472 + 0.361242i 0 −0.850217 − 0.526432i 0
1532.1 0 −0.0922684 + 0.995734i 0.932472 + 0.361242i 0 0 0.602635 + 0.798017i 0 −0.982973 − 0.183750i 0
1658.1 0 0.602635 − 0.798017i −0.850217 + 0.526432i 0 0 0.982973 + 0.183750i 0 −0.273663 − 0.961826i 0
1679.1 0 0.602635 + 0.798017i −0.850217 − 0.526432i 0 0 0.982973 − 0.183750i 0 −0.273663 + 0.961826i 0
1721.1 0 −0.932472 − 0.361242i 0.0922684 + 0.995734i 0 0 0.850217 + 0.526432i 0 0.739009 + 0.673696i 0
1742.1 0 0.850217 − 0.526432i −0.602635 − 0.798017i 0 0 −0.739009 − 0.673696i 0 0.445738 − 0.895163i 0
1994.1 0 0.982973 + 0.183750i 0.739009 + 0.673696i 0 0 0.273663 − 0.961826i 0 0.932472 + 0.361242i 0
2099.1 0 0.982973 − 0.183750i 0.739009 − 0.673696i 0 0 0.273663 + 0.961826i 0 0.932472 − 0.361242i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 125.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
721.v even 34 1 inner
2163.bu odd 34 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2163.1.bu.b yes 16
3.b odd 2 1 CM 2163.1.bu.b yes 16
7.b odd 2 1 2163.1.bu.a ✓ 16
21.c even 2 1 2163.1.bu.a ✓ 16
103.f odd 34 1 2163.1.bu.a ✓ 16
309.k even 34 1 2163.1.bu.a ✓ 16
721.v even 34 1 inner 2163.1.bu.b yes 16
2163.bu odd 34 1 inner 2163.1.bu.b yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2163.1.bu.a ✓ 16 7.b odd 2 1
2163.1.bu.a ✓ 16 21.c even 2 1
2163.1.bu.a ✓ 16 103.f odd 34 1
2163.1.bu.a ✓ 16 309.k even 34 1
2163.1.bu.b yes 16 1.a even 1 1 trivial
2163.1.bu.b yes 16 3.b odd 2 1 CM
2163.1.bu.b yes 16 721.v even 34 1 inner
2163.1.bu.b yes 16 2163.bu odd 34 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{16} - 34T_{13}^{11} + 68T_{13}^{7} + 221T_{13}^{6} + 17T_{13}^{5} - 17T_{13}^{3} + 119T_{13}^{2} - 85T_{13} + 17 \) acting on \(S_{1}^{\mathrm{new}}(2163, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - T^{15} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - T^{15} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} - 34 T^{11} + \cdots + 17 \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( T^{16} + 17 T^{10} + \cdots + 17 \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( T^{16} + 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{16} + 51 T^{9} + \cdots + 17 \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( T^{16} + 17 T^{11} + \cdots + 17 \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( T^{16} - 17 T^{13} + \cdots + 17 \) Copy content Toggle raw display
$67$ \( T^{16} - 17 T^{7} + \cdots + 17 \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} + 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{16} - 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( T^{16} \) Copy content Toggle raw display
$97$ \( T^{16} + 17 T^{15} + \cdots + 17 \) Copy content Toggle raw display
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