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Properties

Label 867.1.g.a
Level $867$
Weight $1$
Character orbit 867.g
Analytic conductor $0.433$
Analytic rank $0$
Dimension $8$
Projective image $D_{3}$
CM discriminant -3
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [867,1,Mod(110,867)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(867, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("867.110");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 867 = 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 867.g (of order \(8\), degree \(4\), 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: \(0.432689365953\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.867.1
Artin image: $S_3\times C_{16}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{48} - \cdots)\)

$q$-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.

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

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/867\mathbb{Z}\right)^\times\).

\(n\) \(290\) \(292\)
\(\chi(n)\) \(-1\) \(-\zeta_{16}^{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
110.1
−0.382683 − 0.923880i
0.382683 + 0.923880i
−0.382683 + 0.923880i
0.382683 − 0.923880i
−0.923880 − 0.382683i
0.923880 + 0.382683i
−0.923880 + 0.382683i
0.923880 − 0.382683i
0 −0.382683 + 0.923880i − 1.00000i 0 0 0.923880 − 0.382683i 0 −0.707107 − 0.707107i 0
110.2 0 0.382683 − 0.923880i − 1.00000i 0 0 −0.923880 + 0.382683i 0 −0.707107 − 0.707107i 0
134.1 0 −0.382683 − 0.923880i 1.00000i 0 0 0.923880 + 0.382683i 0 −0.707107 + 0.707107i 0
134.2 0 0.382683 + 0.923880i 1.00000i 0 0 −0.923880 − 0.382683i 0 −0.707107 + 0.707107i 0
155.1 0 −0.923880 + 0.382683i 1.00000i 0 0 −0.382683 + 0.923880i 0 0.707107 − 0.707107i 0
155.2 0 0.923880 − 0.382683i 1.00000i 0 0 0.382683 − 0.923880i 0 0.707107 − 0.707107i 0
179.1 0 −0.923880 − 0.382683i − 1.00000i 0 0 −0.382683 − 0.923880i 0 0.707107 + 0.707107i 0
179.2 0 0.923880 + 0.382683i − 1.00000i 0 0 0.382683 + 0.923880i 0 0.707107 + 0.707107i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 110.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
17.b even 2 1 inner
17.c even 4 2 inner
17.d even 8 4 inner
51.c odd 2 1 inner
51.f odd 4 2 inner
51.g odd 8 4 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 867.1.g.a 8
3.b odd 2 1 CM 867.1.g.a 8
17.b even 2 1 inner 867.1.g.a 8
17.c even 4 2 inner 867.1.g.a 8
17.d even 8 4 inner 867.1.g.a 8
17.e odd 16 1 867.1.b.a ✓ 1
17.e odd 16 1 867.1.b.b yes 1
17.e odd 16 2 867.1.c.a 2
17.e odd 16 4 867.1.f.a 4
51.c odd 2 1 inner 867.1.g.a 8
51.f odd 4 2 inner 867.1.g.a 8
51.g odd 8 4 inner 867.1.g.a 8
51.i even 16 1 867.1.b.a ✓ 1
51.i even 16 1 867.1.b.b yes 1
51.i even 16 2 867.1.c.a 2
51.i even 16 4 867.1.f.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
867.1.b.a ✓ 1 17.e odd 16 1
867.1.b.a ✓ 1 51.i even 16 1
867.1.b.b yes 1 17.e odd 16 1
867.1.b.b yes 1 51.i even 16 1
867.1.c.a 2 17.e odd 16 2
867.1.c.a 2 51.i even 16 2
867.1.f.a 4 17.e odd 16 4
867.1.f.a 4 51.i even 16 4
867.1.g.a 8 1.a even 1 1 trivial
867.1.g.a 8 3.b odd 2 1 CM
867.1.g.a 8 17.b even 2 1 inner
867.1.g.a 8 17.c even 4 2 inner
867.1.g.a 8 17.d even 8 4 inner
867.1.g.a 8 51.c odd 2 1 inner
867.1.g.a 8 51.f odd 4 2 inner
867.1.g.a 8 51.g odd 8 4 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(867, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 1 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} + 1 \) Copy content Toggle raw display
$37$ \( T^{8} + 1 \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} + 1 \) Copy content Toggle raw display
$67$ \( (T - 1)^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} + 256 \) Copy content Toggle raw display
$79$ \( T^{8} + 256 \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} + 1 \) Copy content Toggle raw display
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