Properties

Label 107.1.b.a
Level $107$
Weight $1$
Character orbit 107.b
Self dual yes
Analytic conductor $0.053$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -107
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [107,1,Mod(106,107)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(107, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("107.106");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 107 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 107.b (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(0.0533999563517\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.107.1
Artin image: $S_3$
Artin field: Galois closure of 3.1.107.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{3} + q^{4}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + q^{4} - q^{11} - q^{12} - q^{13} + q^{16} - q^{19} - q^{23} + q^{25} + q^{27} + 2 q^{29} + q^{33} - q^{37} + q^{39} - q^{41} - q^{44} + 2 q^{47} - q^{48} + q^{49} - q^{52} - q^{53} + q^{57} - q^{61} + q^{64} + q^{69} - q^{75} - q^{76} - q^{79} - q^{81} + 2 q^{83} - 2 q^{87} - q^{89} - q^{92}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/107\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
106.1
0
0 −1.00000 1.00000 0 0 0 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
107.b odd 2 1 CM by \(\Q(\sqrt{-107}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 107.1.b.a 1
3.b odd 2 1 963.1.b.a 1
4.b odd 2 1 1712.1.g.a 1
5.b even 2 1 2675.1.c.a 1
5.c odd 4 2 2675.1.d.a 2
107.b odd 2 1 CM 107.1.b.a 1
321.d even 2 1 963.1.b.a 1
428.b even 2 1 1712.1.g.a 1
535.d odd 2 1 2675.1.c.a 1
535.e even 4 2 2675.1.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
107.1.b.a 1 1.a even 1 1 trivial
107.1.b.a 1 107.b odd 2 1 CM
963.1.b.a 1 3.b odd 2 1
963.1.b.a 1 321.d even 2 1
1712.1.g.a 1 4.b odd 2 1
1712.1.g.a 1 428.b even 2 1
2675.1.c.a 1 5.b even 2 1
2675.1.c.a 1 535.d odd 2 1
2675.1.d.a 2 5.c odd 4 2
2675.1.d.a 2 535.e even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(107, [\chi])\).

Hecke characteristic polynomials

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