Properties

Label 23.31.b.a
Level $23$
Weight $31$
Character orbit 23.b
Self dual yes
Analytic conductor $131.133$
Analytic rank $0$
Dimension $1$
CM discriminant -23
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [23,31,Mod(22,23)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(23, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 31, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("23.22");
 
S:= CuspForms(chi, 31);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 31 \)
Character orbit: \([\chi]\) \(=\) 23.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: \(131.132802376\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 50407 q^{2} + 19799482 q^{3} + 1467123825 q^{4} - 998032489174 q^{6} - 19829206524407 q^{8} + 186128355373675 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 50407 q^{2} + 19799482 q^{3} + 1467123825 q^{4} - 998032489174 q^{6} - 19829206524407 q^{8} + 186128355373675 q^{9} + 29\!\cdots\!50 q^{12}+ \cdots - 11\!\cdots\!43 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/23\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
0
−50407.0 1.97995e7 1.46712e9 0 −9.98032e11 0 −1.98292e13 1.86128e14 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
23.b odd 2 1 CM by \(\Q(\sqrt{-23}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 23.31.b.a 1
23.b odd 2 1 CM 23.31.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.31.b.a 1 1.a even 1 1 trivial
23.31.b.a 1 23.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 50407 \) acting on \(S_{31}^{\mathrm{new}}(23, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 50407 \) Copy content Toggle raw display
$3$ \( T - 19799482 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 96\!\cdots\!82 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T + 26\!\cdots\!07 \) Copy content Toggle raw display
$29$ \( T + 45\!\cdots\!74 \) Copy content Toggle raw display
$31$ \( T - 31\!\cdots\!74 \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T - 31\!\cdots\!74 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T + 17\!\cdots\!82 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T - 13\!\cdots\!98 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T + 32\!\cdots\!26 \) Copy content Toggle raw display
$73$ \( T + 50\!\cdots\!18 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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