Properties

Label 396.4.a.b
Level $396$
Weight $4$
Character orbit 396.a
Self dual yes
Analytic conductor $23.365$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [396,4,Mod(1,396)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(396, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("396.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 396 = 2^{2} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 396.a (trivial)

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: \(23.3647563623\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 12 q^{5} + 26 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 12 q^{5} + 26 q^{7} - 11 q^{11} - 34 q^{13} - 126 q^{17} + 110 q^{19} + 180 q^{23} + 19 q^{25} + 18 q^{29} - 292 q^{31} - 312 q^{35} - 238 q^{37} - 426 q^{41} + 146 q^{43} - 528 q^{47} + 333 q^{49} - 408 q^{53} + 132 q^{55} - 324 q^{59} - 550 q^{61} + 408 q^{65} + 824 q^{67} - 552 q^{71} - 850 q^{73} - 286 q^{77} + 866 q^{79} + 660 q^{83} + 1512 q^{85} - 768 q^{89} - 884 q^{91} - 1320 q^{95} - 286 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 0 −12.0000 0 26.0000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(11\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 396.4.a.b 1
3.b odd 2 1 396.4.a.g yes 1
4.b odd 2 1 1584.4.a.c 1
12.b even 2 1 1584.4.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
396.4.a.b 1 1.a even 1 1 trivial
396.4.a.g yes 1 3.b odd 2 1
1584.4.a.c 1 4.b odd 2 1
1584.4.a.r 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(396))\):

\( T_{5} + 12 \) Copy content Toggle raw display
\( T_{7} - 26 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 12 \) Copy content Toggle raw display
$7$ \( T - 26 \) Copy content Toggle raw display
$11$ \( T + 11 \) Copy content Toggle raw display
$13$ \( T + 34 \) Copy content Toggle raw display
$17$ \( T + 126 \) Copy content Toggle raw display
$19$ \( T - 110 \) Copy content Toggle raw display
$23$ \( T - 180 \) Copy content Toggle raw display
$29$ \( T - 18 \) Copy content Toggle raw display
$31$ \( T + 292 \) Copy content Toggle raw display
$37$ \( T + 238 \) Copy content Toggle raw display
$41$ \( T + 426 \) Copy content Toggle raw display
$43$ \( T - 146 \) Copy content Toggle raw display
$47$ \( T + 528 \) Copy content Toggle raw display
$53$ \( T + 408 \) Copy content Toggle raw display
$59$ \( T + 324 \) Copy content Toggle raw display
$61$ \( T + 550 \) Copy content Toggle raw display
$67$ \( T - 824 \) Copy content Toggle raw display
$71$ \( T + 552 \) Copy content Toggle raw display
$73$ \( T + 850 \) Copy content Toggle raw display
$79$ \( T - 866 \) Copy content Toggle raw display
$83$ \( T - 660 \) Copy content Toggle raw display
$89$ \( T + 768 \) Copy content Toggle raw display
$97$ \( T + 286 \) Copy content Toggle raw display
show more
show less