Properties

Label 396.4.a.c
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: no (minimal twist has level 132)
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)
 
\(f(q)\) \(=\) \( q - 10 q^{5} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 10 q^{5} + 8 q^{7} + 11 q^{11} + 18 q^{13} - 46 q^{17} + 40 q^{19} - 44 q^{23} - 25 q^{25} - 186 q^{29} - 72 q^{31} - 80 q^{35} - 114 q^{37} - 174 q^{41} - 416 q^{43} + 156 q^{47} - 279 q^{49} + 62 q^{53} - 110 q^{55} + 348 q^{59} - 446 q^{61} - 180 q^{65} - 956 q^{67} + 444 q^{71} + 306 q^{73} + 88 q^{77} - 664 q^{79} + 124 q^{83} + 460 q^{85} - 602 q^{89} + 144 q^{91} - 400 q^{95} + 1522 q^{97}+O(q^{100}) \) 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 −10.0000 0 8.00000 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.c 1
3.b odd 2 1 132.4.a.d 1
4.b odd 2 1 1584.4.a.f 1
12.b even 2 1 528.4.a.e 1
24.f even 2 1 2112.4.a.q 1
24.h odd 2 1 2112.4.a.e 1
33.d even 2 1 1452.4.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
132.4.a.d 1 3.b odd 2 1
396.4.a.c 1 1.a even 1 1 trivial
528.4.a.e 1 12.b even 2 1
1452.4.a.i 1 33.d even 2 1
1584.4.a.f 1 4.b odd 2 1
2112.4.a.e 1 24.h odd 2 1
2112.4.a.q 1 24.f even 2 1

Hecke kernels

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

\( T_{5} + 10 \) Copy content Toggle raw display
\( T_{7} - 8 \) 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
$5$ \( T + 10 \) Copy content Toggle raw display
$7$ \( T - 8 \) Copy content Toggle raw display
$11$ \( T - 11 \) Copy content Toggle raw display
$13$ \( T - 18 \) Copy content Toggle raw display
$17$ \( T + 46 \) Copy content Toggle raw display
$19$ \( T - 40 \) Copy content Toggle raw display
$23$ \( T + 44 \) Copy content Toggle raw display
$29$ \( T + 186 \) Copy content Toggle raw display
$31$ \( T + 72 \) Copy content Toggle raw display
$37$ \( T + 114 \) Copy content Toggle raw display
$41$ \( T + 174 \) Copy content Toggle raw display
$43$ \( T + 416 \) Copy content Toggle raw display
$47$ \( T - 156 \) Copy content Toggle raw display
$53$ \( T - 62 \) Copy content Toggle raw display
$59$ \( T - 348 \) Copy content Toggle raw display
$61$ \( T + 446 \) Copy content Toggle raw display
$67$ \( T + 956 \) Copy content Toggle raw display
$71$ \( T - 444 \) Copy content Toggle raw display
$73$ \( T - 306 \) Copy content Toggle raw display
$79$ \( T + 664 \) Copy content Toggle raw display
$83$ \( T - 124 \) Copy content Toggle raw display
$89$ \( T + 602 \) Copy content Toggle raw display
$97$ \( T - 1522 \) Copy content Toggle raw display
show more
show less