Properties

Label 396.4.a.a
Level $396$
Weight $4$
Character orbit 396.a
Self dual yes
Analytic conductor $23.365$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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: \(0\)
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)
 
\(f(q)\) \(=\) \( q - 22 q^{5} - 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 22 q^{5} - 20 q^{7} - 11 q^{11} + 22 q^{13} - 110 q^{17} + 48 q^{19} - 72 q^{23} + 359 q^{25} + 142 q^{29} + 184 q^{31} + 440 q^{35} - 194 q^{37} + 482 q^{41} - 80 q^{43} - 392 q^{47} + 57 q^{49} + 34 q^{53} + 242 q^{55} + 108 q^{59} + 382 q^{61} - 484 q^{65} + 84 q^{67} + 1040 q^{71} - 606 q^{73} + 220 q^{77} - 1292 q^{79} - 356 q^{83} + 2420 q^{85} + 406 q^{89} - 440 q^{91} - 1056 q^{95} + 1090 q^{97}+O(q^{100}) \) 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 −22.0000 0 −20.0000 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.a 1
3.b odd 2 1 132.4.a.c 1
4.b odd 2 1 1584.4.a.a 1
12.b even 2 1 528.4.a.l 1
24.f even 2 1 2112.4.a.a 1
24.h odd 2 1 2112.4.a.n 1
33.d even 2 1 1452.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
132.4.a.c 1 3.b odd 2 1
396.4.a.a 1 1.a even 1 1 trivial
528.4.a.l 1 12.b even 2 1
1452.4.a.c 1 33.d even 2 1
1584.4.a.a 1 4.b odd 2 1
2112.4.a.a 1 24.f even 2 1
2112.4.a.n 1 24.h odd 2 1

Hecke kernels

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

\( T_{5} + 22 \) Copy content Toggle raw display
\( T_{7} + 20 \) 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
$5$ \( T + 22 \) Copy content Toggle raw display
$7$ \( T + 20 \) Copy content Toggle raw display
$11$ \( T + 11 \) Copy content Toggle raw display
$13$ \( T - 22 \) Copy content Toggle raw display
$17$ \( T + 110 \) Copy content Toggle raw display
$19$ \( T - 48 \) Copy content Toggle raw display
$23$ \( T + 72 \) Copy content Toggle raw display
$29$ \( T - 142 \) Copy content Toggle raw display
$31$ \( T - 184 \) Copy content Toggle raw display
$37$ \( T + 194 \) Copy content Toggle raw display
$41$ \( T - 482 \) Copy content Toggle raw display
$43$ \( T + 80 \) Copy content Toggle raw display
$47$ \( T + 392 \) Copy content Toggle raw display
$53$ \( T - 34 \) Copy content Toggle raw display
$59$ \( T - 108 \) Copy content Toggle raw display
$61$ \( T - 382 \) Copy content Toggle raw display
$67$ \( T - 84 \) Copy content Toggle raw display
$71$ \( T - 1040 \) Copy content Toggle raw display
$73$ \( T + 606 \) Copy content Toggle raw display
$79$ \( T + 1292 \) Copy content Toggle raw display
$83$ \( T + 356 \) Copy content Toggle raw display
$89$ \( T - 406 \) Copy content Toggle raw display
$97$ \( T - 1090 \) Copy content Toggle raw display
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