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Properties

Label 336.2.a.a
Level $336$
Weight $2$
Character orbit 336.a
Self dual yes
Analytic conductor $2.683$
Analytic rank $1$
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] = [336,2,Mod(1,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 336.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: \(2.68297350792\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
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)
 
\(f(q)\) \(=\) \( q - q^{3} - 2 q^{5} + q^{7} + q^{9} - 4 q^{11} - 2 q^{13} + 2 q^{15} - 6 q^{17} - 4 q^{19} - q^{21} - q^{25} - q^{27} - 2 q^{29} + 4 q^{33} - 2 q^{35} + 6 q^{37} + 2 q^{39} + 2 q^{41} + 4 q^{43}+ \cdots - 4 q^{99}+O(q^{100}) \) 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 −1.00000 0 −2.00000 0 1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(7\) \( -1 \)

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 336.2.a.a 1
3.b odd 2 1 1008.2.a.l 1
4.b odd 2 1 21.2.a.a ✓ 1
5.b even 2 1 8400.2.a.bn 1
7.b odd 2 1 2352.2.a.v 1
7.c even 3 2 2352.2.q.x 2
7.d odd 6 2 2352.2.q.e 2
8.b even 2 1 1344.2.a.s 1
8.d odd 2 1 1344.2.a.g 1
12.b even 2 1 63.2.a.a 1
16.e even 4 2 5376.2.c.l 2
16.f odd 4 2 5376.2.c.r 2
20.d odd 2 1 525.2.a.d 1
20.e even 4 2 525.2.d.a 2
21.c even 2 1 7056.2.a.p 1
24.f even 2 1 4032.2.a.h 1
24.h odd 2 1 4032.2.a.k 1
28.d even 2 1 147.2.a.a 1
28.f even 6 2 147.2.e.c 2
28.g odd 6 2 147.2.e.b 2
36.f odd 6 2 567.2.f.g 2
36.h even 6 2 567.2.f.b 2
44.c even 2 1 2541.2.a.j 1
52.b odd 2 1 3549.2.a.c 1
56.e even 2 1 9408.2.a.bv 1
56.h odd 2 1 9408.2.a.m 1
60.h even 2 1 1575.2.a.c 1
60.l odd 4 2 1575.2.d.a 2
68.d odd 2 1 6069.2.a.b 1
76.d even 2 1 7581.2.a.d 1
84.h odd 2 1 441.2.a.f 1
84.j odd 6 2 441.2.e.b 2
84.n even 6 2 441.2.e.a 2
132.d odd 2 1 7623.2.a.g 1
140.c even 2 1 3675.2.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.a.a ✓ 1 4.b odd 2 1
63.2.a.a 1 12.b even 2 1
147.2.a.a 1 28.d even 2 1
147.2.e.b 2 28.g odd 6 2
147.2.e.c 2 28.f even 6 2
336.2.a.a 1 1.a even 1 1 trivial
441.2.a.f 1 84.h odd 2 1
441.2.e.a 2 84.n even 6 2
441.2.e.b 2 84.j odd 6 2
525.2.a.d 1 20.d odd 2 1
525.2.d.a 2 20.e even 4 2
567.2.f.b 2 36.h even 6 2
567.2.f.g 2 36.f odd 6 2
1008.2.a.l 1 3.b odd 2 1
1344.2.a.g 1 8.d odd 2 1
1344.2.a.s 1 8.b even 2 1
1575.2.a.c 1 60.h even 2 1
1575.2.d.a 2 60.l odd 4 2
2352.2.a.v 1 7.b odd 2 1
2352.2.q.e 2 7.d odd 6 2
2352.2.q.x 2 7.c even 3 2
2541.2.a.j 1 44.c even 2 1
3549.2.a.c 1 52.b odd 2 1
3675.2.a.n 1 140.c even 2 1
4032.2.a.h 1 24.f even 2 1
4032.2.a.k 1 24.h odd 2 1
5376.2.c.l 2 16.e even 4 2
5376.2.c.r 2 16.f odd 4 2
6069.2.a.b 1 68.d odd 2 1
7056.2.a.p 1 21.c even 2 1
7581.2.a.d 1 76.d even 2 1
7623.2.a.g 1 132.d odd 2 1
8400.2.a.bn 1 5.b even 2 1
9408.2.a.m 1 56.h odd 2 1
9408.2.a.bv 1 56.e even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(336))\):

\( T_{5} + 2 \) Copy content Toggle raw display
\( T_{11} + 4 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T + 2 \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T + 4 \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T + 6 \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 2 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 6 \) Copy content Toggle raw display
$41$ \( T - 2 \) Copy content Toggle raw display
$43$ \( T - 4 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T + 12 \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 6 \) Copy content Toggle raw display
$79$ \( T - 16 \) Copy content Toggle raw display
$83$ \( T - 12 \) Copy content Toggle raw display
$89$ \( T + 14 \) Copy content Toggle raw display
$97$ \( T - 18 \) Copy content Toggle raw display
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