↑

Properties

Label 1344.2.a.a
Level $1344$
Weight $2$
Character orbit 1344.a
Self dual yes
Analytic conductor $10.732$
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] = [1344,2,Mod(1,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, 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("1344.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1344.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: \(10.7318940317\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
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)
 
\(f(q)\) \(=\) \( q - q^{3} - 4 q^{5} + q^{7} + q^{9} + 2 q^{11} + 6 q^{13} + 4 q^{15} - 4 q^{17} - 4 q^{19} - q^{21} - 2 q^{23} + 11 q^{25} - q^{27} + 2 q^{29} - 2 q^{33} - 4 q^{35} - 2 q^{37} - 6 q^{39} - 4 q^{43}+ \cdots + 2 q^{99}+O(q^{100}) \) 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 −4.00000 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 1344.2.a.a 1
3.b odd 2 1 4032.2.a.bn 1
4.b odd 2 1 1344.2.a.k 1
7.b odd 2 1 9408.2.a.df 1
8.b even 2 1 336.2.a.f 1
8.d odd 2 1 84.2.a.a ✓ 1
12.b even 2 1 4032.2.a.bm 1
16.e even 4 2 5376.2.c.p 2
16.f odd 4 2 5376.2.c.q 2
24.f even 2 1 252.2.a.a 1
24.h odd 2 1 1008.2.a.a 1
28.d even 2 1 9408.2.a.bn 1
40.e odd 2 1 2100.2.a.r 1
40.f even 2 1 8400.2.a.e 1
40.k even 4 2 2100.2.k.i 2
56.e even 2 1 588.2.a.d 1
56.h odd 2 1 2352.2.a.a 1
56.j odd 6 2 2352.2.q.z 2
56.k odd 6 2 588.2.i.e 2
56.m even 6 2 588.2.i.d 2
56.p even 6 2 2352.2.q.b 2
72.l even 6 2 2268.2.j.n 2
72.p odd 6 2 2268.2.j.a 2
120.m even 2 1 6300.2.a.w 1
120.q odd 4 2 6300.2.k.g 2
168.e odd 2 1 1764.2.a.k 1
168.i even 2 1 7056.2.a.cd 1
168.v even 6 2 1764.2.k.k 2
168.be odd 6 2 1764.2.k.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.a.a ✓ 1 8.d odd 2 1
252.2.a.a 1 24.f even 2 1
336.2.a.f 1 8.b even 2 1
588.2.a.d 1 56.e even 2 1
588.2.i.d 2 56.m even 6 2
588.2.i.e 2 56.k odd 6 2
1008.2.a.a 1 24.h odd 2 1
1344.2.a.a 1 1.a even 1 1 trivial
1344.2.a.k 1 4.b odd 2 1
1764.2.a.k 1 168.e odd 2 1
1764.2.k.a 2 168.be odd 6 2
1764.2.k.k 2 168.v even 6 2
2100.2.a.r 1 40.e odd 2 1
2100.2.k.i 2 40.k even 4 2
2268.2.j.a 2 72.p odd 6 2
2268.2.j.n 2 72.l even 6 2
2352.2.a.a 1 56.h odd 2 1
2352.2.q.b 2 56.p even 6 2
2352.2.q.z 2 56.j odd 6 2
4032.2.a.bm 1 12.b even 2 1
4032.2.a.bn 1 3.b odd 2 1
5376.2.c.p 2 16.e even 4 2
5376.2.c.q 2 16.f odd 4 2
6300.2.a.w 1 120.m even 2 1
6300.2.k.g 2 120.q odd 4 2
7056.2.a.cd 1 168.i even 2 1
8400.2.a.e 1 40.f even 2 1
9408.2.a.bn 1 28.d even 2 1
9408.2.a.df 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1344))\):

\( T_{5} + 4 \) Copy content Toggle raw display
\( T_{11} - 2 \) Copy content Toggle raw display
\( T_{13} - 6 \) Copy content Toggle raw display
\( T_{19} + 4 \) 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
$5$ \( T + 4 \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T - 2 \) Copy content Toggle raw display
$13$ \( T - 6 \) Copy content Toggle raw display
$17$ \( T + 4 \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T + 2 \) Copy content Toggle raw display
$29$ \( T - 2 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T + 2 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T + 12 \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T + 8 \) Copy content Toggle raw display
$61$ \( T + 6 \) Copy content Toggle raw display
$67$ \( T + 8 \) Copy content Toggle raw display
$71$ \( T + 14 \) Copy content Toggle raw display
$73$ \( T + 2 \) Copy content Toggle raw display
$79$ \( T + 12 \) Copy content Toggle raw display
$83$ \( T + 4 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
show more
show less