Properties

Label 1408.2.a.d
Level $1408$
Weight $2$
Character orbit 1408.a
Self dual yes
Analytic conductor $11.243$
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] = [1408,2,Mod(1,1408)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1408, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1408.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1408 = 2^{7} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1408.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: \(11.2429366046\)
Analytic rank: \(0\)
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)
 
\(f(q)\) \(=\) \( q + 2 q^{5} + 4 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{5} + 4 q^{7} - 3 q^{9} - q^{11} + 6 q^{17} + 4 q^{19} + 2 q^{23} - q^{25} - 4 q^{29} + 2 q^{31} + 8 q^{35} - 10 q^{37} + 6 q^{41} + 12 q^{43} - 6 q^{45} + 6 q^{47} + 9 q^{49} - 2 q^{53} - 2 q^{55} - 4 q^{59} - 8 q^{61} - 12 q^{63} - 4 q^{67} + 2 q^{71} - 14 q^{73} - 4 q^{77} + 16 q^{79} + 9 q^{81} - 12 q^{83} + 12 q^{85} + 14 q^{89} + 8 q^{95} - 2 q^{97} + 3 q^{99}+O(q^{100}) \) 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 0 0 2.00000 0 4.00000 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 1408.2.a.d yes 1
4.b odd 2 1 1408.2.a.c yes 1
8.b even 2 1 1408.2.a.b yes 1
8.d odd 2 1 1408.2.a.a 1
16.e even 4 2 2816.2.c.c 2
16.f odd 4 2 2816.2.c.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1408.2.a.a 1 8.d odd 2 1
1408.2.a.b yes 1 8.b even 2 1
1408.2.a.c yes 1 4.b odd 2 1
1408.2.a.d yes 1 1.a even 1 1 trivial
2816.2.c.c 2 16.e even 4 2
2816.2.c.n 2 16.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1408))\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5} - 2 \) Copy content Toggle raw display
\( T_{7} - 4 \) 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
$5$ \( T - 2 \) Copy content Toggle raw display
$7$ \( T - 4 \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 6 \) Copy content Toggle raw display
$19$ \( T - 4 \) Copy content Toggle raw display
$23$ \( T - 2 \) Copy content Toggle raw display
$29$ \( T + 4 \) Copy content Toggle raw display
$31$ \( T - 2 \) Copy content Toggle raw display
$37$ \( T + 10 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T - 12 \) Copy content Toggle raw display
$47$ \( T - 6 \) Copy content Toggle raw display
$53$ \( T + 2 \) Copy content Toggle raw display
$59$ \( T + 4 \) Copy content Toggle raw display
$61$ \( T + 8 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T - 2 \) Copy content Toggle raw display
$73$ \( T + 14 \) 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
$97$ \( T + 2 \) Copy content Toggle raw display
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