Properties

Label 1008.6.a.q
Level $1008$
Weight $6$
Character orbit 1008.a
Self dual yes
Analytic conductor $161.667$
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] = [1008,6,Mod(1,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1008.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: \(161.666890371\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
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)
 
\(f(q)\) \(=\) \( q - 4 q^{5} + 49 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{5} + 49 q^{7} + 370 q^{11} + 122 q^{13} + 1428 q^{17} - 1724 q^{19} - 2670 q^{23} - 3109 q^{25} - 4302 q^{29} - 3104 q^{31} - 196 q^{35} - 14318 q^{37} + 12272 q^{41} + 21652 q^{43} - 2644 q^{47} + 2401 q^{49} + 24342 q^{53} - 1480 q^{55} - 14088 q^{59} - 24474 q^{61} - 488 q^{65} - 7208 q^{67} + 54302 q^{71} - 48962 q^{73} + 18130 q^{77} + 33332 q^{79} - 4004 q^{83} - 5712 q^{85} - 64752 q^{89} + 5978 q^{91} + 6896 q^{95} + 7038 q^{97}+O(q^{100}) \) 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 −4.00000 0 49.0000 0 0 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 1008.6.a.q 1
3.b odd 2 1 336.6.a.m 1
4.b odd 2 1 504.6.a.d 1
12.b even 2 1 168.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.6.a.b 1 12.b even 2 1
336.6.a.m 1 3.b odd 2 1
504.6.a.d 1 4.b odd 2 1
1008.6.a.q 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1008))\):

\( T_{5} + 4 \) Copy content Toggle raw display
\( T_{11} - 370 \) 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
$5$ \( T + 4 \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T - 370 \) Copy content Toggle raw display
$13$ \( T - 122 \) Copy content Toggle raw display
$17$ \( T - 1428 \) Copy content Toggle raw display
$19$ \( T + 1724 \) Copy content Toggle raw display
$23$ \( T + 2670 \) Copy content Toggle raw display
$29$ \( T + 4302 \) Copy content Toggle raw display
$31$ \( T + 3104 \) Copy content Toggle raw display
$37$ \( T + 14318 \) Copy content Toggle raw display
$41$ \( T - 12272 \) Copy content Toggle raw display
$43$ \( T - 21652 \) Copy content Toggle raw display
$47$ \( T + 2644 \) Copy content Toggle raw display
$53$ \( T - 24342 \) Copy content Toggle raw display
$59$ \( T + 14088 \) Copy content Toggle raw display
$61$ \( T + 24474 \) Copy content Toggle raw display
$67$ \( T + 7208 \) Copy content Toggle raw display
$71$ \( T - 54302 \) Copy content Toggle raw display
$73$ \( T + 48962 \) Copy content Toggle raw display
$79$ \( T - 33332 \) Copy content Toggle raw display
$83$ \( T + 4004 \) Copy content Toggle raw display
$89$ \( T + 64752 \) Copy content Toggle raw display
$97$ \( T - 7038 \) Copy content Toggle raw display
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