Properties

Label 1008.6.a.d
Level $1008$
Weight $6$
Character orbit 1008.a
Self dual yes
Analytic conductor $161.667$
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] = [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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
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)
 
\(f(q)\) \(=\) \( q - 76 q^{5} + 49 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 76 q^{5} + 49 q^{7} + 650 q^{11} + 762 q^{13} + 556 q^{17} + 2452 q^{19} - 2950 q^{23} + 2651 q^{25} + 674 q^{29} + 3024 q^{31} - 3724 q^{35} + 7730 q^{37} + 17016 q^{41} - 21836 q^{43} - 23940 q^{47} + 2401 q^{49} - 15594 q^{53} - 49400 q^{55} + 5608 q^{59} + 150 q^{61} - 57912 q^{65} + 43784 q^{67} - 39178 q^{71} - 23570 q^{73} + 31850 q^{77} + 17892 q^{79} + 38972 q^{83} - 42256 q^{85} - 6024 q^{89} + 37338 q^{91} - 186352 q^{95} + 108430 q^{97}+O(q^{100}) \) 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 −76.0000 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.d 1
3.b odd 2 1 336.6.a.q 1
4.b odd 2 1 126.6.a.a 1
12.b even 2 1 42.6.a.e 1
28.d even 2 1 882.6.a.j 1
60.h even 2 1 1050.6.a.f 1
60.l odd 4 2 1050.6.g.h 2
84.h odd 2 1 294.6.a.k 1
84.j odd 6 2 294.6.e.c 2
84.n even 6 2 294.6.e.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.a.e 1 12.b even 2 1
126.6.a.a 1 4.b odd 2 1
294.6.a.k 1 84.h odd 2 1
294.6.e.c 2 84.j odd 6 2
294.6.e.d 2 84.n even 6 2
336.6.a.q 1 3.b odd 2 1
882.6.a.j 1 28.d even 2 1
1008.6.a.d 1 1.a even 1 1 trivial
1050.6.a.f 1 60.h even 2 1
1050.6.g.h 2 60.l odd 4 2

Hecke kernels

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

\( T_{5} + 76 \) Copy content Toggle raw display
\( T_{11} - 650 \) 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
$5$ \( T + 76 \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T - 650 \) Copy content Toggle raw display
$13$ \( T - 762 \) Copy content Toggle raw display
$17$ \( T - 556 \) Copy content Toggle raw display
$19$ \( T - 2452 \) Copy content Toggle raw display
$23$ \( T + 2950 \) Copy content Toggle raw display
$29$ \( T - 674 \) Copy content Toggle raw display
$31$ \( T - 3024 \) Copy content Toggle raw display
$37$ \( T - 7730 \) Copy content Toggle raw display
$41$ \( T - 17016 \) Copy content Toggle raw display
$43$ \( T + 21836 \) Copy content Toggle raw display
$47$ \( T + 23940 \) Copy content Toggle raw display
$53$ \( T + 15594 \) Copy content Toggle raw display
$59$ \( T - 5608 \) Copy content Toggle raw display
$61$ \( T - 150 \) Copy content Toggle raw display
$67$ \( T - 43784 \) Copy content Toggle raw display
$71$ \( T + 39178 \) Copy content Toggle raw display
$73$ \( T + 23570 \) Copy content Toggle raw display
$79$ \( T - 17892 \) Copy content Toggle raw display
$83$ \( T - 38972 \) Copy content Toggle raw display
$89$ \( T + 6024 \) Copy content Toggle raw display
$97$ \( T - 108430 \) Copy content Toggle raw display
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