Properties

Label 1008.6.a.c
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 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)
 
\(f(q)\) \(=\) \( q - 78 q^{5} - 49 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 78 q^{5} - 49 q^{7} + 444 q^{11} - 442 q^{13} + 126 q^{17} - 2684 q^{19} + 4200 q^{23} + 2959 q^{25} + 5442 q^{29} - 80 q^{31} + 3822 q^{35} - 5434 q^{37} - 7962 q^{41} + 11524 q^{43} - 13920 q^{47} + 2401 q^{49} + 9594 q^{53} - 34632 q^{55} + 27492 q^{59} + 49478 q^{61} + 34476 q^{65} + 59356 q^{67} + 32040 q^{71} - 61846 q^{73} - 21756 q^{77} + 65776 q^{79} + 40188 q^{83} - 9828 q^{85} + 7974 q^{89} + 21658 q^{91} + 209352 q^{95} - 143662 q^{97}+O(q^{100}) \) 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 −78.0000 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.c 1
3.b odd 2 1 336.6.a.r 1
4.b odd 2 1 63.6.a.d 1
12.b even 2 1 21.6.a.a 1
28.d even 2 1 441.6.a.j 1
60.h even 2 1 525.6.a.d 1
60.l odd 4 2 525.6.d.b 2
84.h odd 2 1 147.6.a.b 1
84.j odd 6 2 147.6.e.i 2
84.n even 6 2 147.6.e.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.a 1 12.b even 2 1
63.6.a.d 1 4.b odd 2 1
147.6.a.b 1 84.h odd 2 1
147.6.e.i 2 84.j odd 6 2
147.6.e.j 2 84.n even 6 2
336.6.a.r 1 3.b odd 2 1
441.6.a.j 1 28.d even 2 1
525.6.a.d 1 60.h even 2 1
525.6.d.b 2 60.l odd 4 2
1008.6.a.c 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(1008))\):

\( T_{5} + 78 \) Copy content Toggle raw display
\( T_{11} - 444 \) 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
$5$ \( T + 78 \) Copy content Toggle raw display
$7$ \( T + 49 \) Copy content Toggle raw display
$11$ \( T - 444 \) Copy content Toggle raw display
$13$ \( T + 442 \) Copy content Toggle raw display
$17$ \( T - 126 \) Copy content Toggle raw display
$19$ \( T + 2684 \) Copy content Toggle raw display
$23$ \( T - 4200 \) Copy content Toggle raw display
$29$ \( T - 5442 \) Copy content Toggle raw display
$31$ \( T + 80 \) Copy content Toggle raw display
$37$ \( T + 5434 \) Copy content Toggle raw display
$41$ \( T + 7962 \) Copy content Toggle raw display
$43$ \( T - 11524 \) Copy content Toggle raw display
$47$ \( T + 13920 \) Copy content Toggle raw display
$53$ \( T - 9594 \) Copy content Toggle raw display
$59$ \( T - 27492 \) Copy content Toggle raw display
$61$ \( T - 49478 \) Copy content Toggle raw display
$67$ \( T - 59356 \) Copy content Toggle raw display
$71$ \( T - 32040 \) Copy content Toggle raw display
$73$ \( T + 61846 \) Copy content Toggle raw display
$79$ \( T - 65776 \) Copy content Toggle raw display
$83$ \( T - 40188 \) Copy content Toggle raw display
$89$ \( T - 7974 \) Copy content Toggle raw display
$97$ \( T + 143662 \) Copy content Toggle raw display
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