Properties

Label 56.6.a.a
Level $56$
Weight $6$
Character orbit 56.a
Self dual yes
Analytic conductor $8.981$
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] = [56,6,Mod(1,56)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(56, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("56.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 56.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: \(8.98149390953\)
Analytic rank: \(1\)
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)
 
\(f(q)\) \(=\) \( q - 6 q^{3} + 4 q^{5} + 49 q^{7} - 207 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 6 q^{3} + 4 q^{5} + 49 q^{7} - 207 q^{9} - 240 q^{11} - 744 q^{13} - 24 q^{15} - 1042 q^{17} - 986 q^{19} - 294 q^{21} + 184 q^{23} - 3109 q^{25} + 2700 q^{27} - 734 q^{29} + 5140 q^{31} + 1440 q^{33} + 196 q^{35} - 6054 q^{37} + 4464 q^{39} + 7598 q^{41} + 13016 q^{43} - 828 q^{45} + 14668 q^{47} + 2401 q^{49} + 6252 q^{51} - 14522 q^{53} - 960 q^{55} + 5916 q^{57} - 13362 q^{59} + 9676 q^{61} - 10143 q^{63} - 2976 q^{65} - 62124 q^{67} - 1104 q^{69} - 2112 q^{71} - 28910 q^{73} + 18654 q^{75} - 11760 q^{77} - 101768 q^{79} + 34101 q^{81} - 23922 q^{83} - 4168 q^{85} + 4404 q^{87} + 141674 q^{89} - 36456 q^{91} - 30840 q^{93} - 3944 q^{95} + 99982 q^{97} + 49680 q^{99}+O(q^{100}) \) 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 −6.00000 0 4.00000 0 49.0000 0 −207.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( -1 \)

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 56.6.a.a 1
3.b odd 2 1 504.6.a.e 1
4.b odd 2 1 112.6.a.f 1
7.b odd 2 1 392.6.a.c 1
7.c even 3 2 392.6.i.d 2
7.d odd 6 2 392.6.i.c 2
8.b even 2 1 448.6.a.j 1
8.d odd 2 1 448.6.a.g 1
12.b even 2 1 1008.6.a.p 1
28.d even 2 1 784.6.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.a.a 1 1.a even 1 1 trivial
112.6.a.f 1 4.b odd 2 1
392.6.a.c 1 7.b odd 2 1
392.6.i.c 2 7.d odd 6 2
392.6.i.d 2 7.c even 3 2
448.6.a.g 1 8.d odd 2 1
448.6.a.j 1 8.b even 2 1
504.6.a.e 1 3.b odd 2 1
784.6.a.e 1 28.d even 2 1
1008.6.a.p 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 6 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(56))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 6 \) Copy content Toggle raw display
$5$ \( T - 4 \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T + 240 \) Copy content Toggle raw display
$13$ \( T + 744 \) Copy content Toggle raw display
$17$ \( T + 1042 \) Copy content Toggle raw display
$19$ \( T + 986 \) Copy content Toggle raw display
$23$ \( T - 184 \) Copy content Toggle raw display
$29$ \( T + 734 \) Copy content Toggle raw display
$31$ \( T - 5140 \) Copy content Toggle raw display
$37$ \( T + 6054 \) Copy content Toggle raw display
$41$ \( T - 7598 \) Copy content Toggle raw display
$43$ \( T - 13016 \) Copy content Toggle raw display
$47$ \( T - 14668 \) Copy content Toggle raw display
$53$ \( T + 14522 \) Copy content Toggle raw display
$59$ \( T + 13362 \) Copy content Toggle raw display
$61$ \( T - 9676 \) Copy content Toggle raw display
$67$ \( T + 62124 \) Copy content Toggle raw display
$71$ \( T + 2112 \) Copy content Toggle raw display
$73$ \( T + 28910 \) Copy content Toggle raw display
$79$ \( T + 101768 \) Copy content Toggle raw display
$83$ \( T + 23922 \) Copy content Toggle raw display
$89$ \( T - 141674 \) Copy content Toggle raw display
$97$ \( T - 99982 \) Copy content Toggle raw display
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