Properties

Label 56.6.a.b
Level $56$
Weight $6$
Character orbit 56.a
Self dual yes
Analytic conductor $8.981$
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] = [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: \(0\)
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)
 
\(f(q)\) \(=\) \( q + 30 q^{3} + 32 q^{5} + 49 q^{7} + 657 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 30 q^{3} + 32 q^{5} + 49 q^{7} + 657 q^{9} - 624 q^{11} - 708 q^{13} + 960 q^{15} + 934 q^{17} + 1858 q^{19} + 1470 q^{21} - 1120 q^{23} - 2101 q^{25} + 12420 q^{27} - 1174 q^{29} + 2908 q^{31} - 18720 q^{33} + 1568 q^{35} - 12462 q^{37} - 21240 q^{39} + 2662 q^{41} - 7144 q^{43} + 21024 q^{45} - 7468 q^{47} + 2401 q^{49} + 28020 q^{51} - 27274 q^{53} - 19968 q^{55} + 55740 q^{57} + 2490 q^{59} - 11096 q^{61} + 32193 q^{63} - 22656 q^{65} + 39756 q^{67} - 33600 q^{69} - 69888 q^{71} + 16450 q^{73} - 63030 q^{75} - 30576 q^{77} + 78376 q^{79} + 212949 q^{81} + 109818 q^{83} + 29888 q^{85} - 35220 q^{87} - 56966 q^{89} - 34692 q^{91} + 87240 q^{93} + 59456 q^{95} - 115946 q^{97} - 409968 q^{99}+O(q^{100}) \) 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 30.0000 0 32.0000 0 49.0000 0 657.000 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.b 1
3.b odd 2 1 504.6.a.b 1
4.b odd 2 1 112.6.a.a 1
7.b odd 2 1 392.6.a.a 1
7.c even 3 2 392.6.i.a 2
7.d odd 6 2 392.6.i.f 2
8.b even 2 1 448.6.a.a 1
8.d odd 2 1 448.6.a.p 1
12.b even 2 1 1008.6.a.h 1
28.d even 2 1 784.6.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.a.b 1 1.a even 1 1 trivial
112.6.a.a 1 4.b odd 2 1
392.6.a.a 1 7.b odd 2 1
392.6.i.a 2 7.c even 3 2
392.6.i.f 2 7.d odd 6 2
448.6.a.a 1 8.b even 2 1
448.6.a.p 1 8.d odd 2 1
504.6.a.b 1 3.b odd 2 1
784.6.a.n 1 28.d even 2 1
1008.6.a.h 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 30 \) 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
$3$ \( T - 30 \) Copy content Toggle raw display
$5$ \( T - 32 \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T + 624 \) Copy content Toggle raw display
$13$ \( T + 708 \) Copy content Toggle raw display
$17$ \( T - 934 \) Copy content Toggle raw display
$19$ \( T - 1858 \) Copy content Toggle raw display
$23$ \( T + 1120 \) Copy content Toggle raw display
$29$ \( T + 1174 \) Copy content Toggle raw display
$31$ \( T - 2908 \) Copy content Toggle raw display
$37$ \( T + 12462 \) Copy content Toggle raw display
$41$ \( T - 2662 \) Copy content Toggle raw display
$43$ \( T + 7144 \) Copy content Toggle raw display
$47$ \( T + 7468 \) Copy content Toggle raw display
$53$ \( T + 27274 \) Copy content Toggle raw display
$59$ \( T - 2490 \) Copy content Toggle raw display
$61$ \( T + 11096 \) Copy content Toggle raw display
$67$ \( T - 39756 \) Copy content Toggle raw display
$71$ \( T + 69888 \) Copy content Toggle raw display
$73$ \( T - 16450 \) Copy content Toggle raw display
$79$ \( T - 78376 \) Copy content Toggle raw display
$83$ \( T - 109818 \) Copy content Toggle raw display
$89$ \( T + 56966 \) Copy content Toggle raw display
$97$ \( T + 115946 \) Copy content Toggle raw display
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