Properties

Label 168.2.a.b
Level $168$
Weight $2$
Character orbit 168.a
Self dual yes
Analytic conductor $1.341$
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] = [168,2,Mod(1,168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(168, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("168.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 168.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: \(1.34148675396\)
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)
 
\(f(q)\) \(=\) \( q + q^{3} + 2 q^{5} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + 2 q^{5} - q^{7} + q^{9} - 2 q^{13} + 2 q^{15} + 6 q^{17} - 4 q^{19} - q^{21} - 4 q^{23} - q^{25} + q^{27} + 6 q^{29} - 8 q^{31} - 2 q^{35} - 10 q^{37} - 2 q^{39} - 10 q^{41} + 12 q^{43} + 2 q^{45} - 8 q^{47} + q^{49} + 6 q^{51} + 6 q^{53} - 4 q^{57} + 4 q^{59} - 10 q^{61} - q^{63} - 4 q^{65} + 12 q^{67} - 4 q^{69} + 4 q^{71} + 2 q^{73} - q^{75} + 8 q^{79} + q^{81} + 4 q^{83} + 12 q^{85} + 6 q^{87} + 6 q^{89} + 2 q^{91} - 8 q^{93} - 8 q^{95} + 10 q^{97}+O(q^{100}) \) 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 1.00000 0 2.00000 0 −1.00000 0 1.00000 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 168.2.a.b 1
3.b odd 2 1 504.2.a.b 1
4.b odd 2 1 336.2.a.c 1
5.b even 2 1 4200.2.a.i 1
5.c odd 4 2 4200.2.t.m 2
7.b odd 2 1 1176.2.a.a 1
7.c even 3 2 1176.2.q.b 2
7.d odd 6 2 1176.2.q.j 2
8.b even 2 1 1344.2.a.c 1
8.d odd 2 1 1344.2.a.n 1
12.b even 2 1 1008.2.a.e 1
16.e even 4 2 5376.2.c.bd 2
16.f odd 4 2 5376.2.c.f 2
20.d odd 2 1 8400.2.a.bx 1
21.c even 2 1 3528.2.a.w 1
21.g even 6 2 3528.2.s.h 2
21.h odd 6 2 3528.2.s.v 2
24.f even 2 1 4032.2.a.bj 1
24.h odd 2 1 4032.2.a.be 1
28.d even 2 1 2352.2.a.q 1
28.f even 6 2 2352.2.q.j 2
28.g odd 6 2 2352.2.q.o 2
56.e even 2 1 9408.2.a.bc 1
56.h odd 2 1 9408.2.a.cy 1
84.h odd 2 1 7056.2.a.br 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.a.b 1 1.a even 1 1 trivial
336.2.a.c 1 4.b odd 2 1
504.2.a.b 1 3.b odd 2 1
1008.2.a.e 1 12.b even 2 1
1176.2.a.a 1 7.b odd 2 1
1176.2.q.b 2 7.c even 3 2
1176.2.q.j 2 7.d odd 6 2
1344.2.a.c 1 8.b even 2 1
1344.2.a.n 1 8.d odd 2 1
2352.2.a.q 1 28.d even 2 1
2352.2.q.j 2 28.f even 6 2
2352.2.q.o 2 28.g odd 6 2
3528.2.a.w 1 21.c even 2 1
3528.2.s.h 2 21.g even 6 2
3528.2.s.v 2 21.h odd 6 2
4032.2.a.be 1 24.h odd 2 1
4032.2.a.bj 1 24.f even 2 1
4200.2.a.i 1 5.b even 2 1
4200.2.t.m 2 5.c odd 4 2
5376.2.c.f 2 16.f odd 4 2
5376.2.c.bd 2 16.e even 4 2
7056.2.a.br 1 84.h odd 2 1
8400.2.a.bx 1 20.d odd 2 1
9408.2.a.bc 1 56.e even 2 1
9408.2.a.cy 1 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13} + 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(168))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T - 2 \) Copy content Toggle raw display
$7$ \( T + 1 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T - 6 \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T + 4 \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T + 8 \) Copy content Toggle raw display
$37$ \( T + 10 \) Copy content Toggle raw display
$41$ \( T + 10 \) Copy content Toggle raw display
$43$ \( T - 12 \) Copy content Toggle raw display
$47$ \( T + 8 \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T - 4 \) Copy content Toggle raw display
$61$ \( T + 10 \) Copy content Toggle raw display
$67$ \( T - 12 \) Copy content Toggle raw display
$71$ \( T - 4 \) Copy content Toggle raw display
$73$ \( T - 2 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T - 4 \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T - 10 \) Copy content Toggle raw display
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