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Properties

Label 168.1.s.b
Level $168$
Weight $1$
Character orbit 168.s
Analytic conductor $0.084$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -24
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [168,1,Mod(53,168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(168, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 3, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("168.53");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 168.s (of order \(6\), degree \(2\), minimal)

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: no
Analytic conductor: \(0.0838429221223\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.1176.1
Artin image: $C_6\times S_3$
Artin field: Galois closure of 12.0.458838245376.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{6}^{2} q^{2} + \zeta_{6} q^{3} - \zeta_{6} q^{4} + \zeta_{6}^{2} q^{5} + q^{6} - \zeta_{6} q^{7} - q^{8} + \zeta_{6}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6}^{2} q^{2} + \zeta_{6} q^{3} - \zeta_{6} q^{4} + \zeta_{6}^{2} q^{5} + q^{6} - \zeta_{6} q^{7} - q^{8} + \zeta_{6}^{2} q^{9} + \zeta_{6} q^{10} - \zeta_{6} q^{11} - \zeta_{6}^{2} q^{12} - q^{14} - q^{15} + \zeta_{6}^{2} q^{16} + \zeta_{6} q^{18} + q^{20} - \zeta_{6}^{2} q^{21} - q^{22} - \zeta_{6} q^{24} - q^{27} + \zeta_{6}^{2} q^{28} + q^{29} + \zeta_{6}^{2} q^{30} + \zeta_{6} q^{31} + \zeta_{6} q^{32} - \zeta_{6}^{2} q^{33} + q^{35} + q^{36} - \zeta_{6}^{2} q^{40} - \zeta_{6} q^{42} + \zeta_{6}^{2} q^{44} - \zeta_{6} q^{45} - q^{48} + \zeta_{6}^{2} q^{49} - \zeta_{6} q^{53} + \zeta_{6}^{2} q^{54} + q^{55} + \zeta_{6} q^{56} - \zeta_{6}^{2} q^{58} - \zeta_{6} q^{59} + \zeta_{6} q^{60} + q^{62} + q^{63} + q^{64} - \zeta_{6} q^{66} - \zeta_{6}^{2} q^{70} - \zeta_{6}^{2} q^{72} - \zeta_{6} q^{73} + \zeta_{6}^{2} q^{77} - \zeta_{6}^{2} q^{79} - \zeta_{6} q^{80} - \zeta_{6} q^{81} + q^{83} - q^{84} + \zeta_{6} q^{87} + \zeta_{6} q^{88} - q^{90} + \zeta_{6}^{2} q^{93} + \zeta_{6}^{2} q^{96} - q^{97} + \zeta_{6} q^{98} + q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{3} - q^{4} - q^{5} + 2 q^{6} - q^{7} - 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + q^{3} - q^{4} - q^{5} + 2 q^{6} - q^{7} - 2 q^{8} - q^{9} + q^{10} - q^{11} + q^{12} - 2 q^{14} - 2 q^{15} - q^{16} + q^{18} + 2 q^{20} + q^{21} - 2 q^{22} - q^{24} - 2 q^{27} - q^{28} + 2 q^{29} - q^{30} + q^{31} + q^{32} + q^{33} + 2 q^{35} + 2 q^{36} + q^{40} - q^{42} - q^{44} - q^{45} - 2 q^{48} - q^{49} - q^{53} - q^{54} + 2 q^{55} + q^{56} + q^{58} - q^{59} + q^{60} + 2 q^{62} + 2 q^{63} + 2 q^{64} - q^{66} + q^{70} + q^{72} - 2 q^{73} - q^{77} + q^{79} - q^{80} - q^{81} + 2 q^{83} - 2 q^{84} + q^{87} + q^{88} - 2 q^{90} - q^{93} - q^{96} - 2 q^{97} + q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(85\) \(113\) \(127\)
\(\chi(n)\) \(\zeta_{6}^{2}\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i 0.500000 − 0.866025i −0.500000 + 0.866025i −0.500000 − 0.866025i 1.00000 −0.500000 + 0.866025i −1.00000 −0.500000 − 0.866025i 0.500000 − 0.866025i
149.1 0.500000 − 0.866025i 0.500000 + 0.866025i −0.500000 − 0.866025i −0.500000 + 0.866025i 1.00000 −0.500000 − 0.866025i −1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
7.c even 3 1 inner
168.s odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.1.s.b yes 2
3.b odd 2 1 168.1.s.a ✓ 2
4.b odd 2 1 672.1.ba.a 2
7.b odd 2 1 1176.1.s.b 2
7.c even 3 1 inner 168.1.s.b yes 2
7.c even 3 1 1176.1.n.a 1
7.d odd 6 1 1176.1.n.b 1
7.d odd 6 1 1176.1.s.b 2
8.b even 2 1 168.1.s.a ✓ 2
8.d odd 2 1 672.1.ba.b 2
12.b even 2 1 672.1.ba.b 2
21.c even 2 1 1176.1.s.a 2
21.g even 6 1 1176.1.n.c 1
21.g even 6 1 1176.1.s.a 2
21.h odd 6 1 168.1.s.a ✓ 2
21.h odd 6 1 1176.1.n.d 1
24.f even 2 1 672.1.ba.a 2
24.h odd 2 1 CM 168.1.s.b yes 2
28.g odd 6 1 672.1.ba.a 2
56.h odd 2 1 1176.1.s.a 2
56.j odd 6 1 1176.1.n.c 1
56.j odd 6 1 1176.1.s.a 2
56.k odd 6 1 672.1.ba.b 2
56.p even 6 1 168.1.s.a ✓ 2
56.p even 6 1 1176.1.n.d 1
84.n even 6 1 672.1.ba.b 2
168.i even 2 1 1176.1.s.b 2
168.s odd 6 1 inner 168.1.s.b yes 2
168.s odd 6 1 1176.1.n.a 1
168.v even 6 1 672.1.ba.a 2
168.ba even 6 1 1176.1.n.b 1
168.ba even 6 1 1176.1.s.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.1.s.a ✓ 2 3.b odd 2 1
168.1.s.a ✓ 2 8.b even 2 1
168.1.s.a ✓ 2 21.h odd 6 1
168.1.s.a ✓ 2 56.p even 6 1
168.1.s.b yes 2 1.a even 1 1 trivial
168.1.s.b yes 2 7.c even 3 1 inner
168.1.s.b yes 2 24.h odd 2 1 CM
168.1.s.b yes 2 168.s odd 6 1 inner
672.1.ba.a 2 4.b odd 2 1
672.1.ba.a 2 24.f even 2 1
672.1.ba.a 2 28.g odd 6 1
672.1.ba.a 2 168.v even 6 1
672.1.ba.b 2 8.d odd 2 1
672.1.ba.b 2 12.b even 2 1
672.1.ba.b 2 56.k odd 6 1
672.1.ba.b 2 84.n even 6 1
1176.1.n.a 1 7.c even 3 1
1176.1.n.a 1 168.s odd 6 1
1176.1.n.b 1 7.d odd 6 1
1176.1.n.b 1 168.ba even 6 1
1176.1.n.c 1 21.g even 6 1
1176.1.n.c 1 56.j odd 6 1
1176.1.n.d 1 21.h odd 6 1
1176.1.n.d 1 56.p even 6 1
1176.1.s.a 2 21.c even 2 1
1176.1.s.a 2 21.g even 6 1
1176.1.s.a 2 56.h odd 2 1
1176.1.s.a 2 56.j odd 6 1
1176.1.s.b 2 7.b odd 2 1
1176.1.s.b 2 7.d odd 6 1
1176.1.s.b 2 168.i even 2 1
1176.1.s.b 2 168.ba even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + T_{5} + 1 \) acting on \(S_{1}^{\mathrm{new}}(168, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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