Newform orbit 168.1.s.a

• Label: 168.1.s.a
• 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: $4$

Newspace parameters

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

Self dual:	no
Analytic conductor:	\(0.0838429221223\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
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:	$C_3\times S_3$
Artin field:	Galois closure of 6.0.677376.1

$q$-expansion

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

\(f(q)\)	\(=\)	\( q + \zeta_{6}^{2} q^{2} - \zeta_{6} q^{3} - \zeta_{6} q^{4} - \zeta_{6}^{2} q^{5} + q^{6} - \zeta_{6} q^{7} + q^{8} + \zeta_{6}^{2} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{3} - q^{4} + q^{5} + 2 q^{6} - q^{7} + 2 q^{8} - q^{9}+O(q^{10}) \)

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.

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

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.a	✓	2
3.b	odd	2	1		168.1.s.b	yes	2
4.b	odd	2	1		672.1.ba.b		2
7.b	odd	2	1		1176.1.s.a		2
7.c	even	3	1	inner	168.1.s.a	✓	2
7.c	even	3	1		1176.1.n.d		1
7.d	odd	6	1		1176.1.n.c		1
7.d	odd	6	1		1176.1.s.a		2
8.b	even	2	1		168.1.s.b	yes	2
8.d	odd	2	1		672.1.ba.a		2
12.b	even	2	1		672.1.ba.a		2
21.c	even	2	1		1176.1.s.b		2
21.g	even	6	1		1176.1.n.b		1
21.g	even	6	1		1176.1.s.b		2
21.h	odd	6	1		168.1.s.b	yes	2
21.h	odd	6	1		1176.1.n.a		1
24.f	even	2	1		672.1.ba.b		2
24.h	odd	2	1	CM	168.1.s.a	✓	2
28.g	odd	6	1		672.1.ba.b		2
56.h	odd	2	1		1176.1.s.b		2
56.j	odd	6	1		1176.1.n.b		1
56.j	odd	6	1		1176.1.s.b		2
56.k	odd	6	1		672.1.ba.a		2
56.p	even	6	1		168.1.s.b	yes	2
56.p	even	6	1		1176.1.n.a		1
84.n	even	6	1		672.1.ba.a		2
168.i	even	2	1		1176.1.s.a		2
168.s	odd	6	1	inner	168.1.s.a	✓	2
168.s	odd	6	1		1176.1.n.d		1
168.v	even	6	1		672.1.ba.b		2
168.ba	even	6	1		1176.1.n.c		1
168.ba	even	6	1		1176.1.s.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
168.1.s.a	✓	2	1.a	even	1	1	trivial
168.1.s.a	✓	2	7.c	even	3	1	inner
168.1.s.a	✓	2	24.h	odd	2	1	CM
168.1.s.a	✓	2	168.s	odd	6	1	inner
168.1.s.b	yes	2	3.b	odd	2	1
168.1.s.b	yes	2	8.b	even	2	1
168.1.s.b	yes	2	21.h	odd	6	1
168.1.s.b	yes	2	56.p	even	6	1
672.1.ba.a		2	8.d	odd	2	1
672.1.ba.a		2	12.b	even	2	1
672.1.ba.a		2	56.k	odd	6	1
672.1.ba.a		2	84.n	even	6	1
672.1.ba.b		2	4.b	odd	2	1
672.1.ba.b		2	24.f	even	2	1
672.1.ba.b		2	28.g	odd	6	1
672.1.ba.b		2	168.v	even	6	1
1176.1.n.a		1	21.h	odd	6	1
1176.1.n.a		1	56.p	even	6	1
1176.1.n.b		1	21.g	even	6	1
1176.1.n.b		1	56.j	odd	6	1
1176.1.n.c		1	7.d	odd	6	1
1176.1.n.c		1	168.ba	even	6	1
1176.1.n.d		1	7.c	even	3	1
1176.1.n.d		1	168.s	odd	6	1
1176.1.s.a		2	7.b	odd	2	1
1176.1.s.a		2	7.d	odd	6	1
1176.1.s.a		2	168.i	even	2	1
1176.1.s.a		2	168.ba	even	6	1
1176.1.s.b		2	21.c	even	2	1
1176.1.s.b		2	21.g	even	6	1
1176.1.s.b		2	56.h	odd	2	1
1176.1.s.b		2	56.j	odd	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 \(S_{1}^{\mathrm{new}}(168, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + T + 1 \)
$3$	\( T^{2} + T + 1 \)
$5$	\( T^{2} - T + 1 \)
$7$	\( T^{2} + T + 1 \)
$11$	\( T^{2} - T + 1 \)