Newform orbit 168.2.q.c

• Label: 168.2.q.c
• Level: $168$
• Weight: $2$
• Character orbit: 168.q
• Analytic conductor: $1.341$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	168.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.34148675396\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{-19})\)
Defining polynomial:	\( x^{4} - x^{3} - 4x^{2} - 5x + 25 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + ( - \beta_{2} + 1) q^{3} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{5} + ( - \beta_{3} + \beta_1 - 2) q^{7} - \beta_{2} q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + q^{5} - 6 q^{7} - 2 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{3} + 4\nu^{2} - 4\nu - 5 ) / 20 \)
\(\beta_{3}\)	\(=\)	\( ( -\nu^{3} + 4\nu + 5 ) / 4 \)

Character values

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

\(n\)	\(73\)	\(85\)	\(113\)	\(127\)
\(\chi(n)\)	\(-\beta_{2}\)	\(1\)	\(1\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

25.1

−1.63746	−	1.52274i
2.13746	+	0.656712i
−1.63746	+	1.52274i
2.13746	−	0.656712i

0

0.500000

−

0.866025i

0

−1.63746

−

2.83616i

0

−1.50000

−

2.17945i

0

−0.500000

−

0.866025i

0

25.2

0

0.500000

−

0.866025i

0

2.13746

+

3.70219i

0

−1.50000

+

2.17945i

0

−0.500000

−

0.866025i

0

121.1

0

0.500000

+

0.866025i

0

−1.63746

+

2.83616i

0

−1.50000

+

2.17945i

0

−0.500000

+

0.866025i

0

121.2

0

0.500000

+

0.866025i

0

2.13746

−

3.70219i

0

−1.50000

−

2.17945i

0

−0.500000

+

0.866025i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	168.2.q.c	✓	4
3.b	odd	2	1		504.2.s.i		4
4.b	odd	2	1		336.2.q.g		4
7.b	odd	2	1		1176.2.q.l		4
7.c	even	3	1	inner	168.2.q.c	✓	4
7.c	even	3	1		1176.2.a.k		2
7.d	odd	6	1		1176.2.a.n		2
7.d	odd	6	1		1176.2.q.l		4
8.b	even	2	1		1344.2.q.w		4
8.d	odd	2	1		1344.2.q.x		4
12.b	even	2	1		1008.2.s.r		4
21.c	even	2	1		3528.2.s.bk		4
21.g	even	6	1		3528.2.a.bd		2
21.g	even	6	1		3528.2.s.bk		4
21.h	odd	6	1		504.2.s.i		4
21.h	odd	6	1		3528.2.a.bk		2
28.d	even	2	1		2352.2.q.bf		4
28.f	even	6	1		2352.2.a.ba		2
28.f	even	6	1		2352.2.q.bf		4
28.g	odd	6	1		336.2.q.g		4
28.g	odd	6	1		2352.2.a.bf		2
56.j	odd	6	1		9408.2.a.dj		2
56.k	odd	6	1		1344.2.q.x		4
56.k	odd	6	1		9408.2.a.dp		2
56.m	even	6	1		9408.2.a.dw		2
56.p	even	6	1		1344.2.q.w		4
56.p	even	6	1		9408.2.a.ec		2
84.j	odd	6	1		7056.2.a.ch		2
84.n	even	6	1		1008.2.s.r		4
84.n	even	6	1		7056.2.a.cu		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
168.2.q.c	✓	4	1.a	even	1	1	trivial
168.2.q.c	✓	4	7.c	even	3	1	inner
336.2.q.g		4	4.b	odd	2	1
336.2.q.g		4	28.g	odd	6	1
504.2.s.i		4	3.b	odd	2	1
504.2.s.i		4	21.h	odd	6	1
1008.2.s.r		4	12.b	even	2	1
1008.2.s.r		4	84.n	even	6	1
1176.2.a.k		2	7.c	even	3	1
1176.2.a.n		2	7.d	odd	6	1
1176.2.q.l		4	7.b	odd	2	1
1176.2.q.l		4	7.d	odd	6	1
1344.2.q.w		4	8.b	even	2	1
1344.2.q.w		4	56.p	even	6	1
1344.2.q.x		4	8.d	odd	2	1
1344.2.q.x		4	56.k	odd	6	1
2352.2.a.ba		2	28.f	even	6	1
2352.2.a.bf		2	28.g	odd	6	1
2352.2.q.bf		4	28.d	even	2	1
2352.2.q.bf		4	28.f	even	6	1
3528.2.a.bd		2	21.g	even	6	1
3528.2.a.bk		2	21.h	odd	6	1
3528.2.s.bk		4	21.c	even	2	1
3528.2.s.bk		4	21.g	even	6	1
7056.2.a.ch		2	84.j	odd	6	1
7056.2.a.cu		2	84.n	even	6	1
9408.2.a.dj		2	56.j	odd	6	1
9408.2.a.dp		2	56.k	odd	6	1
9408.2.a.dw		2	56.m	even	6	1
9408.2.a.ec		2	56.p	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - T_{5}^{3} + 15T_{5}^{2} + 14T_{5} + 196 \) acting on \(S_{2}^{\mathrm{new}}(168, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} \)
$3$	\( (T^{2} - T + 1)^{2} \)
$5$	\( T^{4} - T^{3} + 15 T^{2} + 14 T + 196 \)
$7$	\( (T^{2} + 3 T + 7)^{2} \)
$11$	\( T^{4} - T^{3} + 15 T^{2} + 14 T + 196 \)