Newform orbit 162.8.c.l

• Label: 162.8.c.l
• Level: $162$
• Weight: $8$
• Character orbit: 162.c
• Analytic conductor: $50.606$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	162.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(50.6063741284\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 2)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + ( - 8 \zeta_{6} + 8) q^{2} - 64 \zeta_{6} q^{4} + 210 \zeta_{6} q^{5} + (1016 \zeta_{6} - 1016) q^{7} - 512 q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{2} - 64 q^{4} + 210 q^{5} - 1016 q^{7} - 1024 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\)	\(83\)
\(\chi(n)\)	\(-\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

55.1

0.500000	+	0.866025i
0.500000	−	0.866025i

4.00000

−

6.92820i

0

−32.0000

−

55.4256i

105.000

+

181.865i

0

−508.000

+

879.882i

−512.000

0

1680.00

109.1

4.00000

+

6.92820i

0

−32.0000

+

55.4256i

105.000

−

181.865i

0

−508.000

−

879.882i

−512.000

0

1680.00

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	162.8.c.l		2
3.b	odd	2	1		162.8.c.a		2
9.c	even	3	1		2.8.a.a	✓	1
9.c	even	3	1	inner	162.8.c.l		2
9.d	odd	6	1		18.8.a.b		1
9.d	odd	6	1		162.8.c.a		2
36.f	odd	6	1		16.8.a.b		1
36.h	even	6	1		144.8.a.i		1
45.h	odd	6	1		450.8.a.c		1
45.j	even	6	1		50.8.a.g		1
45.k	odd	12	2		50.8.b.c		2
45.l	even	12	2		450.8.c.g		2
63.g	even	3	1		98.8.c.d		2
63.h	even	3	1		98.8.c.d		2
63.k	odd	6	1		98.8.c.e		2
63.l	odd	6	1		98.8.a.a		1
63.t	odd	6	1		98.8.c.e		2
72.j	odd	6	1		576.8.a.g		1
72.l	even	6	1		576.8.a.f		1
72.n	even	6	1		64.8.a.c		1
72.p	odd	6	1		64.8.a.e		1
99.h	odd	6	1		242.8.a.e		1
117.t	even	6	1		338.8.a.d		1
117.y	odd	12	2		338.8.b.d		2
144.v	odd	12	2		256.8.b.f		2
144.x	even	12	2		256.8.b.b		2
153.h	even	6	1		578.8.a.b		1
180.p	odd	6	1		400.8.a.l		1
180.x	even	12	2		400.8.c.j		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2.8.a.a	✓	1	9.c	even	3	1
16.8.a.b		1	36.f	odd	6	1
18.8.a.b		1	9.d	odd	6	1
50.8.a.g		1	45.j	even	6	1
50.8.b.c		2	45.k	odd	12	2
64.8.a.c		1	72.n	even	6	1
64.8.a.e		1	72.p	odd	6	1
98.8.a.a		1	63.l	odd	6	1
98.8.c.d		2	63.g	even	3	1
98.8.c.d		2	63.h	even	3	1
98.8.c.e		2	63.k	odd	6	1
98.8.c.e		2	63.t	odd	6	1
144.8.a.i		1	36.h	even	6	1
162.8.c.a		2	3.b	odd	2	1
162.8.c.a		2	9.d	odd	6	1
162.8.c.l		2	1.a	even	1	1	trivial
162.8.c.l		2	9.c	even	3	1	inner
242.8.a.e		1	99.h	odd	6	1
256.8.b.b		2	144.x	even	12	2
256.8.b.f		2	144.v	odd	12	2
338.8.a.d		1	117.t	even	6	1
338.8.b.d		2	117.y	odd	12	2
400.8.a.l		1	180.p	odd	6	1
400.8.c.j		2	180.x	even	12	2
450.8.a.c		1	45.h	odd	6	1
450.8.c.g		2	45.l	even	12	2
576.8.a.f		1	72.l	even	6	1
576.8.a.g		1	72.j	odd	6	1
578.8.a.b		1	153.h	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 210T_{5} + 44100 \) acting on \(S_{8}^{\mathrm{new}}(162, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 8T + 64 \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 210T + 44100 \)
$7$	\( T^{2} + 1016 T + 1032256 \)
$11$	\( T^{2} + 1092 T + 1192464 \)