Newform orbit 162.2.c.c

• Label: 162.2.c.c
• Level: $162$
• Weight: $2$
• Character orbit: 162.c
• Analytic conductor: $1.294$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.29357651274\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 54)
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.

\(f(q)\)	\(=\)	\( q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} - 3 \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} - q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} - 3 q^{5} + q^{7} - 2 q^{8}+O(q^{10}) \)

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

0.500000

−

0.866025i

0

−0.500000

−

0.866025i

−1.50000

−

2.59808i

0

0.500000

−

0.866025i

−1.00000

0

−3.00000

109.1

0.500000

+

0.866025i

0

−0.500000

+

0.866025i

−1.50000

+

2.59808i

0

0.500000

+

0.866025i

−1.00000

0

−3.00000

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	162.2.c.c		2
3.b	odd	2	1		162.2.c.b		2
4.b	odd	2	1		1296.2.i.c		2
9.c	even	3	1		54.2.a.a	✓	1
9.c	even	3	1	inner	162.2.c.c		2
9.d	odd	6	1		54.2.a.b	yes	1
9.d	odd	6	1		162.2.c.b		2
12.b	even	2	1		1296.2.i.o		2
36.f	odd	6	1		432.2.a.g		1
36.f	odd	6	1		1296.2.i.c		2
36.h	even	6	1		432.2.a.b		1
36.h	even	6	1		1296.2.i.o		2
45.h	odd	6	1		1350.2.a.h		1
45.j	even	6	1		1350.2.a.r		1
45.k	odd	12	2		1350.2.c.b		2
45.l	even	12	2		1350.2.c.k		2
63.l	odd	6	1		2646.2.a.a		1
63.o	even	6	1		2646.2.a.bd		1
72.j	odd	6	1		1728.2.a.y		1
72.l	even	6	1		1728.2.a.z		1
72.n	even	6	1		1728.2.a.c		1
72.p	odd	6	1		1728.2.a.d		1
99.g	even	6	1		6534.2.a.b		1
99.h	odd	6	1		6534.2.a.bc		1
117.n	odd	6	1		9126.2.a.r		1
117.t	even	6	1		9126.2.a.u		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
54.2.a.a	✓	1	9.c	even	3	1
54.2.a.b	yes	1	9.d	odd	6	1
162.2.c.b		2	3.b	odd	2	1
162.2.c.b		2	9.d	odd	6	1
162.2.c.c		2	1.a	even	1	1	trivial
162.2.c.c		2	9.c	even	3	1	inner
432.2.a.b		1	36.h	even	6	1
432.2.a.g		1	36.f	odd	6	1
1296.2.i.c		2	4.b	odd	2	1
1296.2.i.c		2	36.f	odd	6	1
1296.2.i.o		2	12.b	even	2	1
1296.2.i.o		2	36.h	even	6	1
1350.2.a.h		1	45.h	odd	6	1
1350.2.a.r		1	45.j	even	6	1
1350.2.c.b		2	45.k	odd	12	2
1350.2.c.k		2	45.l	even	12	2
1728.2.a.c		1	72.n	even	6	1
1728.2.a.d		1	72.p	odd	6	1
1728.2.a.y		1	72.j	odd	6	1
1728.2.a.z		1	72.l	even	6	1
2646.2.a.a		1	63.l	odd	6	1
2646.2.a.bd		1	63.o	even	6	1
6534.2.a.b		1	99.g	even	6	1
6534.2.a.bc		1	99.h	odd	6	1
9126.2.a.r		1	117.n	odd	6	1
9126.2.a.u		1	117.t	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(162, [\chi])\):

\( T_{5}^{2} + 3T_{5} + 9 \)

\( T_{7}^{2} - T_{7} + 1 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - T + 1 \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 3T + 9 \)
$7$	\( T^{2} - T + 1 \)
$11$	\( T^{2} - 3T + 9 \)