Newform orbit 162.2.c.d

• Label: 162.2.c.d
• 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:	yes
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.

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

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

2.00000

−

3.46410i

−1.00000

0

3.00000

109.1

0.500000

+

0.866025i

0

−0.500000

+

0.866025i

1.50000

−

2.59808i

0

2.00000

+

3.46410i

−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.d		2
3.b	odd	2	1		162.2.c.a		2
4.b	odd	2	1		1296.2.i.n		2
9.c	even	3	1		162.2.a.a	✓	1
9.c	even	3	1	inner	162.2.c.d		2
9.d	odd	6	1		162.2.a.d	yes	1
9.d	odd	6	1		162.2.c.a		2
12.b	even	2	1		1296.2.i.b		2
36.f	odd	6	1		1296.2.a.c		1
36.f	odd	6	1		1296.2.i.n		2
36.h	even	6	1		1296.2.a.l		1
36.h	even	6	1		1296.2.i.b		2
45.h	odd	6	1		4050.2.a.r		1
45.j	even	6	1		4050.2.a.bh		1
45.k	odd	12	2		4050.2.c.g		2
45.l	even	12	2		4050.2.c.n		2
63.l	odd	6	1		7938.2.a.n		1
63.o	even	6	1		7938.2.a.s		1
72.j	odd	6	1		5184.2.a.c		1
72.l	even	6	1		5184.2.a.h		1
72.n	even	6	1		5184.2.a.y		1
72.p	odd	6	1		5184.2.a.bd		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
162.2.a.a	✓	1	9.c	even	3	1
162.2.a.d	yes	1	9.d	odd	6	1
162.2.c.a		2	3.b	odd	2	1
162.2.c.a		2	9.d	odd	6	1
162.2.c.d		2	1.a	even	1	1	trivial
162.2.c.d		2	9.c	even	3	1	inner
1296.2.a.c		1	36.f	odd	6	1
1296.2.a.l		1	36.h	even	6	1
1296.2.i.b		2	12.b	even	2	1
1296.2.i.b		2	36.h	even	6	1
1296.2.i.n		2	4.b	odd	2	1
1296.2.i.n		2	36.f	odd	6	1
4050.2.a.r		1	45.h	odd	6	1
4050.2.a.bh		1	45.j	even	6	1
4050.2.c.g		2	45.k	odd	12	2
4050.2.c.n		2	45.l	even	12	2
5184.2.a.c		1	72.j	odd	6	1
5184.2.a.h		1	72.l	even	6	1
5184.2.a.y		1	72.n	even	6	1
5184.2.a.bd		1	72.p	odd	6	1
7938.2.a.n		1	63.l	odd	6	1
7938.2.a.s		1	63.o	even	6	1

Hecke kernels

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

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

\( T_{7}^{2} - 4T_{7} + 16 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - T + 1 \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 3T + 9 \)
$7$	\( T^{2} - 4T + 16 \)
$11$	\( T^{2} \)