Newform orbit 162.4.c.h

• Label: 162.4.c.h
• Level: $162$
• Weight: $4$
• Character orbit: 162.c
• Analytic conductor: $9.558$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(9.55830942093\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\)	\(=\)	\( q + ( - 2 \zeta_{6} + 2) q^{2} - 4 \zeta_{6} q^{4} + 12 \zeta_{6} q^{5} + ( - 7 \zeta_{6} + 7) q^{7} - 8 q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 4 q^{4} + 12 q^{5} + 7 q^{7} - 16 q^{8}+O(q^{10}) \)

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

1.00000

−

1.73205i

0

−2.00000

−

3.46410i

6.00000

+

10.3923i

0

3.50000

−

6.06218i

−8.00000

0

24.0000

109.1

1.00000

+

1.73205i

0

−2.00000

+

3.46410i

6.00000

−

10.3923i

0

3.50000

+

6.06218i

−8.00000

0

24.0000

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	162.4.c.h		2
3.b	odd	2	1		162.4.c.a		2
9.c	even	3	1		54.4.a.a	✓	1
9.c	even	3	1	inner	162.4.c.h		2
9.d	odd	6	1		54.4.a.d	yes	1
9.d	odd	6	1		162.4.c.a		2
36.f	odd	6	1		432.4.a.b		1
36.h	even	6	1		432.4.a.m		1
45.h	odd	6	1		1350.4.a.h		1
45.j	even	6	1		1350.4.a.v		1
45.k	odd	12	2		1350.4.c.a		2
45.l	even	12	2		1350.4.c.t		2
72.j	odd	6	1		1728.4.a.e		1
72.l	even	6	1		1728.4.a.f		1
72.n	even	6	1		1728.4.a.ba		1
72.p	odd	6	1		1728.4.a.bb		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
54.4.a.a	✓	1	9.c	even	3	1
54.4.a.d	yes	1	9.d	odd	6	1
162.4.c.a		2	3.b	odd	2	1
162.4.c.a		2	9.d	odd	6	1
162.4.c.h		2	1.a	even	1	1	trivial
162.4.c.h		2	9.c	even	3	1	inner
432.4.a.b		1	36.f	odd	6	1
432.4.a.m		1	36.h	even	6	1
1350.4.a.h		1	45.h	odd	6	1
1350.4.a.v		1	45.j	even	6	1
1350.4.c.a		2	45.k	odd	12	2
1350.4.c.t		2	45.l	even	12	2
1728.4.a.e		1	72.j	odd	6	1
1728.4.a.f		1	72.l	even	6	1
1728.4.a.ba		1	72.n	even	6	1
1728.4.a.bb		1	72.p	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 12T_{5} + 144 \) acting on \(S_{4}^{\mathrm{new}}(162, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 2T + 4 \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 12T + 144 \)
$7$	\( T^{2} - 7T + 49 \)
$11$	\( T^{2} - 60T + 3600 \)