Newform orbit 162.10.c.i

• Label: 162.10.c.i
• Level: $162$
• Weight: $10$
• Character orbit: 162.c
• Analytic conductor: $83.436$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	162.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(83.4358054585\)
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.

\(f(q)\)	\(=\)	\( q + ( - 16 \zeta_{6} + 16) q^{2} - 256 \zeta_{6} q^{4} + 870 \zeta_{6} q^{5} + ( - 952 \zeta_{6} + 952) q^{7} - 4096 q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 16 q^{2} - 256 q^{4} + 870 q^{5} + 952 q^{7} - 8192 q^{8}+O(q^{10}) \)

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

8.00000

−

13.8564i

0

−128.000

−

221.703i

435.000

+

753.442i

0

476.000

−

824.456i

−4096.00

0

13920.0

109.1

8.00000

+

13.8564i

0

−128.000

+

221.703i

435.000

−

753.442i

0

476.000

+

824.456i

−4096.00

0

13920.0

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	162.10.c.i		2
3.b	odd	2	1		162.10.c.b		2
9.c	even	3	1		18.10.a.a		1
9.c	even	3	1	inner	162.10.c.i		2
9.d	odd	6	1		2.10.a.a	✓	1
9.d	odd	6	1		162.10.c.b		2
36.f	odd	6	1		144.10.a.d		1
36.h	even	6	1		16.10.a.d		1
45.h	odd	6	1		50.10.a.c		1
45.l	even	12	2		50.10.b.a		2
63.i	even	6	1		98.10.c.b		2
63.j	odd	6	1		98.10.c.c		2
63.n	odd	6	1		98.10.c.c		2
63.o	even	6	1		98.10.a.c		1
63.s	even	6	1		98.10.c.b		2
72.j	odd	6	1		64.10.a.h		1
72.l	even	6	1		64.10.a.b		1
99.g	even	6	1		242.10.a.a		1
117.n	odd	6	1		338.10.a.a		1
144.u	even	12	2		256.10.b.e		2
144.w	odd	12	2		256.10.b.g		2
180.n	even	6	1		400.10.a.b		1
180.v	odd	12	2		400.10.c.d		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2.10.a.a	✓	1	9.d	odd	6	1
16.10.a.d		1	36.h	even	6	1
18.10.a.a		1	9.c	even	3	1
50.10.a.c		1	45.h	odd	6	1
50.10.b.a		2	45.l	even	12	2
64.10.a.b		1	72.l	even	6	1
64.10.a.h		1	72.j	odd	6	1
98.10.a.c		1	63.o	even	6	1
98.10.c.b		2	63.i	even	6	1
98.10.c.b		2	63.s	even	6	1
98.10.c.c		2	63.j	odd	6	1
98.10.c.c		2	63.n	odd	6	1
144.10.a.d		1	36.f	odd	6	1
162.10.c.b		2	3.b	odd	2	1
162.10.c.b		2	9.d	odd	6	1
162.10.c.i		2	1.a	even	1	1	trivial
162.10.c.i		2	9.c	even	3	1	inner
242.10.a.a		1	99.g	even	6	1
256.10.b.e		2	144.u	even	12	2
256.10.b.g		2	144.w	odd	12	2
338.10.a.a		1	117.n	odd	6	1
400.10.a.b		1	180.n	even	6	1
400.10.c.d		2	180.v	odd	12	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 870T_{5} + 756900 \) acting on \(S_{10}^{\mathrm{new}}(162, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 16T + 256 \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 870T + 756900 \)
$7$	\( T^{2} - 952T + 906304 \)
$11$	T^2 + 56148*T + 3152597904