Newform orbit 243.2.c.b

• Label: 243.2.c.b
• Level: $243$
• Weight: $2$
• Character orbit: 243.c
• Analytic conductor: $1.940$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 243 = 3^{5} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	243.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.94036476912\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{3}]$

$q$-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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/243\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(-\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

82.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

0

1.00000

−

1.73205i

0

0

2.00000

+

3.46410i

0

0

0

163.1

0

0

1.00000

+

1.73205i

0

0

2.00000

−

3.46410i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
9.c	even	3	1	inner
9.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	243.2.c.b		2
3.b	odd	2	1	CM	243.2.c.b		2
9.c	even	3	1		243.2.a.a	✓	1
9.c	even	3	1	inner	243.2.c.b		2
9.d	odd	6	1		243.2.a.a	✓	1
9.d	odd	6	1	inner	243.2.c.b		2
27.e	even	9	6		729.2.e.e		6
27.f	odd	18	6		729.2.e.e		6
36.f	odd	6	1		3888.2.a.n		1
36.h	even	6	1		3888.2.a.n		1
45.h	odd	6	1		6075.2.a.w		1
45.j	even	6	1		6075.2.a.w		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
243.2.a.a	✓	1	9.c	even	3	1
243.2.a.a	✓	1	9.d	odd	6	1
243.2.c.b		2	1.a	even	1	1	trivial
243.2.c.b		2	3.b	odd	2	1	CM
243.2.c.b		2	9.c	even	3	1	inner
243.2.c.b		2	9.d	odd	6	1	inner
729.2.e.e		6	27.e	even	9	6
729.2.e.e		6	27.f	odd	18	6
3888.2.a.n		1	36.f	odd	6	1
3888.2.a.n		1	36.h	even	6	1
6075.2.a.w		1	45.h	odd	6	1
6075.2.a.w		1	45.j	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(243, [\chi])\):

\( T_{2} \)

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

Hecke characteristic polynomials

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