Newform orbit 361.2.c.b

• Label: 361.2.c.b
• Level: $361$
• Weight: $2$
• Character orbit: 361.c
• Analytic conductor: $2.883$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -19
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 361 = 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	361.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.88259951297\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\)	\(=\)	\( q + 2 \zeta_{6} q^{4} + ( - \zeta_{6} + 1) q^{5} + 3 q^{7} + 3 \zeta_{6} q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + q^{5} + 6 q^{7} + 3 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/361\mathbb{Z}\right)^\times\).

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

68.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

0

1.00000

+

1.73205i

0.500000

−

0.866025i

0

3.00000

0

1.50000

+

2.59808i

0

292.1

0

0

1.00000

−

1.73205i

0.500000

+

0.866025i

0

3.00000

0

1.50000

−

2.59808i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.b	odd	2	1	CM by \(\Q(\sqrt{-19}) \)
19.c	even	3	1	inner
19.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	361.2.c.b		2
19.b	odd	2	1	CM	361.2.c.b		2
19.c	even	3	1		361.2.a.a	✓	1
19.c	even	3	1	inner	361.2.c.b		2
19.d	odd	6	1		361.2.a.a	✓	1
19.d	odd	6	1	inner	361.2.c.b		2
19.e	even	9	6		361.2.e.c		6
19.f	odd	18	6		361.2.e.c		6
57.f	even	6	1		3249.2.a.e		1
57.h	odd	6	1		3249.2.a.e		1
76.f	even	6	1		5776.2.a.i		1
76.g	odd	6	1		5776.2.a.i		1
95.h	odd	6	1		9025.2.a.f		1
95.i	even	6	1		9025.2.a.f		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
361.2.a.a	✓	1	19.c	even	3	1
361.2.a.a	✓	1	19.d	odd	6	1
361.2.c.b		2	1.a	even	1	1	trivial
361.2.c.b		2	19.b	odd	2	1	CM
361.2.c.b		2	19.c	even	3	1	inner
361.2.c.b		2	19.d	odd	6	1	inner
361.2.e.c		6	19.e	even	9	6
361.2.e.c		6	19.f	odd	18	6
3249.2.a.e		1	57.f	even	6	1
3249.2.a.e		1	57.h	odd	6	1
5776.2.a.i		1	76.f	even	6	1
5776.2.a.i		1	76.g	odd	6	1
9025.2.a.f		1	95.h	odd	6	1
9025.2.a.f		1	95.i	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(361, [\chi])\):

\( T_{2} \)

\( T_{3} \)

Hecke characteristic polynomials

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