Newform orbit 364.2.k.b

• Label: 364.2.k.b
• Level: $364$
• Weight: $2$
• Character orbit: 364.k
• Analytic conductor: $2.907$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 364 = 2^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	364.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.90655463357\)
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:	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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/364\mathbb{Z}\right)^\times\).

\(n\)	\(157\)	\(183\)	\(197\)
\(\chi(n)\)	\(1\)	\(1\)	\(-\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

29.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

0

0

3.00000

0

−0.500000

+

0.866025i

0

1.50000

−

2.59808i

0

113.1

0

0

0

3.00000

0

−0.500000

−

0.866025i

0

1.50000

+

2.59808i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	364.2.k.b	✓	2
3.b	odd	2	1		3276.2.z.a		2
4.b	odd	2	1		1456.2.s.d		2
7.b	odd	2	1		2548.2.k.b		2
7.c	even	3	1		2548.2.i.c		2
7.c	even	3	1		2548.2.l.c		2
7.d	odd	6	1		2548.2.i.f		2
7.d	odd	6	1		2548.2.l.f		2
13.c	even	3	1	inner	364.2.k.b	✓	2
13.c	even	3	1		4732.2.a.f		1
13.e	even	6	1		4732.2.a.b		1
13.f	odd	12	2		4732.2.g.d		2
39.i	odd	6	1		3276.2.z.a		2
52.j	odd	6	1		1456.2.s.d		2
91.g	even	3	1		2548.2.i.c		2
91.h	even	3	1		2548.2.l.c		2
91.m	odd	6	1		2548.2.i.f		2
91.n	odd	6	1		2548.2.k.b		2
91.v	odd	6	1		2548.2.l.f		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
364.2.k.b	✓	2	1.a	even	1	1	trivial
364.2.k.b	✓	2	13.c	even	3	1	inner
1456.2.s.d		2	4.b	odd	2	1
1456.2.s.d		2	52.j	odd	6	1
2548.2.i.c		2	7.c	even	3	1
2548.2.i.c		2	91.g	even	3	1
2548.2.i.f		2	7.d	odd	6	1
2548.2.i.f		2	91.m	odd	6	1
2548.2.k.b		2	7.b	odd	2	1
2548.2.k.b		2	91.n	odd	6	1
2548.2.l.c		2	7.c	even	3	1
2548.2.l.c		2	91.h	even	3	1
2548.2.l.f		2	7.d	odd	6	1
2548.2.l.f		2	91.v	odd	6	1
3276.2.z.a		2	3.b	odd	2	1
3276.2.z.a		2	39.i	odd	6	1
4732.2.a.b		1	13.e	even	6	1
4732.2.a.f		1	13.c	even	3	1
4732.2.g.d		2	13.f	odd	12	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(364, [\chi])\):

\( T_{3} \)

\( T_{5} - 3 \)

Hecke characteristic polynomials

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