Newform orbit 360.3.p.b

• Label: 360.3.p.b
• Level: $360$
• Weight: $3$
• Character orbit: 360.p
• Self dual: yes
• Analytic conductor: $9.809$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -40
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	360.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(9.80928951697\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\)

\(=\)

\( q + 2 q^{2} + 4 q^{4} - 5 q^{5} + 6 q^{7} + 8 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\)	\(181\)	\(217\)	\(271\)	\(281\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

19.1

0

2.00000

0

4.00000

−5.00000

0

6.00000

8.00000

0

−10.0000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
40.e	odd	2	1	CM by \(\Q(\sqrt{-10}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	360.3.p.b		1
3.b	odd	2	1		40.3.e.a	✓	1
4.b	odd	2	1		1440.3.p.a		1
5.b	even	2	1		360.3.p.a		1
8.b	even	2	1		1440.3.p.b		1
8.d	odd	2	1		360.3.p.a		1
12.b	even	2	1		160.3.e.b		1
15.d	odd	2	1		40.3.e.b	yes	1
15.e	even	4	2		200.3.g.c		2
20.d	odd	2	1		1440.3.p.b		1
24.f	even	2	1		40.3.e.b	yes	1
24.h	odd	2	1		160.3.e.a		1
40.e	odd	2	1	CM	360.3.p.b		1
40.f	even	2	1		1440.3.p.a		1
48.i	odd	4	2		1280.3.h.b		2
48.k	even	4	2		1280.3.h.c		2
60.h	even	2	1		160.3.e.a		1
60.l	odd	4	2		800.3.g.c		2
120.i	odd	2	1		160.3.e.b		1
120.m	even	2	1		40.3.e.a	✓	1
120.q	odd	4	2		200.3.g.c		2
120.w	even	4	2		800.3.g.c		2
240.t	even	4	2		1280.3.h.b		2
240.bm	odd	4	2		1280.3.h.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
40.3.e.a	✓	1	3.b	odd	2	1
40.3.e.a	✓	1	120.m	even	2	1
40.3.e.b	yes	1	15.d	odd	2	1
40.3.e.b	yes	1	24.f	even	2	1
160.3.e.a		1	24.h	odd	2	1
160.3.e.a		1	60.h	even	2	1
160.3.e.b		1	12.b	even	2	1
160.3.e.b		1	120.i	odd	2	1
200.3.g.c		2	15.e	even	4	2
200.3.g.c		2	120.q	odd	4	2
360.3.p.a		1	5.b	even	2	1
360.3.p.a		1	8.d	odd	2	1
360.3.p.b		1	1.a	even	1	1	trivial
360.3.p.b		1	40.e	odd	2	1	CM
800.3.g.c		2	60.l	odd	4	2
800.3.g.c		2	120.w	even	4	2
1280.3.h.b		2	48.i	odd	4	2
1280.3.h.b		2	240.t	even	4	2
1280.3.h.c		2	48.k	even	4	2
1280.3.h.c		2	240.bm	odd	4	2
1440.3.p.a		1	4.b	odd	2	1
1440.3.p.a		1	40.f	even	2	1
1440.3.p.b		1	8.b	even	2	1
1440.3.p.b		1	20.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(360, [\chi])\):

\( T_{7} - 6 \)

\( T_{23} - 26 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T - 2 \)
$3$	\( T \)
$5$	\( T + 5 \)
$7$	\( T - 6 \)
$11$	\( T - 18 \)