Newform orbit 588.3.c.c

• Label: 588.3.c.c
• Level: $588$
• Weight: $3$
• Character orbit: 588.c
• Self dual: yes
• Analytic conductor: $16.022$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -3
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	588.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(16.0218395444\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 12)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\)

\(=\)

\( q + 3 q^{3} + 9 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(295\)	\(493\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

197.1

0

0

3.00000

0

0

0

0

0

9.00000

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	588.3.c.c		1
3.b	odd	2	1	CM	588.3.c.c		1
7.b	odd	2	1		12.3.c.a	✓	1
7.c	even	3	2		588.3.p.b		2
7.d	odd	6	2		588.3.p.c		2
21.c	even	2	1		12.3.c.a	✓	1
21.g	even	6	2		588.3.p.c		2
21.h	odd	6	2		588.3.p.b		2
28.d	even	2	1		48.3.e.a		1
35.c	odd	2	1		300.3.g.b		1
35.f	even	4	2		300.3.b.a		2
56.e	even	2	1		192.3.e.a		1
56.h	odd	2	1		192.3.e.b		1
63.l	odd	6	2		324.3.g.b		2
63.o	even	6	2		324.3.g.b		2
77.b	even	2	1		1452.3.e.b		1
84.h	odd	2	1		48.3.e.a		1
105.g	even	2	1		300.3.g.b		1
105.k	odd	4	2		300.3.b.a		2
112.j	even	4	2		768.3.h.b		2
112.l	odd	4	2		768.3.h.a		2
140.c	even	2	1		1200.3.l.b		1
140.j	odd	4	2		1200.3.c.c		2
168.e	odd	2	1		192.3.e.a		1
168.i	even	2	1		192.3.e.b		1
231.h	odd	2	1		1452.3.e.b		1
252.s	odd	6	2		1296.3.q.b		2
252.bi	even	6	2		1296.3.q.b		2
336.v	odd	4	2		768.3.h.b		2
336.y	even	4	2		768.3.h.a		2
420.o	odd	2	1		1200.3.l.b		1
420.w	even	4	2		1200.3.c.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
12.3.c.a	✓	1	7.b	odd	2	1
12.3.c.a	✓	1	21.c	even	2	1
48.3.e.a		1	28.d	even	2	1
48.3.e.a		1	84.h	odd	2	1
192.3.e.a		1	56.e	even	2	1
192.3.e.a		1	168.e	odd	2	1
192.3.e.b		1	56.h	odd	2	1
192.3.e.b		1	168.i	even	2	1
300.3.b.a		2	35.f	even	4	2
300.3.b.a		2	105.k	odd	4	2
300.3.g.b		1	35.c	odd	2	1
300.3.g.b		1	105.g	even	2	1
324.3.g.b		2	63.l	odd	6	2
324.3.g.b		2	63.o	even	6	2
588.3.c.c		1	1.a	even	1	1	trivial
588.3.c.c		1	3.b	odd	2	1	CM
588.3.p.b		2	7.c	even	3	2
588.3.p.b		2	21.h	odd	6	2
588.3.p.c		2	7.d	odd	6	2
588.3.p.c		2	21.g	even	6	2
768.3.h.a		2	112.l	odd	4	2
768.3.h.a		2	336.y	even	4	2
768.3.h.b		2	112.j	even	4	2
768.3.h.b		2	336.v	odd	4	2
1200.3.c.c		2	140.j	odd	4	2
1200.3.c.c		2	420.w	even	4	2
1200.3.l.b		1	140.c	even	2	1
1200.3.l.b		1	420.o	odd	2	1
1296.3.q.b		2	252.s	odd	6	2
1296.3.q.b		2	252.bi	even	6	2
1452.3.e.b		1	77.b	even	2	1
1452.3.e.b		1	231.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(588, [\chi])\):

\( T_{5} \)

\( T_{13} - 22 \)

Hecke characteristic polynomials

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