Newform orbit 588.3.g.b

• Label: 588.3.g.b
• Level: $588$
• Weight: $3$
• Character orbit: 588.g
• Analytic conductor: $16.022$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(16.0218395444\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 12)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + ( - \beta - 1) q^{2} - \beta q^{3} + (2 \beta - 2) q^{4} + 2 q^{5} + (\beta - 3) q^{6} + 8 q^{8} - 3 q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 4 q^{4} + 4 q^{5} - 6 q^{6} + 16 q^{8} - 6 q^{9}+O(q^{10}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

295.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−1.00000

−

1.73205i

−

1.73205i

−2.00000

+

3.46410i

2.00000

−3.00000

+

1.73205i

0

8.00000

−3.00000

−2.00000

−

3.46410i

295.2

−1.00000

+

1.73205i

1.73205i

−2.00000

−

3.46410i

2.00000

−3.00000

−

1.73205i

0

8.00000

−3.00000

−2.00000

+

3.46410i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	588.3.g.b		2
4.b	odd	2	1	inner	588.3.g.b		2
7.b	odd	2	1		12.3.d.a	✓	2
21.c	even	2	1		36.3.d.c		2
28.d	even	2	1		12.3.d.a	✓	2
35.c	odd	2	1		300.3.c.b		2
35.f	even	4	2		300.3.f.a		4
56.e	even	2	1		192.3.g.b		2
56.h	odd	2	1		192.3.g.b		2
63.l	odd	6	1		324.3.f.d		2
63.l	odd	6	1		324.3.f.j		2
63.o	even	6	1		324.3.f.a		2
63.o	even	6	1		324.3.f.g		2
84.h	odd	2	1		36.3.d.c		2
105.g	even	2	1		900.3.c.e		2
105.k	odd	4	2		900.3.f.c		4
112.j	even	4	2		768.3.b.c		4
112.l	odd	4	2		768.3.b.c		4
140.c	even	2	1		300.3.c.b		2
140.j	odd	4	2		300.3.f.a		4
168.e	odd	2	1		576.3.g.e		2
168.i	even	2	1		576.3.g.e		2
252.s	odd	6	1		324.3.f.a		2
252.s	odd	6	1		324.3.f.g		2
252.bi	even	6	1		324.3.f.d		2
252.bi	even	6	1		324.3.f.j		2
336.v	odd	4	2		2304.3.b.l		4
336.y	even	4	2		2304.3.b.l		4
420.o	odd	2	1		900.3.c.e		2
420.w	even	4	2		900.3.f.c		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
12.3.d.a	✓	2	7.b	odd	2	1
12.3.d.a	✓	2	28.d	even	2	1
36.3.d.c		2	21.c	even	2	1
36.3.d.c		2	84.h	odd	2	1
192.3.g.b		2	56.e	even	2	1
192.3.g.b		2	56.h	odd	2	1
300.3.c.b		2	35.c	odd	2	1
300.3.c.b		2	140.c	even	2	1
300.3.f.a		4	35.f	even	4	2
300.3.f.a		4	140.j	odd	4	2
324.3.f.a		2	63.o	even	6	1
324.3.f.a		2	252.s	odd	6	1
324.3.f.d		2	63.l	odd	6	1
324.3.f.d		2	252.bi	even	6	1
324.3.f.g		2	63.o	even	6	1
324.3.f.g		2	252.s	odd	6	1
324.3.f.j		2	63.l	odd	6	1
324.3.f.j		2	252.bi	even	6	1
576.3.g.e		2	168.e	odd	2	1
576.3.g.e		2	168.i	even	2	1
588.3.g.b		2	1.a	even	1	1	trivial
588.3.g.b		2	4.b	odd	2	1	inner
768.3.b.c		4	112.j	even	4	2
768.3.b.c		4	112.l	odd	4	2
900.3.c.e		2	105.g	even	2	1
900.3.c.e		2	420.o	odd	2	1
900.3.f.c		4	105.k	odd	4	2
900.3.f.c		4	420.w	even	4	2
2304.3.b.l		4	336.v	odd	4	2
2304.3.b.l		4	336.y	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 2 \) acting on \(S_{3}^{\mathrm{new}}(588, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 2T + 4 \)
$3$	\( T^{2} + 3 \)
$5$	\( (T - 2)^{2} \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 48 \)