Newform orbit 588.3.m.b

• Label: 588.3.m.b
• Level: $588$
• Weight: $3$
• Character orbit: 588.m
• 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.m (of order \(6\), degree \(2\), not 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:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + ( - \zeta_{6} - 1) q^{3} + ( - 4 \zeta_{6} + 8) q^{5} + 3 \zeta_{6} q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} + 12 q^{5} + 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(197\)	\(295\)	\(493\)
\(\chi(n)\)	\(1\)	\(1\)	\(\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

313.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

−1.50000

+

0.866025i

0

6.00000

+

3.46410i

0

0

0

1.50000

−

2.59808i

0

325.1

0

−1.50000

−

0.866025i

0

6.00000

−

3.46410i

0

0

0

1.50000

+

2.59808i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	588.3.m.b		2
3.b	odd	2	1		1764.3.z.a		2
7.b	odd	2	1		588.3.m.c		2
7.c	even	3	1		84.3.d.a	✓	2
7.c	even	3	1		588.3.m.c		2
7.d	odd	6	1		84.3.d.a	✓	2
7.d	odd	6	1	inner	588.3.m.b		2
21.c	even	2	1		1764.3.z.g		2
21.g	even	6	1		252.3.d.a		2
21.g	even	6	1		1764.3.z.a		2
21.h	odd	6	1		252.3.d.a		2
21.h	odd	6	1		1764.3.z.g		2
28.f	even	6	1		336.3.f.b		2
28.g	odd	6	1		336.3.f.b		2
35.i	odd	6	1		2100.3.j.c		2
35.j	even	6	1		2100.3.j.c		2
35.k	even	12	2		2100.3.p.b		4
35.l	odd	12	2		2100.3.p.b		4
56.j	odd	6	1		1344.3.f.a		2
56.k	odd	6	1		1344.3.f.d		2
56.m	even	6	1		1344.3.f.d		2
56.p	even	6	1		1344.3.f.a		2
84.j	odd	6	1		1008.3.f.f		2
84.n	even	6	1		1008.3.f.f		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
84.3.d.a	✓	2	7.c	even	3	1
84.3.d.a	✓	2	7.d	odd	6	1
252.3.d.a		2	21.g	even	6	1
252.3.d.a		2	21.h	odd	6	1
336.3.f.b		2	28.f	even	6	1
336.3.f.b		2	28.g	odd	6	1
588.3.m.b		2	1.a	even	1	1	trivial
588.3.m.b		2	7.d	odd	6	1	inner
588.3.m.c		2	7.b	odd	2	1
588.3.m.c		2	7.c	even	3	1
1008.3.f.f		2	84.j	odd	6	1
1008.3.f.f		2	84.n	even	6	1
1344.3.f.a		2	56.j	odd	6	1
1344.3.f.a		2	56.p	even	6	1
1344.3.f.d		2	56.k	odd	6	1
1344.3.f.d		2	56.m	even	6	1
1764.3.z.a		2	3.b	odd	2	1
1764.3.z.a		2	21.g	even	6	1
1764.3.z.g		2	21.c	even	2	1
1764.3.z.g		2	21.h	odd	6	1
2100.3.j.c		2	35.i	odd	6	1
2100.3.j.c		2	35.j	even	6	1
2100.3.p.b		4	35.k	even	12	2
2100.3.p.b		4	35.l	odd	12	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 12T_{5} + 48 \) acting on \(S_{3}^{\mathrm{new}}(588, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 3T + 3 \)
$5$	\( T^{2} - 12T + 48 \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 18T + 324 \)