Newform orbit 585.2.c.b

• Label: 585.2.c.b
• Level: $585$
• Weight: $2$
• Character orbit: 585.c
• Analytic conductor: $4.671$
• Analytic rank: $0$
• Dimension: $6$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	585.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.67124851824\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	6.0.350464.1
Defining polynomial:	\(x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2\)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + ( \beta_{3} - \beta_{5} ) q^{2} + ( -2 - \beta_{1} - \beta_{2} ) q^{4} + ( -\beta_{1} + \beta_{4} ) q^{5} + ( -\beta_{3} + \beta_{4} - \beta_{5} ) q^{7} + ( -4 \beta_{3} + 3 \beta_{4} ) q^{8} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 10 q^{4} + O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2\):

\(\beta_{0}\)	\(=\)	\( 1 \)
\(\beta_{1}\)	\(=\)	\((\)\( -\nu^{5} + 8 \nu^{4} - 4 \nu^{3} - \nu^{2} + 2 \nu + 38 \)\()/23\)
\(\beta_{2}\)	\(=\)	\((\)\( -5 \nu^{5} + 17 \nu^{4} - 20 \nu^{3} - 5 \nu^{2} + 10 \nu + 29 \)\()/23\)
\(\beta_{3}\)	\(=\)	\((\)\( 7 \nu^{5} - 10 \nu^{4} + 5 \nu^{3} + 30 \nu^{2} + 32 \nu - 13 \)\()/23\)
\(\beta_{4}\)	\(=\)	\((\)\( -11 \nu^{5} + 19 \nu^{4} - 21 \nu^{3} - 11 \nu^{2} - 70 \nu + 27 \)\()/23\)
\(\beta_{5}\)	\(=\)	\((\)\( -14 \nu^{5} + 20 \nu^{4} - 10 \nu^{3} - 37 \nu^{2} - 64 \nu + 26 \)\()/23\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).

\(n\)	\(326\)	\(352\)	\(496\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

469.1

0.403032	−	0.403032i
−0.854638	+	0.854638i
1.45161	+	1.45161i
1.45161	−	1.45161i
−0.854638	−	0.854638i
0.403032	+	0.403032i

−

2.67513i

0

−5.15633

−1.67513

+

1.48119i

0

0.806063i

8.44358i

0

3.96239

+

4.48119i

469.2

−

1.53919i

0

−0.369102

−0.539189

−

2.17009i

0

−

1.70928i

−

2.51026i

0

−3.34017

+

0.829914i

469.3

−

1.21432i

0

0.525428

2.21432

+

0.311108i

0

−

2.90321i

−

3.06668i

0

0.377784

−

2.68889i

469.4

1.21432i

0

0.525428

2.21432

−

0.311108i

0

2.90321i

3.06668i

0

0.377784

+

2.68889i

469.5

1.53919i

0

−0.369102

−0.539189

+

2.17009i

0

1.70928i

2.51026i

0

−3.34017

−

0.829914i

469.6

2.67513i

0

−5.15633

−1.67513

−

1.48119i

0

−

0.806063i

−

8.44358i

0

3.96239

−

4.48119i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	585.2.c.b		6
3.b	odd	2	1		65.2.b.a	✓	6
5.b	even	2	1	inner	585.2.c.b		6
5.c	odd	4	1		2925.2.a.bf		3
5.c	odd	4	1		2925.2.a.bj		3
12.b	even	2	1		1040.2.d.c		6
15.d	odd	2	1		65.2.b.a	✓	6
15.e	even	4	1		325.2.a.j		3
15.e	even	4	1		325.2.a.k		3
39.d	odd	2	1		845.2.b.c		6
39.f	even	4	1		845.2.d.a		6
39.f	even	4	1		845.2.d.b		6
39.h	odd	6	2		845.2.n.g		12
39.i	odd	6	2		845.2.n.f		12
39.k	even	12	2		845.2.l.d		12
39.k	even	12	2		845.2.l.e		12
60.h	even	2	1		1040.2.d.c		6
60.l	odd	4	1		5200.2.a.cb		3
60.l	odd	4	1		5200.2.a.cj		3
195.e	odd	2	1		845.2.b.c		6
195.n	even	4	1		845.2.d.a		6
195.n	even	4	1		845.2.d.b		6
195.s	even	4	1		4225.2.a.ba		3
195.s	even	4	1		4225.2.a.bh		3
195.x	odd	6	2		845.2.n.f		12
195.y	odd	6	2		845.2.n.g		12
195.bh	even	12	2		845.2.l.d		12
195.bh	even	12	2		845.2.l.e		12

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
65.2.b.a	✓	6	3.b	odd	2	1
65.2.b.a	✓	6	15.d	odd	2	1
325.2.a.j		3	15.e	even	4	1
325.2.a.k		3	15.e	even	4	1
585.2.c.b		6	1.a	even	1	1	trivial
585.2.c.b		6	5.b	even	2	1	inner
845.2.b.c		6	39.d	odd	2	1
845.2.b.c		6	195.e	odd	2	1
845.2.d.a		6	39.f	even	4	1
845.2.d.a		6	195.n	even	4	1
845.2.d.b		6	39.f	even	4	1
845.2.d.b		6	195.n	even	4	1
845.2.l.d		12	39.k	even	12	2
845.2.l.d		12	195.bh	even	12	2
845.2.l.e		12	39.k	even	12	2
845.2.l.e		12	195.bh	even	12	2
845.2.n.f		12	39.i	odd	6	2
845.2.n.f		12	195.x	odd	6	2
845.2.n.g		12	39.h	odd	6	2
845.2.n.g		12	195.y	odd	6	2
1040.2.d.c		6	12.b	even	2	1
1040.2.d.c		6	60.h	even	2	1
2925.2.a.bf		3	5.c	odd	4	1
2925.2.a.bj		3	5.c	odd	4	1
4225.2.a.ba		3	195.s	even	4	1
4225.2.a.bh		3	195.s	even	4	1
5200.2.a.cb		3	60.l	odd	4	1
5200.2.a.cj		3	60.l	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 11 T_{2}^{4} + 31 T_{2}^{2} + 25 \) acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( 25 + 31 T^{2} + 11 T^{4} + T^{6} \)
$3$	\( T^{6} \)
$5$	\( 125 - 5 T^{2} - 16 T^{3} - T^{4} + T^{6} \)
$7$	\( 16 + 32 T^{2} + 12 T^{4} + T^{6} \)
$11$	\( ( 2 + 8 T - 6 T^{2} + T^{3} )^{2} \)