Newform orbit 240.2.o.b

• Label: 240.2.o.b
• Level: $240$
• Weight: $2$
• Character orbit: 240.o
• Analytic conductor: $1.916$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	240.o (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.91640964851\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\Q(\zeta_{24})\)
Defining polynomial:	\( x^{8} - x^{4} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{10}\cdot 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta_{2} q^{3} - \beta_1 q^{5} + (\beta_{4} - \beta_1 + 1) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9}+O(q^{10}) \)

Basis of coefficient ring

\(\beta_{1}\)	\(=\)	\( -\zeta_{24}^{6} - \zeta_{24}^{5} - \zeta_{24}^{3} + 2\zeta_{24}^{2} + \zeta_{24} \)
\(\beta_{2}\)	\(=\)	\( \zeta_{24}^{6} - \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \)
\(\beta_{3}\)	\(=\)	\( 4\zeta_{24}^{4} - 2 \)
\(\beta_{4}\)	\(=\)	\( -\zeta_{24}^{6} + \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24}^{2} - \zeta_{24} \)
\(\beta_{5}\)	\(=\)	\( 2\zeta_{24}^{7} - 2\zeta_{24}^{6} - 2\zeta_{24}^{5} + 2\zeta_{24}^{4} - 1 \)
\(\beta_{6}\)	\(=\)	\( 4\zeta_{24}^{7} + 2\zeta_{24}^{5} - 2\zeta_{24}^{3} + 2\zeta_{24} \)
\(\beta_{7}\)	\(=\)	\( -2\zeta_{24}^{7} - 2\zeta_{24}^{6} - 2\zeta_{24}^{4} + 2\zeta_{24}^{3} + 2\zeta_{24} + 1 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(97\)	\(161\)	\(181\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

239.1

−0.965926	+	0.258819i
0.258819	−	0.965926i
−0.965926	−	0.258819i
0.258819	+	0.965926i
0.965926	−	0.258819i
−0.258819	+	0.965926i
0.965926	+	0.258819i
−0.258819	−	0.965926i

0

−1.41421

−

1.00000i

0

−1.73205

+

1.41421i

0

0

0

1.00000

+

2.82843i

0

239.2

0

−1.41421

−

1.00000i

0

1.73205

+

1.41421i

0

0

0

1.00000

+

2.82843i

0

239.3

0

−1.41421

+

1.00000i

0

−1.73205

−

1.41421i

0

0

0

1.00000

−

2.82843i

0

239.4

0

−1.41421

+

1.00000i

0

1.73205

−

1.41421i

0

0

0

1.00000

−

2.82843i

0

239.5

0

1.41421

−

1.00000i

0

−1.73205

−

1.41421i

0

0

0

1.00000

−

2.82843i

0

239.6

0

1.41421

−

1.00000i

0

1.73205

−

1.41421i

0

0

0

1.00000

−

2.82843i

0

239.7

0

1.41421

+

1.00000i

0

−1.73205

+

1.41421i

0

0

0

1.00000

+

2.82843i

0

239.8

0

1.41421

+

1.00000i

0

1.73205

+

1.41421i

0

0

0

1.00000

+

2.82843i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
5.b	even	2	1	inner
12.b	even	2	1	inner
15.d	odd	2	1	inner
20.d	odd	2	1	inner
60.h	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	240.2.o.b	✓	8
3.b	odd	2	1	inner	240.2.o.b	✓	8
4.b	odd	2	1	inner	240.2.o.b	✓	8
5.b	even	2	1	inner	240.2.o.b	✓	8
5.c	odd	4	1		1200.2.h.j		4
5.c	odd	4	1		1200.2.h.n		4
8.b	even	2	1		960.2.o.d		8
8.d	odd	2	1		960.2.o.d		8
12.b	even	2	1	inner	240.2.o.b	✓	8
15.d	odd	2	1	inner	240.2.o.b	✓	8
15.e	even	4	1		1200.2.h.j		4
15.e	even	4	1		1200.2.h.n		4
20.d	odd	2	1	inner	240.2.o.b	✓	8
20.e	even	4	1		1200.2.h.j		4
20.e	even	4	1		1200.2.h.n		4
24.f	even	2	1		960.2.o.d		8
24.h	odd	2	1		960.2.o.d		8
40.e	odd	2	1		960.2.o.d		8
40.f	even	2	1		960.2.o.d		8
60.h	even	2	1	inner	240.2.o.b	✓	8
60.l	odd	4	1		1200.2.h.j		4
60.l	odd	4	1		1200.2.h.n		4
120.i	odd	2	1		960.2.o.d		8
120.m	even	2	1		960.2.o.d		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
240.2.o.b	✓	8	1.a	even	1	1	trivial
240.2.o.b	✓	8	3.b	odd	2	1	inner
240.2.o.b	✓	8	4.b	odd	2	1	inner
240.2.o.b	✓	8	5.b	even	2	1	inner
240.2.o.b	✓	8	12.b	even	2	1	inner
240.2.o.b	✓	8	15.d	odd	2	1	inner
240.2.o.b	✓	8	20.d	odd	2	1	inner
240.2.o.b	✓	8	60.h	even	2	1	inner
960.2.o.d		8	8.b	even	2	1
960.2.o.d		8	8.d	odd	2	1
960.2.o.d		8	24.f	even	2	1
960.2.o.d		8	24.h	odd	2	1
960.2.o.d		8	40.e	odd	2	1
960.2.o.d		8	40.f	even	2	1
960.2.o.d		8	120.i	odd	2	1
960.2.o.d		8	120.m	even	2	1
1200.2.h.j		4	5.c	odd	4	1
1200.2.h.j		4	15.e	even	4	1
1200.2.h.j		4	20.e	even	4	1
1200.2.h.j		4	60.l	odd	4	1
1200.2.h.n		4	5.c	odd	4	1
1200.2.h.n		4	15.e	even	4	1
1200.2.h.n		4	20.e	even	4	1
1200.2.h.n		4	60.l	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} \) acting on \(S_{2}^{\mathrm{new}}(240, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{8} \)
$3$	\( (T^{4} - 2 T^{2} + 9)^{2} \)
$5$	\( (T^{4} - 2 T^{2} + 25)^{2} \)
$7$	\( T^{8} \)
$11$	\( (T^{2} - 24)^{4} \)