Newform orbit 260.2.i.b

• Label: 260.2.i.b
• Level: $260$
• Weight: $2$
• Character orbit: 260.i
• Analytic conductor: $2.076$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 260 = 2^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	260.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.07611045255\)
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + (\zeta_{6} - 1) q^{3} + q^{5} + \zeta_{6} q^{7} + 2 \zeta_{6} q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 2 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/260\mathbb{Z}\right)^\times\).

\(n\)	\(41\)	\(131\)	\(157\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

61.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

−0.500000

+

0.866025i

0

1.00000

0

0.500000

+

0.866025i

0

1.00000

+

1.73205i

0

81.1

0

−0.500000

−

0.866025i

0

1.00000

0

0.500000

−

0.866025i

0

1.00000

−

1.73205i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	260.2.i.b	✓	2
3.b	odd	2	1		2340.2.q.b		2
4.b	odd	2	1		1040.2.q.j		2
5.b	even	2	1		1300.2.i.e		2
5.c	odd	4	2		1300.2.bb.a		4
13.c	even	3	1	inner	260.2.i.b	✓	2
13.c	even	3	1		3380.2.a.h		1
13.e	even	6	1		3380.2.a.g		1
13.f	odd	12	2		3380.2.f.e		2
39.i	odd	6	1		2340.2.q.b		2
52.j	odd	6	1		1040.2.q.j		2
65.n	even	6	1		1300.2.i.e		2
65.q	odd	12	2		1300.2.bb.a		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
260.2.i.b	✓	2	1.a	even	1	1	trivial
260.2.i.b	✓	2	13.c	even	3	1	inner
1040.2.q.j		2	4.b	odd	2	1
1040.2.q.j		2	52.j	odd	6	1
1300.2.i.e		2	5.b	even	2	1
1300.2.i.e		2	65.n	even	6	1
1300.2.bb.a		4	5.c	odd	4	2
1300.2.bb.a		4	65.q	odd	12	2
2340.2.q.b		2	3.b	odd	2	1
2340.2.q.b		2	39.i	odd	6	1
3380.2.a.g		1	13.e	even	6	1
3380.2.a.h		1	13.c	even	3	1
3380.2.f.e		2	13.f	odd	12	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(260, [\chi])\):

\( T_{3}^{2} + T_{3} + 1 \)

\( T_{19}^{2} - 7T_{19} + 49 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + T + 1 \)
$5$	\( (T - 1)^{2} \)
$7$	\( T^{2} - T + 1 \)
$11$	\( T^{2} + 3T + 9 \)