Newform orbit 289.2.b.a

• Label: 289.2.b.a
• Level: $289$
• Weight: $2$
• Character orbit: 289.b
• Analytic conductor: $2.308$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 289 = 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	289.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.30767661842\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-1}) \)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 17)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + q^{2} - q^{4} + \beta q^{5} + 2 \beta q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} + 6 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/289\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(-1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

288.1

	−	1.00000i
		1.00000i

1.00000

0

−1.00000

−

2.00000i

0

−

4.00000i

−3.00000

3.00000

−

2.00000i

288.2

1.00000

0

−1.00000

2.00000i

0

4.00000i

−3.00000

3.00000

2.00000i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	289.2.b.a		2
17.b	even	2	1	inner	289.2.b.a		2
17.c	even	4	1		17.2.a.a	✓	1
17.c	even	4	1		289.2.a.a		1
17.d	even	8	4		289.2.c.a		4
17.e	odd	16	8		289.2.d.d		8
51.f	odd	4	1		153.2.a.c		1
51.f	odd	4	1		2601.2.a.g		1
68.f	odd	4	1		272.2.a.b		1
68.f	odd	4	1		4624.2.a.d		1
85.f	odd	4	1		425.2.b.b		2
85.i	odd	4	1		425.2.b.b		2
85.j	even	4	1		425.2.a.d		1
85.j	even	4	1		7225.2.a.g		1
119.f	odd	4	1		833.2.a.a		1
119.m	odd	12	2		833.2.e.a		2
119.n	even	12	2		833.2.e.b		2
136.i	even	4	1		1088.2.a.i		1
136.j	odd	4	1		1088.2.a.h		1
187.f	odd	4	1		2057.2.a.e		1
204.l	even	4	1		2448.2.a.o		1
221.k	even	4	1		2873.2.a.c		1
255.i	odd	4	1		3825.2.a.d		1
323.g	odd	4	1		6137.2.a.b		1
340.n	odd	4	1		6800.2.a.n		1
357.l	even	4	1		7497.2.a.l		1
391.f	odd	4	1		8993.2.a.a		1
408.q	even	4	1		9792.2.a.i		1
408.t	odd	4	1		9792.2.a.n		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
17.2.a.a	✓	1	17.c	even	4	1
153.2.a.c		1	51.f	odd	4	1
272.2.a.b		1	68.f	odd	4	1
289.2.a.a		1	17.c	even	4	1
289.2.b.a		2	1.a	even	1	1	trivial
289.2.b.a		2	17.b	even	2	1	inner
289.2.c.a		4	17.d	even	8	4
289.2.d.d		8	17.e	odd	16	8
425.2.a.d		1	85.j	even	4	1
425.2.b.b		2	85.f	odd	4	1
425.2.b.b		2	85.i	odd	4	1
833.2.a.a		1	119.f	odd	4	1
833.2.e.a		2	119.m	odd	12	2
833.2.e.b		2	119.n	even	12	2
1088.2.a.h		1	136.j	odd	4	1
1088.2.a.i		1	136.i	even	4	1
2057.2.a.e		1	187.f	odd	4	1
2448.2.a.o		1	204.l	even	4	1
2601.2.a.g		1	51.f	odd	4	1
2873.2.a.c		1	221.k	even	4	1
3825.2.a.d		1	255.i	odd	4	1
4624.2.a.d		1	68.f	odd	4	1
6137.2.a.b		1	323.g	odd	4	1
6800.2.a.n		1	340.n	odd	4	1
7225.2.a.g		1	85.j	even	4	1
7497.2.a.l		1	357.l	even	4	1
8993.2.a.a		1	391.f	odd	4	1
9792.2.a.i		1	408.q	even	4	1
9792.2.a.n		1	408.t	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(289, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T - 1)^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 4 \)
$7$	\( T^{2} + 16 \)
$11$	\( T^{2} \)