Newform orbit 288.3.g.b

• Label: 288.3.g.b
• Level: $288$
• Weight: $3$
• Character orbit: 288.g
• Analytic conductor: $7.847$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	288.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.84743161358\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-1}) \)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 32)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4i\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\)	\(37\)	\(65\)	\(127\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

127.1

	−	1.00000i
		1.00000i

0

0

0

−2.00000

0

−

8.00000i

0

0

0

127.2

0

0

0

−2.00000

0

8.00000i

0

0

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	288.3.g.b		2
3.b	odd	2	1		32.3.c.a	✓	2
4.b	odd	2	1	inner	288.3.g.b		2
8.b	even	2	1		576.3.g.g		2
8.d	odd	2	1		576.3.g.g		2
12.b	even	2	1		32.3.c.a	✓	2
15.d	odd	2	1		800.3.b.a		2
15.e	even	4	1		800.3.h.a		2
15.e	even	4	1		800.3.h.b		2
16.e	even	4	1		2304.3.b.c		2
16.e	even	4	1		2304.3.b.g		2
16.f	odd	4	1		2304.3.b.c		2
16.f	odd	4	1		2304.3.b.g		2
21.c	even	2	1		1568.3.d.b		2
24.f	even	2	1		64.3.c.b		2
24.h	odd	2	1		64.3.c.b		2
48.i	odd	4	1		256.3.d.a		2
48.i	odd	4	1		256.3.d.c		2
48.k	even	4	1		256.3.d.a		2
48.k	even	4	1		256.3.d.c		2
60.h	even	2	1		800.3.b.a		2
60.l	odd	4	1		800.3.h.a		2
60.l	odd	4	1		800.3.h.b		2
84.h	odd	2	1		1568.3.d.b		2
120.i	odd	2	1		1600.3.b.e		2
120.m	even	2	1		1600.3.b.e		2
120.q	odd	4	1		1600.3.h.a		2
120.q	odd	4	1		1600.3.h.c		2
120.w	even	4	1		1600.3.h.a		2
120.w	even	4	1		1600.3.h.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
32.3.c.a	✓	2	3.b	odd	2	1
32.3.c.a	✓	2	12.b	even	2	1
64.3.c.b		2	24.f	even	2	1
64.3.c.b		2	24.h	odd	2	1
256.3.d.a		2	48.i	odd	4	1
256.3.d.a		2	48.k	even	4	1
256.3.d.c		2	48.i	odd	4	1
256.3.d.c		2	48.k	even	4	1
288.3.g.b		2	1.a	even	1	1	trivial
288.3.g.b		2	4.b	odd	2	1	inner
576.3.g.g		2	8.b	even	2	1
576.3.g.g		2	8.d	odd	2	1
800.3.b.a		2	15.d	odd	2	1
800.3.b.a		2	60.h	even	2	1
800.3.h.a		2	15.e	even	4	1
800.3.h.a		2	60.l	odd	4	1
800.3.h.b		2	15.e	even	4	1
800.3.h.b		2	60.l	odd	4	1
1568.3.d.b		2	21.c	even	2	1
1568.3.d.b		2	84.h	odd	2	1
1600.3.b.e		2	120.i	odd	2	1
1600.3.b.e		2	120.m	even	2	1
1600.3.h.a		2	120.q	odd	4	1
1600.3.h.a		2	120.w	even	4	1
1600.3.h.c		2	120.q	odd	4	1
1600.3.h.c		2	120.w	even	4	1
2304.3.b.c		2	16.e	even	4	1
2304.3.b.c		2	16.f	odd	4	1
2304.3.b.g		2	16.e	even	4	1
2304.3.b.g		2	16.f	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 2 \) acting on \(S_{3}^{\mathrm{new}}(288, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( (T + 2)^{2} \)
$7$	\( T^{2} + 64 \)
$11$	\( T^{2} + 16 \)