Newform orbit 288.2.i.f

• Label: 288.2.i.f
• Level: $288$
• Weight: $2$
• Character orbit: 288.i
• Analytic conductor: $2.300$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29969157821\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{3})\)
Coefficient field:	8.0.170772624.1
Defining polynomial:	\( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{2}\cdot 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + (\beta_{5} - \beta_1) q^{3} + (\beta_{7} - \beta_{4} + 1) q^{5} + ( - 2 \beta_{5} - \beta_{3} - \beta_{2}) q^{7} + (\beta_{7} + \beta_{6}) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{5} - 6 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 \) :

\(\beta_{1}\)	\(=\)	\( ( -\nu^{7} + \nu^{6} + \nu^{5} + 2\nu^{3} + 4\nu^{2} + 4\nu - 8 ) / 8 \)
\(\beta_{2}\)	\(=\)	\( ( -\nu^{7} - \nu^{6} - \nu^{5} + 6\nu^{4} - 2\nu^{3} + 8\nu^{2} + 4\nu + 8 ) / 8 \)
\(\beta_{3}\)	\(=\)	\( ( -\nu^{7} + \nu^{6} - 3\nu^{5} - 6\nu^{3} + 4\nu + 8 ) / 8 \)
\(\beta_{4}\)	\(=\)	\( ( -3\nu^{7} + 5\nu^{6} - 7\nu^{5} + 6\nu^{4} - 10\nu^{3} + 24\nu^{2} - 28\nu + 32 ) / 8 \)
\(\beta_{5}\)	\(=\)	\( ( -2\nu^{7} + 3\nu^{6} - 5\nu^{5} + 3\nu^{4} - 4\nu^{3} + 16\nu^{2} - 16\nu + 20 ) / 4 \)
\(\beta_{6}\)	\(=\)	\( ( \nu^{7} - 4\nu^{6} + 6\nu^{5} - 5\nu^{4} + 6\nu^{3} - 14\nu^{2} + 20\nu - 24 ) / 4 \)
\(\beta_{7}\)	\(=\)	\( ( 5\nu^{7} - 7\nu^{6} + 13\nu^{5} - 10\nu^{4} + 10\nu^{3} - 28\nu^{2} + 44\nu - 56 ) / 8 \)

Character values

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

\(n\)	\(37\)	\(65\)	\(127\)
\(\chi(n)\)	\(1\)	\(-1 + \beta_{4}\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

97.1

0.335728	−	1.37379i
1.41203	+	0.0786378i
0.774115	+	1.18353i
−1.02187	+	0.977642i
0.335728	+	1.37379i
1.41203	−	0.0786378i
0.774115	−	1.18353i
−1.02187	−	0.977642i

0

−1.35760

+

1.07561i

0

−1.18614

−

2.05446i

0

−1.10489

+

1.91373i

0

0.686141

−

2.92048i

0

97.2

0

−0.637910

−

1.61030i

0

1.68614

+

2.92048i

0

−2.35143

+

4.07279i

0

−2.18614

+

2.05446i

0

97.3

0

0.637910

+

1.61030i

0

1.68614

+

2.92048i

0

2.35143

−

4.07279i

0

−2.18614

+

2.05446i

0

97.4

0

1.35760

−

1.07561i

0

−1.18614

−

2.05446i

0

1.10489

−

1.91373i

0

0.686141

−

2.92048i

0

193.1

0

−1.35760

−

1.07561i

0

−1.18614

+

2.05446i

0

−1.10489

−

1.91373i

0

0.686141

+

2.92048i

0

193.2

0

−0.637910

+

1.61030i

0

1.68614

−

2.92048i

0

−2.35143

−

4.07279i

0

−2.18614

−

2.05446i

0

193.3

0

0.637910

−

1.61030i

0

1.68614

−

2.92048i

0

2.35143

+

4.07279i

0

−2.18614

−

2.05446i

0

193.4

0

1.35760

+

1.07561i

0

−1.18614

+

2.05446i

0

1.10489

+

1.91373i

0

0.686141

+

2.92048i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
9.c	even	3	1	inner
36.f	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	288.2.i.f	✓	8
3.b	odd	2	1		864.2.i.f		8
4.b	odd	2	1	inner	288.2.i.f	✓	8
8.b	even	2	1		576.2.i.n		8
8.d	odd	2	1		576.2.i.n		8
9.c	even	3	1	inner	288.2.i.f	✓	8
9.c	even	3	1		2592.2.a.u		4
9.d	odd	6	1		864.2.i.f		8
9.d	odd	6	1		2592.2.a.x		4
12.b	even	2	1		864.2.i.f		8
24.f	even	2	1		1728.2.i.n		8
24.h	odd	2	1		1728.2.i.n		8
36.f	odd	6	1	inner	288.2.i.f	✓	8
36.f	odd	6	1		2592.2.a.u		4
36.h	even	6	1		864.2.i.f		8
36.h	even	6	1		2592.2.a.x		4
72.j	odd	6	1		1728.2.i.n		8
72.j	odd	6	1		5184.2.a.cc		4
72.l	even	6	1		1728.2.i.n		8
72.l	even	6	1		5184.2.a.cc		4
72.n	even	6	1		576.2.i.n		8
72.n	even	6	1		5184.2.a.cf		4
72.p	odd	6	1		576.2.i.n		8
72.p	odd	6	1		5184.2.a.cf		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
288.2.i.f	✓	8	1.a	even	1	1	trivial
288.2.i.f	✓	8	4.b	odd	2	1	inner
288.2.i.f	✓	8	9.c	even	3	1	inner
288.2.i.f	✓	8	36.f	odd	6	1	inner
576.2.i.n		8	8.b	even	2	1
576.2.i.n		8	8.d	odd	2	1
576.2.i.n		8	72.n	even	6	1
576.2.i.n		8	72.p	odd	6	1
864.2.i.f		8	3.b	odd	2	1
864.2.i.f		8	9.d	odd	6	1
864.2.i.f		8	12.b	even	2	1
864.2.i.f		8	36.h	even	6	1
1728.2.i.n		8	24.f	even	2	1
1728.2.i.n		8	24.h	odd	2	1
1728.2.i.n		8	72.j	odd	6	1
1728.2.i.n		8	72.l	even	6	1
2592.2.a.u		4	9.c	even	3	1
2592.2.a.u		4	36.f	odd	6	1
2592.2.a.x		4	9.d	odd	6	1
2592.2.a.x		4	36.h	even	6	1
5184.2.a.cc		4	72.j	odd	6	1
5184.2.a.cc		4	72.l	even	6	1
5184.2.a.cf		4	72.n	even	6	1
5184.2.a.cf		4	72.p	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(288, [\chi])\):

\( T_{5}^{4} - T_{5}^{3} + 9T_{5}^{2} + 8T_{5} + 64 \)

\( T_{7}^{8} + 27T_{7}^{6} + 621T_{7}^{4} + 2916T_{7}^{2} + 11664 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{8} \)
$3$	T^8 + 3*T^6 + 12*T^4 + 27*T^2 + 81
$5$	(T^4 - T^3 + 9*T^2 + 8*T + 64)^2
$7$	T^8 + 27*T^6 + 621*T^4 + 2916*T^2 + 11664
$11$	T^8 + 36*T^6 + 1269*T^4 + 972*T^2 + 729