Newform orbit 144.2.i.b

• Label: 144.2.i.b
• Level: 144
• Weight: 2
• Character orbit: 144.i
• Analytic conductor: 1.150
• Analytic rank: 0
• Dimension: 2
• CM: no
• Inner twists: 2

Newspace parameters

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	144.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.14984578911\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\(x^{2} - x + 1\)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 72)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\)	\(37\)	\(65\)	\(127\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

49.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

−

1.73205i

0

0.500000

−

0.866025i

0

−1.50000

−

2.59808i

0

−3.00000

0

97.1

0

1.73205i

0

0.500000

+

0.866025i

0

−1.50000

+

2.59808i

0

−3.00000

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	144.2.i.b		2
3.b	odd	2	1		432.2.i.a		2
4.b	odd	2	1		72.2.i.a	✓	2
8.b	even	2	1		576.2.i.c		2
8.d	odd	2	1		576.2.i.d		2
9.c	even	3	1	inner	144.2.i.b		2
9.c	even	3	1		1296.2.a.e		1
9.d	odd	6	1		432.2.i.a		2
9.d	odd	6	1		1296.2.a.i		1
12.b	even	2	1		216.2.i.a		2
24.f	even	2	1		1728.2.i.h		2
24.h	odd	2	1		1728.2.i.g		2
36.f	odd	6	1		72.2.i.a	✓	2
36.f	odd	6	1		648.2.a.a		1
36.h	even	6	1		216.2.i.a		2
36.h	even	6	1		648.2.a.c		1
72.j	odd	6	1		1728.2.i.g		2
72.j	odd	6	1		5184.2.a.n		1
72.l	even	6	1		1728.2.i.h		2
72.l	even	6	1		5184.2.a.i		1
72.n	even	6	1		576.2.i.c		2
72.n	even	6	1		5184.2.a.x		1
72.p	odd	6	1		576.2.i.d		2
72.p	odd	6	1		5184.2.a.s		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
72.2.i.a	✓	2	4.b	odd	2	1
72.2.i.a	✓	2	36.f	odd	6	1
144.2.i.b		2	1.a	even	1	1	trivial
144.2.i.b		2	9.c	even	3	1	inner
216.2.i.a		2	12.b	even	2	1
216.2.i.a		2	36.h	even	6	1
432.2.i.a		2	3.b	odd	2	1
432.2.i.a		2	9.d	odd	6	1
576.2.i.c		2	8.b	even	2	1
576.2.i.c		2	72.n	even	6	1
576.2.i.d		2	8.d	odd	2	1
576.2.i.d		2	72.p	odd	6	1
648.2.a.a		1	36.f	odd	6	1
648.2.a.c		1	36.h	even	6	1
1296.2.a.e		1	9.c	even	3	1
1296.2.a.i		1	9.d	odd	6	1
1728.2.i.g		2	24.h	odd	2	1
1728.2.i.g		2	72.j	odd	6	1
1728.2.i.h		2	24.f	even	2	1
1728.2.i.h		2	72.l	even	6	1
5184.2.a.i		1	72.l	even	6	1
5184.2.a.n		1	72.j	odd	6	1
5184.2.a.s		1	72.p	odd	6	1
5184.2.a.x		1	72.n	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(144, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	1
$3$	\( 1 + 3 T^{2} \)
$5$	\( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \)
$7$	\( 1 + 3 T + 2 T^{2} + 21 T^{3} + 49 T^{4} \)
$11$	\( 1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4} \)