Newform orbit 144.2.c.a

• Label: 144.2.c.a
• Level: $144$
• Weight: $2$
• Character orbit: 144.c
• Analytic conductor: $1.150$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -4
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	144.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.14984578911\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-2}) \)
Defining polynomial:	\( x^{2} + 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

143.1

	−	1.41421i
		1.41421i

0

0

0

−

4.24264i

0

0

0

0

0

143.2

0

0

0

4.24264i

0

0

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
3.b	odd	2	1	inner
12.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	144.2.c.a	✓	2
3.b	odd	2	1	inner	144.2.c.a	✓	2
4.b	odd	2	1	CM	144.2.c.a	✓	2
5.b	even	2	1		3600.2.h.b		2
5.c	odd	4	2		3600.2.o.a		4
7.b	odd	2	1		7056.2.h.b		2
8.b	even	2	1		576.2.c.a		2
8.d	odd	2	1		576.2.c.a		2
9.c	even	3	2		1296.2.s.h		4
9.d	odd	6	2		1296.2.s.h		4
12.b	even	2	1	inner	144.2.c.a	✓	2
15.d	odd	2	1		3600.2.h.b		2
15.e	even	4	2		3600.2.o.a		4
16.e	even	4	2		2304.2.f.f		4
16.f	odd	4	2		2304.2.f.f		4
20.d	odd	2	1		3600.2.h.b		2
20.e	even	4	2		3600.2.o.a		4
21.c	even	2	1		7056.2.h.b		2
24.f	even	2	1		576.2.c.a		2
24.h	odd	2	1		576.2.c.a		2
28.d	even	2	1		7056.2.h.b		2
36.f	odd	6	2		1296.2.s.h		4
36.h	even	6	2		1296.2.s.h		4
48.i	odd	4	2		2304.2.f.f		4
48.k	even	4	2		2304.2.f.f		4
60.h	even	2	1		3600.2.h.b		2
60.l	odd	4	2		3600.2.o.a		4
84.h	odd	2	1		7056.2.h.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
144.2.c.a	✓	2	1.a	even	1	1	trivial
144.2.c.a	✓	2	3.b	odd	2	1	inner
144.2.c.a	✓	2	4.b	odd	2	1	CM
144.2.c.a	✓	2	12.b	even	2	1	inner
576.2.c.a		2	8.b	even	2	1
576.2.c.a		2	8.d	odd	2	1
576.2.c.a		2	24.f	even	2	1
576.2.c.a		2	24.h	odd	2	1
1296.2.s.h		4	9.c	even	3	2
1296.2.s.h		4	9.d	odd	6	2
1296.2.s.h		4	36.f	odd	6	2
1296.2.s.h		4	36.h	even	6	2
2304.2.f.f		4	16.e	even	4	2
2304.2.f.f		4	16.f	odd	4	2
2304.2.f.f		4	48.i	odd	4	2
2304.2.f.f		4	48.k	even	4	2
3600.2.h.b		2	5.b	even	2	1
3600.2.h.b		2	15.d	odd	2	1
3600.2.h.b		2	20.d	odd	2	1
3600.2.h.b		2	60.h	even	2	1
3600.2.o.a		4	5.c	odd	4	2
3600.2.o.a		4	15.e	even	4	2
3600.2.o.a		4	20.e	even	4	2
3600.2.o.a		4	60.l	odd	4	2
7056.2.h.b		2	7.b	odd	2	1
7056.2.h.b		2	21.c	even	2	1
7056.2.h.b		2	28.d	even	2	1
7056.2.h.b		2	84.h	odd	2	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(144, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 18 \)
$7$	\( T^{2} \)
$11$	\( T^{2} \)