Newform orbit 144.2.k.a

• Label: 144.2.k.a
• Level: $144$
• Weight: $2$
• Character orbit: 144.k
• 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.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.14984578911\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-1}) \)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + (i + 1) q^{2} + 2 i q^{4} + (i + 1) q^{5} - 2 i q^{7} + (2 i - 2) q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{5} - 4 q^{8}+O(q^{10}) \)

Character values

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

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

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

37.1

		1.00000i
	−	1.00000i

1.00000

+

1.00000i

0

2.00000i

1.00000

+

1.00000i

0

−

2.00000i

−2.00000

+

2.00000i

0

2.00000i

109.1

1.00000

−

1.00000i

0

−

2.00000i

1.00000

−

1.00000i

0

2.00000i

−2.00000

−

2.00000i

0

−

2.00000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
16.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	144.2.k.a		2
3.b	odd	2	1		16.2.e.a	✓	2
4.b	odd	2	1		576.2.k.a		2
8.b	even	2	1		1152.2.k.b		2
8.d	odd	2	1		1152.2.k.a		2
12.b	even	2	1		64.2.e.a		2
15.d	odd	2	1		400.2.l.c		2
15.e	even	4	1		400.2.q.a		2
15.e	even	4	1		400.2.q.b		2
16.e	even	4	1	inner	144.2.k.a		2
16.e	even	4	1		1152.2.k.b		2
16.f	odd	4	1		576.2.k.a		2
16.f	odd	4	1		1152.2.k.a		2
21.c	even	2	1		784.2.m.b		2
21.g	even	6	2		784.2.x.c		4
21.h	odd	6	2		784.2.x.f		4
24.f	even	2	1		128.2.e.a		2
24.h	odd	2	1		128.2.e.b		2
32.g	even	8	2		9216.2.a.d		2
32.h	odd	8	2		9216.2.a.s		2
48.i	odd	4	1		16.2.e.a	✓	2
48.i	odd	4	1		128.2.e.b		2
48.k	even	4	1		64.2.e.a		2
48.k	even	4	1		128.2.e.a		2
60.h	even	2	1		1600.2.l.a		2
60.l	odd	4	1		1600.2.q.a		2
60.l	odd	4	1		1600.2.q.b		2
96.o	even	8	2		1024.2.a.e		2
96.o	even	8	2		1024.2.b.b		2
96.p	odd	8	2		1024.2.a.b		2
96.p	odd	8	2		1024.2.b.e		2
240.t	even	4	1		1600.2.l.a		2
240.z	odd	4	1		1600.2.q.a		2
240.bb	even	4	1		400.2.q.b		2
240.bd	odd	4	1		1600.2.q.b		2
240.bf	even	4	1		400.2.q.a		2
240.bm	odd	4	1		400.2.l.c		2
336.y	even	4	1		784.2.m.b		2
336.bo	even	12	2		784.2.x.c		4
336.bt	odd	12	2		784.2.x.f		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
16.2.e.a	✓	2	3.b	odd	2	1
16.2.e.a	✓	2	48.i	odd	4	1
64.2.e.a		2	12.b	even	2	1
64.2.e.a		2	48.k	even	4	1
128.2.e.a		2	24.f	even	2	1
128.2.e.a		2	48.k	even	4	1
128.2.e.b		2	24.h	odd	2	1
128.2.e.b		2	48.i	odd	4	1
144.2.k.a		2	1.a	even	1	1	trivial
144.2.k.a		2	16.e	even	4	1	inner
400.2.l.c		2	15.d	odd	2	1
400.2.l.c		2	240.bm	odd	4	1
400.2.q.a		2	15.e	even	4	1
400.2.q.a		2	240.bf	even	4	1
400.2.q.b		2	15.e	even	4	1
400.2.q.b		2	240.bb	even	4	1
576.2.k.a		2	4.b	odd	2	1
576.2.k.a		2	16.f	odd	4	1
784.2.m.b		2	21.c	even	2	1
784.2.m.b		2	336.y	even	4	1
784.2.x.c		4	21.g	even	6	2
784.2.x.c		4	336.bo	even	12	2
784.2.x.f		4	21.h	odd	6	2
784.2.x.f		4	336.bt	odd	12	2
1024.2.a.b		2	96.p	odd	8	2
1024.2.a.e		2	96.o	even	8	2
1024.2.b.b		2	96.o	even	8	2
1024.2.b.e		2	96.p	odd	8	2
1152.2.k.a		2	8.d	odd	2	1
1152.2.k.a		2	16.f	odd	4	1
1152.2.k.b		2	8.b	even	2	1
1152.2.k.b		2	16.e	even	4	1
1600.2.l.a		2	60.h	even	2	1
1600.2.l.a		2	240.t	even	4	1
1600.2.q.a		2	60.l	odd	4	1
1600.2.q.a		2	240.z	odd	4	1
1600.2.q.b		2	60.l	odd	4	1
1600.2.q.b		2	240.bd	odd	4	1
9216.2.a.d		2	32.g	even	8	2
9216.2.a.s		2	32.h	odd	8	2

Hecke kernels

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

Hecke characteristic polynomials

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