Newform orbit 120.2.m.a

• Label: 120.2.m.a
• Level: $120$
• Weight: $2$
• Character orbit: 120.m
• Analytic conductor: $0.958$
• Analytic rank: $0$
• Dimension: $4$
• CM discriminant: -15
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	120.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.958204824255\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\sqrt{3}, \sqrt{-5})\)
Defining polynomial:	\( x^{4} + x^{2} + 4 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + (\beta_{3} - \beta_1) q^{3} + \beta_{2} q^{4} + (\beta_{3} + \beta_1) q^{5} + ( - \beta_{2} - 2) q^{6} + (2 \beta_{3} - \beta_1) q^{8} + 3 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{4} - 6 q^{6} + 12 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + x^{2} + 4 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{3} + \nu ) / 2 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/120\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(41\)	\(61\)	\(97\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(-1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

59.1

−0.866025	−	1.11803i
−0.866025	+	1.11803i
0.866025	−	1.11803i
0.866025	+	1.11803i

−0.866025

−

1.11803i

1.73205

−0.500000

+

1.93649i

−

2.23607i

−1.50000

−

1.93649i

0

2.59808

−

1.11803i

3.00000

−2.50000

+

1.93649i

59.2

−0.866025

+

1.11803i

1.73205

−0.500000

−

1.93649i

2.23607i

−1.50000

+

1.93649i

0

2.59808

+

1.11803i

3.00000

−2.50000

−

1.93649i

59.3

0.866025

−

1.11803i

−1.73205

−0.500000

−

1.93649i

−

2.23607i

−1.50000

+

1.93649i

0

−2.59808

−

1.11803i

3.00000

−2.50000

−

1.93649i

59.4

0.866025

+

1.11803i

−1.73205

−0.500000

+

1.93649i

2.23607i

−1.50000

−

1.93649i

0

−2.59808

+

1.11803i

3.00000

−2.50000

+

1.93649i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by \(\Q(\sqrt{-15}) \)
3.b	odd	2	1	inner
5.b	even	2	1	inner
8.d	odd	2	1	inner
24.f	even	2	1	inner
40.e	odd	2	1	inner
120.m	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	120.2.m.a	✓	4
3.b	odd	2	1	inner	120.2.m.a	✓	4
4.b	odd	2	1		480.2.m.a		4
5.b	even	2	1	inner	120.2.m.a	✓	4
5.c	odd	4	2		600.2.b.c		4
8.b	even	2	1		480.2.m.a		4
8.d	odd	2	1	inner	120.2.m.a	✓	4
12.b	even	2	1		480.2.m.a		4
15.d	odd	2	1	CM	120.2.m.a	✓	4
15.e	even	4	2		600.2.b.c		4
20.d	odd	2	1		480.2.m.a		4
20.e	even	4	2		2400.2.b.c		4
24.f	even	2	1	inner	120.2.m.a	✓	4
24.h	odd	2	1		480.2.m.a		4
40.e	odd	2	1	inner	120.2.m.a	✓	4
40.f	even	2	1		480.2.m.a		4
40.i	odd	4	2		2400.2.b.c		4
40.k	even	4	2		600.2.b.c		4
60.h	even	2	1		480.2.m.a		4
60.l	odd	4	2		2400.2.b.c		4
120.i	odd	2	1		480.2.m.a		4
120.m	even	2	1	inner	120.2.m.a	✓	4
120.q	odd	4	2		600.2.b.c		4
120.w	even	4	2		2400.2.b.c		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
120.2.m.a	✓	4	1.a	even	1	1	trivial
120.2.m.a	✓	4	3.b	odd	2	1	inner
120.2.m.a	✓	4	5.b	even	2	1	inner
120.2.m.a	✓	4	8.d	odd	2	1	inner
120.2.m.a	✓	4	15.d	odd	2	1	CM
120.2.m.a	✓	4	24.f	even	2	1	inner
120.2.m.a	✓	4	40.e	odd	2	1	inner
120.2.m.a	✓	4	120.m	even	2	1	inner
480.2.m.a		4	4.b	odd	2	1
480.2.m.a		4	8.b	even	2	1
480.2.m.a		4	12.b	even	2	1
480.2.m.a		4	20.d	odd	2	1
480.2.m.a		4	24.h	odd	2	1
480.2.m.a		4	40.f	even	2	1
480.2.m.a		4	60.h	even	2	1
480.2.m.a		4	120.i	odd	2	1
600.2.b.c		4	5.c	odd	4	2
600.2.b.c		4	15.e	even	4	2
600.2.b.c		4	40.k	even	4	2
600.2.b.c		4	120.q	odd	4	2
2400.2.b.c		4	20.e	even	4	2
2400.2.b.c		4	40.i	odd	4	2
2400.2.b.c		4	60.l	odd	4	2
2400.2.b.c		4	120.w	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} \) acting on \(S_{2}^{\mathrm{new}}(120, [\chi])\).

Hecke characteristic polynomials

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