Newform orbit 120.2.b.b

• Label: 120.2.b.b
• Level: $120$
• Weight: $2$
• Character orbit: 120.b
• Analytic conductor: $0.958$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(0.958204824255\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	8.0.1649659456.5
Defining polynomial:	\( x^{8} - x^{7} - 2x^{5} + 4x^{4} - 4x^{3} - 8x + 16 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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 + b1 * q^2 - b4 * q^3 + (b4 + b3 + b1) * q^4 - q^5 + (-b7 + b3) * q^6 + (-b5 + b3 - b2 + b1) * q^7 + (b7 + b6 + b5 - b3 + 1) * q^8 + (-b7 - b5 - b1) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} + q^{4} - 8 q^{5} + q^{6} + 7 q^{8}+O(q^{10}) \)

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

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{6} + \nu^{5} - 4 ) / 4 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{7} + \nu^{6} + 4\nu^{2} - 8\nu - 8 ) / 8 \)
\(\beta_{4}\)	\(=\)	\( ( -\nu^{7} - \nu^{6} + 4\nu^{2} + 8 ) / 8 \)
\(\beta_{5}\)	\(=\)	\( ( \nu^{7} + \nu^{6} - 4\nu^{4} + 4\nu^{3} + 4\nu^{2} - 16 ) / 8 \)
\(\beta_{6}\)	\(=\)	\( ( \nu^{7} - \nu^{6} + 2\nu^{5} + 4\nu^{3} - 4\nu^{2} - 8 ) / 8 \)
\(\beta_{7}\)	\(=\)	\( ( -\nu^{7} + \nu^{6} - 2\nu^{5} + 4\nu^{4} + 4\nu^{2} - 8\nu + 8 ) / 8 \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

11.1

−1.13622	−	0.842022i
−1.13622	+	0.842022i
−0.578647	−	1.29041i
−0.578647	+	1.29041i
0.814732	−	1.15595i
0.814732	+	1.15595i
1.40014	−	0.199044i
1.40014	+	0.199044i

−1.13622

−

0.842022i

−1.71822

−

0.218455i

0.581998

+

1.91345i

−1.00000

1.76833

+

1.69499i

3.64426i

0.949886

−

2.66415i

2.90455

+

0.750707i

1.13622

+

0.842022i

11.2

−1.13622

+

0.842022i

−1.71822

+

0.218455i

0.581998

−

1.91345i

−1.00000

1.76833

−

1.69499i

−

3.64426i

0.949886

+

2.66415i

2.90455

−

0.750707i

1.13622

−

0.842022i

11.3

−0.578647

−

1.29041i

0.751690

−

1.56044i

−1.33034

+

1.49339i

−1.00000

−2.44857

−

0.0670494i

−

4.28591i

2.69688

+

0.852541i

−1.86993

−

2.34593i

0.578647

+

1.29041i

11.4

−0.578647

+

1.29041i

0.751690

+

1.56044i

−1.33034

−

1.49339i

−1.00000

−2.44857

+

0.0670494i

4.28591i

2.69688

−

0.852541i

−1.86993

+

2.34593i

0.578647

−

1.29041i

11.5

0.814732

−

1.15595i

1.48716

+

0.887900i

−0.672424

−

1.88357i

−1.00000

2.23800

−

0.995672i

−

0.797253i

−2.72515

−

0.757320i

1.42327

+

2.64089i

−0.814732

+

1.15595i

11.6

0.814732

+

1.15595i

1.48716

−

0.887900i

−0.672424

+

1.88357i

−1.00000

2.23800

+

0.995672i

0.797253i

−2.72515

+

0.757320i

1.42327

−

2.64089i

−0.814732

−

1.15595i

11.7

1.40014

−

0.199044i

−0.520627

−

1.65195i

1.92076

−

0.557378i

−1.00000

−1.05776

−

2.20933i

1.92736i

2.57839

−

1.16272i

−2.45790

+

1.72010i

−1.40014

+

0.199044i

11.8

1.40014

+

0.199044i

−0.520627

+

1.65195i

1.92076

+

0.557378i

−1.00000

−1.05776

+

2.20933i

−

1.92736i

2.57839

+

1.16272i

−2.45790

−

1.72010i

−1.40014

−

0.199044i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
24.f	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.b.b	yes	8
3.b	odd	2	1		120.2.b.a	✓	8
4.b	odd	2	1		480.2.b.a		8
5.b	even	2	1		600.2.b.e		8
5.c	odd	4	2		600.2.m.d		16
8.b	even	2	1		480.2.b.b		8
8.d	odd	2	1		120.2.b.a	✓	8
12.b	even	2	1		480.2.b.b		8
15.d	odd	2	1		600.2.b.f		8
15.e	even	4	2		600.2.m.c		16
20.d	odd	2	1		2400.2.b.e		8
20.e	even	4	2		2400.2.m.c		16
24.f	even	2	1	inner	120.2.b.b	yes	8
24.h	odd	2	1		480.2.b.a		8
40.e	odd	2	1		600.2.b.f		8
40.f	even	2	1		2400.2.b.f		8
40.i	odd	4	2		2400.2.m.d		16
40.k	even	4	2		600.2.m.c		16
60.h	even	2	1		2400.2.b.f		8
60.l	odd	4	2		2400.2.m.d		16
120.i	odd	2	1		2400.2.b.e		8
120.m	even	2	1		600.2.b.e		8
120.q	odd	4	2		600.2.m.d		16
120.w	even	4	2		2400.2.m.c		16

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
120.2.b.a	✓	8	3.b	odd	2	1
120.2.b.a	✓	8	8.d	odd	2	1
120.2.b.b	yes	8	1.a	even	1	1	trivial
120.2.b.b	yes	8	24.f	even	2	1	inner
480.2.b.a		8	4.b	odd	2	1
480.2.b.a		8	24.h	odd	2	1
480.2.b.b		8	8.b	even	2	1
480.2.b.b		8	12.b	even	2	1
600.2.b.e		8	5.b	even	2	1
600.2.b.e		8	120.m	even	2	1
600.2.b.f		8	15.d	odd	2	1
600.2.b.f		8	40.e	odd	2	1
600.2.m.c		16	15.e	even	4	2
600.2.m.c		16	40.k	even	4	2
600.2.m.d		16	5.c	odd	4	2
600.2.m.d		16	120.q	odd	4	2
2400.2.b.e		8	20.d	odd	2	1
2400.2.b.e		8	120.i	odd	2	1
2400.2.b.f		8	40.f	even	2	1
2400.2.b.f		8	60.h	even	2	1
2400.2.m.c		16	20.e	even	4	2
2400.2.m.c		16	120.w	even	4	2
2400.2.m.d		16	40.i	odd	4	2
2400.2.m.d		16	60.l	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23}^{4} + 2T_{23}^{3} - 44T_{23}^{2} - 188T_{23} - 192 \) acting on \(S_{2}^{\mathrm{new}}(120, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^8 - T^7 - 2*T^5 + 4*T^4 - 4*T^3 - 8*T + 16
$3$	T^8 + 4*T^5 - 2*T^4 + 12*T^3 + 81
$5$	\( (T + 1)^{8} \)
$7$	T^8 + 36*T^6 + 384*T^4 + 1136*T^2 + 576
$11$	T^8 + 48*T^6 + 672*T^4 + 2560*T^2 + 256