Newform orbit 180.2.e.a

• Label: 180.2.e.a
• Level: $180$
• Weight: $2$
• Character orbit: 180.e
• Analytic conductor: $1.437$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	180.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.43730723638\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	8.0.18939904.2
Defining polynomial:	\( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\)	\(=\)	\( q + (\beta_{6} + \beta_{2}) q^{2} + (\beta_{7} - \beta_{5} + \beta_{4}) q^{4} - \beta_{6} q^{5} + (2 \beta_{7} + \beta_{5} + \beta_{4} - 2) q^{7} + (\beta_{6} - 2 \beta_{3} + \beta_{2}) q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{4}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) :

\(\beta_{1}\)	\(=\)	\( \nu^{7} - 3\nu^{6} + 11\nu^{5} - 17\nu^{4} + 26\nu^{3} - 19\nu^{2} + 12\nu - 3 \)
\(\beta_{2}\)	\(=\)	\( -\nu^{7} + 4\nu^{6} - 14\nu^{5} + 28\nu^{4} - 43\nu^{3} + 43\nu^{2} - 28\nu + 8 \)
\(\beta_{3}\)	\(=\)	\( -4\nu^{7} + 14\nu^{6} - 49\nu^{5} + 88\nu^{4} - 129\nu^{3} + 115\nu^{2} - 66\nu + 18 \)
\(\beta_{4}\)	\(=\)	\( -5\nu^{7} + 17\nu^{6} - 60\nu^{5} + 105\nu^{4} - 155\nu^{3} + 133\nu^{2} - 78\nu + 20 \)
\(\beta_{5}\)	\(=\)	\( -5\nu^{7} + 18\nu^{6} - 63\nu^{5} + 115\nu^{4} - 170\nu^{3} + 152\nu^{2} - 90\nu + 24 \)
\(\beta_{6}\)	\(=\)	\( -8\nu^{7} + 28\nu^{6} - 98\nu^{5} + 175\nu^{4} - 256\nu^{3} + 223\nu^{2} - 126\nu + 31 \)
\(\beta_{7}\)	\(=\)	\( -14\nu^{7} + 49\nu^{6} - 171\nu^{5} + 305\nu^{4} - 445\nu^{3} + 387\nu^{2} - 220\nu + 55 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/180\mathbb{Z}\right)^\times\).

\(n\)	\(37\)	\(91\)	\(101\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

71.1

0.500000	+	0.691860i
0.500000	−	0.691860i
0.500000	−	1.44392i
0.500000	+	1.44392i
0.500000	+	0.0297061i
0.500000	−	0.0297061i
0.500000	−	2.10607i
0.500000	+	2.10607i

−1.39897

−

0.207107i

0

1.91421

+

0.579471i

−

1.00000i

0

−

1.63899i

−2.55791

−

1.20711i

0

−0.207107

+

1.39897i

71.2

−1.39897

+

0.207107i

0

1.91421

−

0.579471i

1.00000i

0

1.63899i

−2.55791

+

1.20711i

0

−0.207107

−

1.39897i

71.3

−0.736813

−

1.20711i

0

−0.914214

+

1.77882i

1.00000i

0

5.03127i

2.82083

−

0.207107i

0

1.20711

−

0.736813i

71.4

−0.736813

+

1.20711i

0

−0.914214

−

1.77882i

−

1.00000i

0

−

5.03127i

2.82083

+

0.207107i

0

1.20711

+

0.736813i

71.5

0.736813

−

1.20711i

0

−0.914214

−

1.77882i

1.00000i

0

−

5.03127i

−2.82083

−

0.207107i

0

1.20711

+

0.736813i

71.6

0.736813

+

1.20711i

0

−0.914214

+

1.77882i

−

1.00000i

0

5.03127i

−2.82083

+

0.207107i

0

1.20711

−

0.736813i

71.7

1.39897

−

0.207107i

0

1.91421

−

0.579471i

−

1.00000i

0

1.63899i

2.55791

−

1.20711i

0

−0.207107

−

1.39897i

71.8

1.39897

+

0.207107i

0

1.91421

+

0.579471i

1.00000i

0

−

1.63899i

2.55791

+

1.20711i

0

−0.207107

+

1.39897i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.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	180.2.e.a	✓	8
3.b	odd	2	1	inner	180.2.e.a	✓	8
4.b	odd	2	1	inner	180.2.e.a	✓	8
5.b	even	2	1		900.2.e.d		8
5.c	odd	4	1		900.2.h.b		8
5.c	odd	4	1		900.2.h.c		8
8.b	even	2	1		2880.2.h.e		8
8.d	odd	2	1		2880.2.h.e		8
12.b	even	2	1	inner	180.2.e.a	✓	8
15.d	odd	2	1		900.2.e.d		8
15.e	even	4	1		900.2.h.b		8
15.e	even	4	1		900.2.h.c		8
20.d	odd	2	1		900.2.e.d		8
20.e	even	4	1		900.2.h.b		8
20.e	even	4	1		900.2.h.c		8
24.f	even	2	1		2880.2.h.e		8
24.h	odd	2	1		2880.2.h.e		8
60.h	even	2	1		900.2.e.d		8
60.l	odd	4	1		900.2.h.b		8
60.l	odd	4	1		900.2.h.c		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
180.2.e.a	✓	8	1.a	even	1	1	trivial
180.2.e.a	✓	8	3.b	odd	2	1	inner
180.2.e.a	✓	8	4.b	odd	2	1	inner
180.2.e.a	✓	8	12.b	even	2	1	inner
900.2.e.d		8	5.b	even	2	1
900.2.e.d		8	15.d	odd	2	1
900.2.e.d		8	20.d	odd	2	1
900.2.e.d		8	60.h	even	2	1
900.2.h.b		8	5.c	odd	4	1
900.2.h.b		8	15.e	even	4	1
900.2.h.b		8	20.e	even	4	1
900.2.h.b		8	60.l	odd	4	1
900.2.h.c		8	5.c	odd	4	1
900.2.h.c		8	15.e	even	4	1
900.2.h.c		8	20.e	even	4	1
900.2.h.c		8	60.l	odd	4	1
2880.2.h.e		8	8.b	even	2	1
2880.2.h.e		8	8.d	odd	2	1
2880.2.h.e		8	24.f	even	2	1
2880.2.h.e		8	24.h	odd	2	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(180, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^8 - 2*T^6 + T^4 - 8*T^2 + 16
$3$	\( T^{8} \)
$5$	\( (T^{2} + 1)^{4} \)
$7$	\( (T^{4} + 28 T^{2} + 68)^{2} \)
$11$	\( (T^{4} - 20 T^{2} + 68)^{2} \)