Newform orbit 160.2.a.c

• Label: 160.2.a.c
• Level: $160$
• Weight: $2$
• Character orbit: 160.a
• Self dual: yes
• Analytic conductor: $1.278$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	160.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(1.27760643234\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\)	\(=\)	\( q + \beta q^{3} + q^{5} - \beta q^{7} + 5 q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 10 q^{9}+O(q^{10}) \)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

−1.41421
1.41421

0

−2.82843

0

1.00000

0

2.82843

0

5.00000

0

1.2

0

2.82843

0

1.00000

0

−2.82843

0

5.00000

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(1\)
\(5\)	\(-1\)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	160.2.a.c	✓	2
3.b	odd	2	1		1440.2.a.o		2
4.b	odd	2	1	inner	160.2.a.c	✓	2
5.b	even	2	1		800.2.a.m		2
5.c	odd	4	2		800.2.c.f		4
7.b	odd	2	1		7840.2.a.bf		2
8.b	even	2	1		320.2.a.g		2
8.d	odd	2	1		320.2.a.g		2
12.b	even	2	1		1440.2.a.o		2
15.d	odd	2	1		7200.2.a.cm		2
15.e	even	4	2		7200.2.f.bh		4
16.e	even	4	2		1280.2.d.l		4
16.f	odd	4	2		1280.2.d.l		4
20.d	odd	2	1		800.2.a.m		2
20.e	even	4	2		800.2.c.f		4
24.f	even	2	1		2880.2.a.bk		2
24.h	odd	2	1		2880.2.a.bk		2
28.d	even	2	1		7840.2.a.bf		2
40.e	odd	2	1		1600.2.a.bc		2
40.f	even	2	1		1600.2.a.bc		2
40.i	odd	4	2		1600.2.c.n		4
40.k	even	4	2		1600.2.c.n		4
60.h	even	2	1		7200.2.a.cm		2
60.l	odd	4	2		7200.2.f.bh		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
160.2.a.c	✓	2	1.a	even	1	1	trivial
160.2.a.c	✓	2	4.b	odd	2	1	inner
320.2.a.g		2	8.b	even	2	1
320.2.a.g		2	8.d	odd	2	1
800.2.a.m		2	5.b	even	2	1
800.2.a.m		2	20.d	odd	2	1
800.2.c.f		4	5.c	odd	4	2
800.2.c.f		4	20.e	even	4	2
1280.2.d.l		4	16.e	even	4	2
1280.2.d.l		4	16.f	odd	4	2
1440.2.a.o		2	3.b	odd	2	1
1440.2.a.o		2	12.b	even	2	1
1600.2.a.bc		2	40.e	odd	2	1
1600.2.a.bc		2	40.f	even	2	1
1600.2.c.n		4	40.i	odd	4	2
1600.2.c.n		4	40.k	even	4	2
2880.2.a.bk		2	24.f	even	2	1
2880.2.a.bk		2	24.h	odd	2	1
7200.2.a.cm		2	15.d	odd	2	1
7200.2.a.cm		2	60.h	even	2	1
7200.2.f.bh		4	15.e	even	4	2
7200.2.f.bh		4	60.l	odd	4	2
7840.2.a.bf		2	7.b	odd	2	1
7840.2.a.bf		2	28.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 8 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(160))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 8 \)
$5$	\( (T - 1)^{2} \)
$7$	\( T^{2} - 8 \)
$11$	\( T^{2} - 32 \)