Newform orbit 320.2.n.a

• Label: 320.2.n.a
• Level: $320$
• Weight: $2$
• Character orbit: 320.n
• Analytic conductor: $2.555$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	320.n (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.55521286468\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-1}) \)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 160)
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.

\(f(q)\)	\(=\)	\( q + ( - 2 i - 2) q^{3} + ( - i + 2) q^{5} + ( - 2 i + 2) q^{7} + 5 i q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} + 4 q^{5} + 4 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\)	\(191\)	\(257\)	\(261\)
\(\chi(n)\)	\(-1\)	\(i\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

63.1

	−	1.00000i
		1.00000i

0

−2.00000

+

2.00000i

0

2.00000

+

1.00000i

0

2.00000

+

2.00000i

0

−

5.00000i

0

127.1

0

−2.00000

−

2.00000i

0

2.00000

−

1.00000i

0

2.00000

−

2.00000i

0

5.00000i

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	320.2.n.a		2
4.b	odd	2	1		320.2.n.h		2
5.b	even	2	1		1600.2.n.n		2
5.c	odd	4	1		320.2.n.h		2
5.c	odd	4	1		1600.2.n.a		2
8.b	even	2	1		160.2.n.f	yes	2
8.d	odd	2	1		160.2.n.a	✓	2
16.e	even	4	1		1280.2.o.b		2
16.e	even	4	1		1280.2.o.o		2
16.f	odd	4	1		1280.2.o.a		2
16.f	odd	4	1		1280.2.o.p		2
20.d	odd	2	1		1600.2.n.a		2
20.e	even	4	1	inner	320.2.n.a		2
20.e	even	4	1		1600.2.n.n		2
24.f	even	2	1		1440.2.x.i		2
24.h	odd	2	1		1440.2.x.j		2
40.e	odd	2	1		800.2.n.j		2
40.f	even	2	1		800.2.n.a		2
40.i	odd	4	1		160.2.n.a	✓	2
40.i	odd	4	1		800.2.n.j		2
40.k	even	4	1		160.2.n.f	yes	2
40.k	even	4	1		800.2.n.a		2
80.i	odd	4	1		1280.2.o.p		2
80.j	even	4	1		1280.2.o.o		2
80.s	even	4	1		1280.2.o.b		2
80.t	odd	4	1		1280.2.o.a		2
120.q	odd	4	1		1440.2.x.j		2
120.w	even	4	1		1440.2.x.i		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
160.2.n.a	✓	2	8.d	odd	2	1
160.2.n.a	✓	2	40.i	odd	4	1
160.2.n.f	yes	2	8.b	even	2	1
160.2.n.f	yes	2	40.k	even	4	1
320.2.n.a		2	1.a	even	1	1	trivial
320.2.n.a		2	20.e	even	4	1	inner
320.2.n.h		2	4.b	odd	2	1
320.2.n.h		2	5.c	odd	4	1
800.2.n.a		2	40.f	even	2	1
800.2.n.a		2	40.k	even	4	1
800.2.n.j		2	40.e	odd	2	1
800.2.n.j		2	40.i	odd	4	1
1280.2.o.a		2	16.f	odd	4	1
1280.2.o.a		2	80.t	odd	4	1
1280.2.o.b		2	16.e	even	4	1
1280.2.o.b		2	80.s	even	4	1
1280.2.o.o		2	16.e	even	4	1
1280.2.o.o		2	80.j	even	4	1
1280.2.o.p		2	16.f	odd	4	1
1280.2.o.p		2	80.i	odd	4	1
1440.2.x.i		2	24.f	even	2	1
1440.2.x.i		2	120.w	even	4	1
1440.2.x.j		2	24.h	odd	2	1
1440.2.x.j		2	120.q	odd	4	1
1600.2.n.a		2	5.c	odd	4	1
1600.2.n.a		2	20.d	odd	2	1
1600.2.n.n		2	5.b	even	2	1
1600.2.n.n		2	20.e	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(320, [\chi])\):

\( T_{3}^{2} + 4T_{3} + 8 \)

\( T_{7}^{2} - 4T_{7} + 8 \)

\( T_{13}^{2} - 2T_{13} + 2 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 4T + 8 \)
$5$	\( T^{2} - 4T + 5 \)
$7$	\( T^{2} - 4T + 8 \)
$11$	\( T^{2} \)