Newform orbit 3200.1.e.a

• Label: 3200.1.e.a
• Level: $3200$
• Weight: $1$
• Character orbit: 3200.e
• Analytic conductor: $1.597$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{2}$
• CM/RM discs: -4, -8, 8
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3200 = 2^{7} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3200.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.59700804043\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 128)
Projective image:	\(D_{2}\)
Projective field:	Galois closure of \(\Q(\zeta_{8})\)
Artin image:	$D_4:C_2$
Artin field:	Galois closure of 8.0.4096000000.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\)	\(=\)	\( q + q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times\).

\(n\)	\(901\)	\(1151\)	\(2177\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1599.1

		1.00000i
	−	1.00000i

0

0

0

0

0

0

0

1.00000

0

1599.2

0

0

0

0

0

0

0

1.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
8.b	even	2	1	RM by \(\Q(\sqrt{2}) \)
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
5.b	even	2	1	inner
20.d	odd	2	1	inner
40.e	odd	2	1	inner
40.f	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3200.1.e.a		2
4.b	odd	2	1	CM	3200.1.e.a		2
5.b	even	2	1	inner	3200.1.e.a		2
5.c	odd	4	1		128.1.d.a	✓	1
5.c	odd	4	1		3200.1.g.a		1
8.b	even	2	1	RM	3200.1.e.a		2
8.d	odd	2	1	CM	3200.1.e.a		2
15.e	even	4	1		1152.1.b.a		1
20.d	odd	2	1	inner	3200.1.e.a		2
20.e	even	4	1		128.1.d.a	✓	1
20.e	even	4	1		3200.1.g.a		1
40.e	odd	2	1	inner	3200.1.e.a		2
40.f	even	2	1	inner	3200.1.e.a		2
40.i	odd	4	1		128.1.d.a	✓	1
40.i	odd	4	1		3200.1.g.a		1
40.k	even	4	1		128.1.d.a	✓	1
40.k	even	4	1		3200.1.g.a		1
60.l	odd	4	1		1152.1.b.a		1
80.i	odd	4	1		256.1.c.a		1
80.j	even	4	1		256.1.c.a		1
80.s	even	4	1		256.1.c.a		1
80.t	odd	4	1		256.1.c.a		1
120.q	odd	4	1		1152.1.b.a		1
120.w	even	4	1		1152.1.b.a		1
160.u	even	8	2		1024.1.f.b		2
160.v	odd	8	2		1024.1.f.b		2
160.ba	even	8	2		1024.1.f.b		2
160.bb	odd	8	2		1024.1.f.b		2
240.z	odd	4	1		2304.1.g.b		1
240.bb	even	4	1		2304.1.g.b		1
240.bd	odd	4	1		2304.1.g.b		1
240.bf	even	4	1		2304.1.g.b		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
128.1.d.a	✓	1	5.c	odd	4	1
128.1.d.a	✓	1	20.e	even	4	1
128.1.d.a	✓	1	40.i	odd	4	1
128.1.d.a	✓	1	40.k	even	4	1
256.1.c.a		1	80.i	odd	4	1
256.1.c.a		1	80.j	even	4	1
256.1.c.a		1	80.s	even	4	1
256.1.c.a		1	80.t	odd	4	1
1024.1.f.b		2	160.u	even	8	2
1024.1.f.b		2	160.v	odd	8	2
1024.1.f.b		2	160.ba	even	8	2
1024.1.f.b		2	160.bb	odd	8	2
1152.1.b.a		1	15.e	even	4	1
1152.1.b.a		1	60.l	odd	4	1
1152.1.b.a		1	120.q	odd	4	1
1152.1.b.a		1	120.w	even	4	1
2304.1.g.b		1	240.z	odd	4	1
2304.1.g.b		1	240.bb	even	4	1
2304.1.g.b		1	240.bd	odd	4	1
2304.1.g.b		1	240.bf	even	4	1
3200.1.e.a		2	1.a	even	1	1	trivial
3200.1.e.a		2	4.b	odd	2	1	CM
3200.1.e.a		2	5.b	even	2	1	inner
3200.1.e.a		2	8.b	even	2	1	RM
3200.1.e.a		2	8.d	odd	2	1	CM
3200.1.e.a		2	20.d	odd	2	1	inner
3200.1.e.a		2	40.e	odd	2	1	inner
3200.1.e.a		2	40.f	even	2	1	inner
3200.1.g.a		1	5.c	odd	4	1
3200.1.g.a		1	20.e	even	4	1
3200.1.g.a		1	40.i	odd	4	1
3200.1.g.a		1	40.k	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{1}^{\mathrm{new}}(3200, [\chi])\).

Hecke characteristic polynomials

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