Newform orbit 1600.3.h.c

• Label: 1600.3.h.c
• Level: $1600$
• Weight: $3$
• Character orbit: 1600.h
• Analytic conductor: $43.597$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1600.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(43.5968422976\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-1}) \)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 32)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\)	\(=\)	\( q + 4 q^{3} - 8 q^{7} + 7 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{3} - 16 q^{7} + 14 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\)	\(577\)	\(901\)	\(1151\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1\)

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

4.00000

0

0

0

−8.00000

0

7.00000

0

1599.2

0

4.00000

0

0

0

−8.00000

0

7.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1600.3.h.c		2
4.b	odd	2	1		1600.3.h.a		2
5.b	even	2	1		1600.3.h.a		2
5.c	odd	4	1		64.3.c.b		2
5.c	odd	4	1		1600.3.b.e		2
8.b	even	2	1		800.3.h.a		2
8.d	odd	2	1		800.3.h.b		2
15.e	even	4	1		576.3.g.g		2
20.d	odd	2	1	inner	1600.3.h.c		2
20.e	even	4	1		64.3.c.b		2
20.e	even	4	1		1600.3.b.e		2
40.e	odd	2	1		800.3.h.a		2
40.f	even	2	1		800.3.h.b		2
40.i	odd	4	1		32.3.c.a	✓	2
40.i	odd	4	1		800.3.b.a		2
40.k	even	4	1		32.3.c.a	✓	2
40.k	even	4	1		800.3.b.a		2
60.l	odd	4	1		576.3.g.g		2
80.i	odd	4	1		256.3.d.a		2
80.j	even	4	1		256.3.d.a		2
80.s	even	4	1		256.3.d.c		2
80.t	odd	4	1		256.3.d.c		2
120.q	odd	4	1		288.3.g.b		2
120.w	even	4	1		288.3.g.b		2
240.z	odd	4	1		2304.3.b.c		2
240.bb	even	4	1		2304.3.b.g		2
240.bd	odd	4	1		2304.3.b.g		2
240.bf	even	4	1		2304.3.b.c		2
280.s	even	4	1		1568.3.d.b		2
280.y	odd	4	1		1568.3.d.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
32.3.c.a	✓	2	40.i	odd	4	1
32.3.c.a	✓	2	40.k	even	4	1
64.3.c.b		2	5.c	odd	4	1
64.3.c.b		2	20.e	even	4	1
256.3.d.a		2	80.i	odd	4	1
256.3.d.a		2	80.j	even	4	1
256.3.d.c		2	80.s	even	4	1
256.3.d.c		2	80.t	odd	4	1
288.3.g.b		2	120.q	odd	4	1
288.3.g.b		2	120.w	even	4	1
576.3.g.g		2	15.e	even	4	1
576.3.g.g		2	60.l	odd	4	1
800.3.b.a		2	40.i	odd	4	1
800.3.b.a		2	40.k	even	4	1
800.3.h.a		2	8.b	even	2	1
800.3.h.a		2	40.e	odd	2	1
800.3.h.b		2	8.d	odd	2	1
800.3.h.b		2	40.f	even	2	1
1568.3.d.b		2	280.s	even	4	1
1568.3.d.b		2	280.y	odd	4	1
1600.3.b.e		2	5.c	odd	4	1
1600.3.b.e		2	20.e	even	4	1
1600.3.h.a		2	4.b	odd	2	1
1600.3.h.a		2	5.b	even	2	1
1600.3.h.c		2	1.a	even	1	1	trivial
1600.3.h.c		2	20.d	odd	2	1	inner
2304.3.b.c		2	240.z	odd	4	1
2304.3.b.c		2	240.bf	even	4	1
2304.3.b.g		2	240.bb	even	4	1
2304.3.b.g		2	240.bd	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1600, [\chi])\):

\( T_{3} - 4 \)

\( T_{7} + 8 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( (T - 4)^{2} \)
$5$	\( T^{2} \)
$7$	\( (T + 8)^{2} \)
$11$	\( T^{2} + 16 \)