Newform orbit 800.2.c.e

• Label: 800.2.c.e
• Level: $800$
• Weight: $2$
• Character orbit: 800.c
• Analytic conductor: $6.388$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -4
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$800 = 2^{5} \cdot 5^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	800.c (of order $2$, degree $1$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$6.38803216170$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1})$
Defining polynomial:	$x^{2} + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 32)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

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

$f(q)$	$=$	$q + 3 q^{9} + 3 \beta q^{13} - \beta q^{17} + 10 q^{29} + \beta q^{37} + 10 q^{41} + 7 q^{49} + 7 \beta q^{53} - 10 q^{61} - 3 \beta q^{73} + 9 q^{81} - 10 q^{89} - 9 \beta q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 6 q^{9} + 20 q^{29} + 20 q^{41} + 14 q^{49} - 20 q^{61} + 18 q^{81} - 20 q^{89}+O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$1$	$1$	$-1$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

449.1

	−	1.00000i
		1.00000i

0

0

0

0

0

0

0

3.00000

0

449.2

0

0

0

0

0

0

0

3.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1})$
5.b	even	2	1	inner
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	800.2.c.e		2
3.b	odd	2	1		7200.2.f.m		2
4.b	odd	2	1	CM	800.2.c.e		2
5.b	even	2	1	inner	800.2.c.e		2
5.c	odd	4	1		32.2.a.a	✓	1
5.c	odd	4	1		800.2.a.d		1
8.b	even	2	1		1600.2.c.l		2
8.d	odd	2	1		1600.2.c.l		2
12.b	even	2	1		7200.2.f.m		2
15.d	odd	2	1		7200.2.f.m		2
15.e	even	4	1		288.2.a.d		1
15.e	even	4	1		7200.2.a.v		1
20.d	odd	2	1	inner	800.2.c.e		2
20.e	even	4	1		32.2.a.a	✓	1
20.e	even	4	1		800.2.a.d		1
35.f	even	4	1		1568.2.a.e		1
35.k	even	12	2		1568.2.i.f		2
35.l	odd	12	2		1568.2.i.g		2
40.e	odd	2	1		1600.2.c.l		2
40.f	even	2	1		1600.2.c.l		2
40.i	odd	4	1		64.2.a.a		1
40.i	odd	4	1		1600.2.a.n		1
40.k	even	4	1		64.2.a.a		1
40.k	even	4	1		1600.2.a.n		1
45.k	odd	12	2		2592.2.i.t		2
45.l	even	12	2		2592.2.i.e		2
55.e	even	4	1		3872.2.a.f		1
60.h	even	2	1		7200.2.f.m		2
60.l	odd	4	1		288.2.a.d		1
60.l	odd	4	1		7200.2.a.v		1
65.h	odd	4	1		5408.2.a.g		1
80.i	odd	4	1		256.2.b.b		2
80.j	even	4	1		256.2.b.b		2
80.s	even	4	1		256.2.b.b		2
80.t	odd	4	1		256.2.b.b		2
85.g	odd	4	1		9248.2.a.f		1
120.q	odd	4	1		576.2.a.c		1
120.w	even	4	1		576.2.a.c		1
140.j	odd	4	1		1568.2.a.e		1
140.w	even	12	2		1568.2.i.g		2
140.x	odd	12	2		1568.2.i.f		2
160.u	even	8	2		1024.2.e.j		4
160.v	odd	8	2		1024.2.e.j		4
160.ba	even	8	2		1024.2.e.j		4
160.bb	odd	8	2		1024.2.e.j		4
180.v	odd	12	2		2592.2.i.e		2
180.x	even	12	2		2592.2.i.t		2
220.i	odd	4	1		3872.2.a.f		1
240.z	odd	4	1		2304.2.d.j		2
240.bb	even	4	1		2304.2.d.j		2
240.bd	odd	4	1		2304.2.d.j		2
240.bf	even	4	1		2304.2.d.j		2
260.p	even	4	1		5408.2.a.g		1
280.s	even	4	1		3136.2.a.m		1
280.y	odd	4	1		3136.2.a.m		1
340.r	even	4	1		9248.2.a.f		1
440.t	even	4	1		7744.2.a.v		1
440.w	odd	4	1		7744.2.a.v		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
32.2.a.a	✓	1	5.c	odd	4	1
32.2.a.a	✓	1	20.e	even	4	1
64.2.a.a		1	40.i	odd	4	1
64.2.a.a		1	40.k	even	4	1
256.2.b.b		2	80.i	odd	4	1
256.2.b.b		2	80.j	even	4	1
256.2.b.b		2	80.s	even	4	1
256.2.b.b		2	80.t	odd	4	1
288.2.a.d		1	15.e	even	4	1
288.2.a.d		1	60.l	odd	4	1
576.2.a.c		1	120.q	odd	4	1
576.2.a.c		1	120.w	even	4	1
800.2.a.d		1	5.c	odd	4	1
800.2.a.d		1	20.e	even	4	1
800.2.c.e		2	1.a	even	1	1	trivial
800.2.c.e		2	4.b	odd	2	1	CM
800.2.c.e		2	5.b	even	2	1	inner
800.2.c.e		2	20.d	odd	2	1	inner
1024.2.e.j		4	160.u	even	8	2
1024.2.e.j		4	160.v	odd	8	2
1024.2.e.j		4	160.ba	even	8	2
1024.2.e.j		4	160.bb	odd	8	2
1568.2.a.e		1	35.f	even	4	1
1568.2.a.e		1	140.j	odd	4	1
1568.2.i.f		2	35.k	even	12	2
1568.2.i.f		2	140.x	odd	12	2
1568.2.i.g		2	35.l	odd	12	2
1568.2.i.g		2	140.w	even	12	2
1600.2.a.n		1	40.i	odd	4	1
1600.2.a.n		1	40.k	even	4	1
1600.2.c.l		2	8.b	even	2	1
1600.2.c.l		2	8.d	odd	2	1
1600.2.c.l		2	40.e	odd	2	1
1600.2.c.l		2	40.f	even	2	1
2304.2.d.j		2	240.z	odd	4	1
2304.2.d.j		2	240.bb	even	4	1
2304.2.d.j		2	240.bd	odd	4	1
2304.2.d.j		2	240.bf	even	4	1
2592.2.i.e		2	45.l	even	12	2
2592.2.i.e		2	180.v	odd	12	2
2592.2.i.t		2	45.k	odd	12	2
2592.2.i.t		2	180.x	even	12	2
3136.2.a.m		1	280.s	even	4	1
3136.2.a.m		1	280.y	odd	4	1
3872.2.a.f		1	55.e	even	4	1
3872.2.a.f		1	220.i	odd	4	1
5408.2.a.g		1	65.h	odd	4	1
5408.2.a.g		1	260.p	even	4	1
7200.2.a.v		1	15.e	even	4	1
7200.2.a.v		1	60.l	odd	4	1
7200.2.f.m		2	3.b	odd	2	1
7200.2.f.m		2	12.b	even	2	1
7200.2.f.m		2	15.d	odd	2	1
7200.2.f.m		2	60.h	even	2	1
7744.2.a.v		1	440.t	even	4	1
7744.2.a.v		1	440.w	odd	4	1
9248.2.a.f		1	85.g	odd	4	1
9248.2.a.f		1	340.r	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(800, [\chi])$:

$T_{3}$

$T_{11}$

Hecke characteristic polynomials

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