Embedded newform 800.1.p.b.257.1

• Label: 800.1.p.b.257.1
• Level: $800$
• Weight: $1$
• Character: 800.257
• Analytic conductor: $0.399$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{4}$
• CM discriminant: -4
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$800 = 2^{5} \cdot 5^{2}$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	800.p (of order $4$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$0.399252010106$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
Defining polynomial:	$x^{2} + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 160)
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.2.2000.1
Artin image:	$C_4\wr C_2$
Artin field:	Galois closure of 8.0.8192000.1

Embedding invariants

Embedding label			257.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	800.257
Dual form			800.1.p.b.193.1

$q$-expansion

$f(q)$	$=$	$q+1.00000i q^{9} +(1.00000 - 1.00000i) q^{13} +(1.00000 + 1.00000i) q^{17} +(-1.00000 - 1.00000i) q^{37} -1.00000i q^{49} +(-1.00000 + 1.00000i) q^{53} +(-1.00000 + 1.00000i) q^{73} -1.00000 q^{81} +(-1.00000 - 1.00000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{13} + 2 q^{17} - 2 q^{37} - 2 q^{53} - 2 q^{73} - 2 q^{81} - 2 q^{97}+O(q^{100})$

Character values

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

$n$	$101$	$351$	$577$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

(See $a_n$ instead)

$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$
$2$		0			0
							$3$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
0.707107	+	0.707107i	$0.250000\pi$
$5$		0			0
							$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
−0.707107	+	0.707107i	$0.750000\pi$
$11$		0			0			−	1.00000i	$-0.5\pi$
									1.00000i	$0.5\pi$
$13$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
							1.00000			$0$
$17$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
									1.00000i	$0.5\pi$
$19$		0			0		1.00000			$0$
							−1.00000			$\pi$
$23$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
							0.707107	+	0.707107i	$0.250000\pi$
$29$		0			0		1.00000			$0$
							−1.00000			$\pi$
$31$		0			0			−	1.00000i	$-0.5\pi$
									1.00000i	$0.5\pi$
$37$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
							−1.00000			$\pi$
$41$		0			0			−	1.00000i	$-0.5\pi$
									1.00000i	$0.5\pi$
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
							0.707107	+	0.707107i	$0.250000\pi$
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
							−0.707107	+	0.707107i	$0.750000\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.1.p.b.257.1		2
4.3	odd	2	CM	800.1.p.b.257.1		2
5.2	odd	4		160.1.p.a.33.1	✓	2
5.3	odd	4	inner	800.1.p.b.193.1		2
5.4	even	2		160.1.p.a.97.1	yes	2
8.3	odd	2		1600.1.p.b.257.1		2
8.5	even	2		1600.1.p.b.257.1		2
15.2	even	4		1440.1.bh.b.1153.1		2
15.14	odd	2		1440.1.bh.b.577.1		2
20.3	even	4	inner	800.1.p.b.193.1		2
20.7	even	4		160.1.p.a.33.1	✓	2
20.19	odd	2		160.1.p.a.97.1	yes	2
40.3	even	4		1600.1.p.b.193.1		2
40.13	odd	4		1600.1.p.b.193.1		2
40.19	odd	2		320.1.p.a.257.1		2
40.27	even	4		320.1.p.a.193.1		2
40.29	even	2		320.1.p.a.257.1		2
40.37	odd	4		320.1.p.a.193.1		2
60.47	odd	4		1440.1.bh.b.1153.1		2
60.59	even	2		1440.1.bh.b.577.1		2
80.19	odd	4		1280.1.m.a.897.1		2
80.27	even	4		1280.1.m.a.1153.1		2
80.29	even	4		1280.1.m.a.897.1		2
80.37	odd	4		1280.1.m.a.1153.1		2
80.59	odd	4		1280.1.m.b.897.1		2
80.67	even	4		1280.1.m.b.1153.1		2
80.69	even	4		1280.1.m.b.897.1		2
80.77	odd	4		1280.1.m.b.1153.1		2
120.29	odd	2		2880.1.bh.b.577.1		2
120.59	even	2		2880.1.bh.b.577.1		2
120.77	even	4		2880.1.bh.b.1153.1		2
120.107	odd	4		2880.1.bh.b.1153.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.1.p.a.33.1	✓	2	5.2	odd	4
160.1.p.a.33.1	✓	2	20.7	even	4
160.1.p.a.97.1	yes	2	5.4	even	2
160.1.p.a.97.1	yes	2	20.19	odd	2
320.1.p.a.193.1		2	40.27	even	4
320.1.p.a.193.1		2	40.37	odd	4
320.1.p.a.257.1		2	40.19	odd	2
320.1.p.a.257.1		2	40.29	even	2
800.1.p.b.193.1		2	5.3	odd	4	inner
800.1.p.b.193.1		2	20.3	even	4	inner
800.1.p.b.257.1		2	1.1	even	1	trivial
800.1.p.b.257.1		2	4.3	odd	2	CM
1280.1.m.a.897.1		2	80.19	odd	4
1280.1.m.a.897.1		2	80.29	even	4
1280.1.m.a.1153.1		2	80.27	even	4
1280.1.m.a.1153.1		2	80.37	odd	4
1280.1.m.b.897.1		2	80.59	odd	4
1280.1.m.b.897.1		2	80.69	even	4
1280.1.m.b.1153.1		2	80.67	even	4
1280.1.m.b.1153.1		2	80.77	odd	4
1440.1.bh.b.577.1		2	15.14	odd	2
1440.1.bh.b.577.1		2	60.59	even	2
1440.1.bh.b.1153.1		2	15.2	even	4
1440.1.bh.b.1153.1		2	60.47	odd	4
1600.1.p.b.193.1		2	40.3	even	4
1600.1.p.b.193.1		2	40.13	odd	4
1600.1.p.b.257.1		2	8.3	odd	2
1600.1.p.b.257.1		2	8.5	even	2
2880.1.bh.b.577.1		2	120.29	odd	2
2880.1.bh.b.577.1		2	120.59	even	2
2880.1.bh.b.1153.1		2	120.77	even	4
2880.1.bh.b.1153.1		2	120.107	odd	4