Embedded newform 1280.1.h.c.1279.2

• Label: 1280.1.h.c.1279.2
• Level: $1280$
• Weight: $1$
• Character: 1280.1279
• Self dual: yes
• Analytic conductor: $0.639$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{4}$
• CM discriminant: -20
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$1280 = 2^{8} \cdot 5$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	1280.h (of order $2$, degree $1$, not minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	$0.638803216170$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
Defining polynomial:	$x^{2} - 2$
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 640)
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.2.1600.1
Artin image:	$D_8$
Artin field:	Galois closure of 8.0.2097152000.6

Embedding invariants

Embedding label			1279.2
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	1280.1279

$q$-expansion

$f(q)$	$=$	$q+1.41421 q^{3} +1.00000 q^{5} -1.41421 q^{7} +1.00000 q^{9} +1.41421 q^{15} -2.00000 q^{21} +1.41421 q^{23} +1.00000 q^{25} -1.41421 q^{35} -1.41421 q^{43} +1.00000 q^{45} -1.41421 q^{47} +1.00000 q^{49} -2.00000 q^{61} -1.41421 q^{63} -1.41421 q^{67} +2.00000 q^{69} +1.41421 q^{75} -1.00000 q^{81} -1.41421 q^{83} -2.00000 q^{89} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{5} + 2 q^{9} - 4 q^{21} + 2 q^{25} + 2 q^{45} + 2 q^{49} - 4 q^{61} + 4 q^{69} - 2 q^{81} - 4 q^{89}+O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$-1$	$1$	$-1$

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
			$3$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
0.707107	+	0.707107i				$0.250000\pi$
$5$	1.00000	1.00000
			$7$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
−0.707107	+	0.707107i				$0.750000\pi$
$11$	0	0	1.00000			$0$
			−1.00000			$\pi$
$13$	0	0	1.00000			$0$
			−1.00000			$\pi$
$17$	0	0	1.00000			$0$
			−1.00000			$\pi$
$19$	0	0	1.00000			$0$
			−1.00000			$\pi$
$23$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
			0.707107	+	0.707107i	$0.250000\pi$
$29$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$31$	0	0	1.00000			$0$
			−1.00000			$\pi$
$37$	0	0	1.00000			$0$
			−1.00000			$\pi$
$41$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$43$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
			−0.707107	+	0.707107i	$0.750000\pi$
$47$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
			−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	1280.1.h.c.1279.2		2
4.3	odd	2	inner	1280.1.h.c.1279.1		2
5.4	even	2	inner	1280.1.h.c.1279.1		2
8.3	odd	2		1280.1.h.a.1279.2		2
8.5	even	2		1280.1.h.a.1279.1		2
16.3	odd	4		640.1.e.c.319.1	✓	4
16.5	even	4		640.1.e.c.319.2	yes	4
16.11	odd	4		640.1.e.c.319.4	yes	4
16.13	even	4		640.1.e.c.319.3	yes	4
20.19	odd	2	CM	1280.1.h.c.1279.2		2
40.19	odd	2		1280.1.h.a.1279.1		2
40.29	even	2		1280.1.h.a.1279.2		2
80.3	even	4		3200.1.g.e.2751.4		4
80.13	odd	4		3200.1.g.e.2751.1		4
80.19	odd	4		640.1.e.c.319.3	yes	4
80.27	even	4		3200.1.g.e.2751.3		4
80.29	even	4		640.1.e.c.319.1	✓	4
80.37	odd	4		3200.1.g.e.2751.2		4
80.43	even	4		3200.1.g.e.2751.2		4
80.53	odd	4		3200.1.g.e.2751.3		4
80.59	odd	4		640.1.e.c.319.2	yes	4
80.67	even	4		3200.1.g.e.2751.1		4
80.69	even	4		640.1.e.c.319.4	yes	4
80.77	odd	4		3200.1.g.e.2751.4		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.1.e.c.319.1	✓	4	16.3	odd	4
640.1.e.c.319.1	✓	4	80.29	even	4
640.1.e.c.319.2	yes	4	16.5	even	4
640.1.e.c.319.2	yes	4	80.59	odd	4
640.1.e.c.319.3	yes	4	16.13	even	4
640.1.e.c.319.3	yes	4	80.19	odd	4
640.1.e.c.319.4	yes	4	16.11	odd	4
640.1.e.c.319.4	yes	4	80.69	even	4
1280.1.h.a.1279.1		2	8.5	even	2
1280.1.h.a.1279.1		2	40.19	odd	2
1280.1.h.a.1279.2		2	8.3	odd	2
1280.1.h.a.1279.2		2	40.29	even	2
1280.1.h.c.1279.1		2	4.3	odd	2	inner
1280.1.h.c.1279.1		2	5.4	even	2	inner
1280.1.h.c.1279.2		2	1.1	even	1	trivial
1280.1.h.c.1279.2		2	20.19	odd	2	CM
3200.1.g.e.2751.1		4	80.13	odd	4
3200.1.g.e.2751.1		4	80.67	even	4
3200.1.g.e.2751.2		4	80.37	odd	4
3200.1.g.e.2751.2		4	80.43	even	4
3200.1.g.e.2751.3		4	80.27	even	4
3200.1.g.e.2751.3		4	80.53	odd	4
3200.1.g.e.2751.4		4	80.3	even	4
3200.1.g.e.2751.4		4	80.77	odd	4