Embedded newform 32.2.g.b.13.1

• Label: 32.2.g.b.13.1
• Level: $32$
• Weight: $2$
• Character: 32.13
• Analytic conductor: $0.256$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	32.g (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.255521286468\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{8})\)
Coefficient field:	8.0.18939904.2
Defining polynomial:	\( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			13.1
Root			\(0.500000 - 2.10607i\) of defining polynomial
Character	\(\chi\)	\(=\)	32.13
Dual form			32.2.g.b.5.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.26330 + 0.635665i) q^{2} +(-1.07947 + 2.60607i) q^{3} +(1.19186 - 1.60607i) q^{4} +(0.707107 - 0.292893i) q^{5} +(-0.292893 - 3.97844i) q^{6} +(1.68554 - 1.68554i) q^{7} +(-0.484753 + 2.78658i) q^{8} +(-3.50504 - 3.50504i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} - 4 q^{3} + 4 q^{4} - 8 q^{6} - 8 q^{7} - 4 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{7}{8}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.26330	+	0.635665i	−0.893289	+	0.449483i
							\(3\)	−1.07947	+	2.60607i	−0.623233	+	1.50462i	0.224653	+	0.974439i	\(0.427875\pi\)
−0.847886	+	0.530178i	\(0.822125\pi\)
\(5\)	0.707107	−	0.292893i	0.316228	−	0.130986i	−0.218924	−	0.975742i	\(-0.570255\pi\)
							0.535151	+	0.844756i	\(0.320255\pi\)
\(7\)	1.68554	−	1.68554i	0.637076	−	0.637076i	−0.312757	−	0.949833i	\(-0.601253\pi\)
							0.949833	+	0.312757i	\(0.101253\pi\)
\(11\)	−0.334743	−	0.808140i	−0.100929	−	0.243664i	0.865347	−	0.501173i	\(-0.167098\pi\)
							−0.966276	+	0.257510i	\(0.917098\pi\)
\(13\)	1.09083	+	0.451835i	0.302541	+	0.125316i	0.528789	−	0.848753i	\(-0.322646\pi\)
							−0.226249	+	0.974070i	\(0.572646\pi\)
\(17\)			0.224777i			0.0545165i	0.999628	+	0.0272583i	\(0.00867765\pi\)
							−0.999628	+	0.0272583i	\(0.991322\pi\)
\(19\)	−2.87740	−	1.19186i	−0.660122	−	0.273431i	0.0273681	−	0.999625i	\(-0.491287\pi\)
							−0.687490	+	0.726194i	\(0.741287\pi\)
\(23\)	−3.68554	−	3.68554i	−0.768489	−	0.768489i	0.209351	−	0.977840i	\(-0.432865\pi\)
							−0.977840	+	0.209351i	\(0.932865\pi\)
\(29\)	2.34610	−	5.66398i	0.435659	−	1.05177i	−0.541773	−	0.840525i	\(-0.682247\pi\)
							0.977432	−	0.211250i	\(-0.0677534\pi\)
\(31\)	6.82843			1.22642			0.613211	−	0.789919i	\(-0.289878\pi\)
							0.613211	+	0.789919i	\(0.289878\pi\)
\(37\)	−9.87613	+	4.09083i	−1.62363	+	0.672528i	−0.994496	−	0.104773i	\(-0.966588\pi\)
							−0.629129	+	0.777301i	\(0.716588\pi\)
\(41\)	6.37109	+	6.37109i	0.994997	+	0.994997i	0.999988	−	0.00499079i	\(-0.00158862\pi\)
							−0.00499079	+	0.999988i	\(0.501589\pi\)
\(43\)	1.90790	+	4.60607i	0.290952	+	0.702420i	0.999996	−	0.00274415i	\(-0.000873491\pi\)
							−0.709045	+	0.705164i	\(0.750873\pi\)
\(47\)			0.542661i			0.0791552i	0.999216	+	0.0395776i	\(0.0126012\pi\)
							−0.999216	+	0.0395776i	\(0.987399\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.2.g.b.13.1	yes	8
3.2	odd	2		288.2.v.b.109.2		8
4.3	odd	2		128.2.g.b.17.2		8
5.2	odd	4		800.2.ba.d.749.1		8
5.3	odd	4		800.2.ba.c.749.2		8
5.4	even	2		800.2.y.b.301.2		8
8.3	odd	2		256.2.g.c.33.1		8
8.5	even	2		256.2.g.d.33.2		8
12.11	even	2		1152.2.v.b.145.1		8
16.3	odd	4		512.2.g.g.321.2		8
16.5	even	4		512.2.g.h.321.2		8
16.11	odd	4		512.2.g.f.321.1		8
16.13	even	4		512.2.g.e.321.1		8
32.3	odd	8		512.2.g.g.193.2		8
32.5	even	8	inner	32.2.g.b.5.1	✓	8
32.11	odd	8		256.2.g.c.225.1		8
32.13	even	8		512.2.g.h.193.2		8
32.19	odd	8		512.2.g.f.193.1		8
32.21	even	8		256.2.g.d.225.2		8
32.27	odd	8		128.2.g.b.113.2		8
32.29	even	8		512.2.g.e.193.1		8
64.5	even	16		4096.2.a.k.1.1		8
64.27	odd	16		4096.2.a.q.1.1		8
64.37	even	16		4096.2.a.k.1.8		8
64.59	odd	16		4096.2.a.q.1.8		8
96.5	odd	8		288.2.v.b.37.2		8
96.59	even	8		1152.2.v.b.1009.1		8
160.37	odd	8		800.2.ba.c.549.2		8
160.69	even	8		800.2.y.b.101.2		8
160.133	odd	8		800.2.ba.d.549.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.2.g.b.5.1	✓	8	32.5	even	8	inner
32.2.g.b.13.1	yes	8	1.1	even	1	trivial
128.2.g.b.17.2		8	4.3	odd	2
128.2.g.b.113.2		8	32.27	odd	8
256.2.g.c.33.1		8	8.3	odd	2
256.2.g.c.225.1		8	32.11	odd	8
256.2.g.d.33.2		8	8.5	even	2
256.2.g.d.225.2		8	32.21	even	8
288.2.v.b.37.2		8	96.5	odd	8
288.2.v.b.109.2		8	3.2	odd	2
512.2.g.e.193.1		8	32.29	even	8
512.2.g.e.321.1		8	16.13	even	4
512.2.g.f.193.1		8	32.19	odd	8
512.2.g.f.321.1		8	16.11	odd	4
512.2.g.g.193.2		8	32.3	odd	8
512.2.g.g.321.2		8	16.3	odd	4
512.2.g.h.193.2		8	32.13	even	8
512.2.g.h.321.2		8	16.5	even	4
800.2.y.b.101.2		8	160.69	even	8
800.2.y.b.301.2		8	5.4	even	2
800.2.ba.c.549.2		8	160.37	odd	8
800.2.ba.c.749.2		8	5.3	odd	4
800.2.ba.d.549.1		8	160.133	odd	8
800.2.ba.d.749.1		8	5.2	odd	4
1152.2.v.b.145.1		8	12.11	even	2
1152.2.v.b.1009.1		8	96.59	even	8
4096.2.a.k.1.1		8	64.5	even	16
4096.2.a.k.1.8		8	64.37	even	16
4096.2.a.q.1.1		8	64.27	odd	16
4096.2.a.q.1.8		8	64.59	odd	16