Embedded newform 32.4.g.a.21.2

• Label: 32.4.g.a.21.2
• Level: $32$
• Weight: $4$
• Character: 32.21
• Analytic conductor: $1.888$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.88806112018\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(11\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			21.2
Character	\(\chi\)	\(=\)	32.21
Dual form			32.4.g.a.29.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.58440 + 1.14930i) q^{2} +(-5.56908 + 2.30679i) q^{3} +(5.35820 - 5.94051i) q^{4} +(6.28381 - 15.1704i) q^{5} +(11.7415 - 12.3622i) q^{6} +(-16.6573 - 16.6573i) q^{7} +(-7.02028 + 21.5108i) q^{8} +(6.60151 - 6.60151i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.58440	+	1.14930i	−0.913722	+	0.406340i
							\(3\)	−5.56908	+	2.30679i	−1.07177	+	0.443942i	−0.847615	−	0.530612i	\(-0.821962\pi\)
−0.224155	+	0.974553i	\(0.571962\pi\)
\(5\)	6.28381	−	15.1704i	0.562041	−	1.35689i	−0.346091	−	0.938201i	\(-0.612491\pi\)
							0.908131	−	0.418685i	\(-0.137509\pi\)
\(7\)	−16.6573	−	16.6573i	−0.899411	−	0.899411i	0.0959733	−	0.995384i	\(-0.469404\pi\)
							−0.995384	+	0.0959733i	\(0.969404\pi\)
\(11\)	3.11257	+	1.28927i	0.0853158	+	0.0353390i	0.424933	−	0.905225i	\(-0.360298\pi\)
							−0.339617	+	0.940564i	\(0.610298\pi\)
\(13\)	−28.8862	−	69.7374i	−0.616275	−	1.48782i	−0.855998	−	0.516979i	\(-0.827057\pi\)
							0.239723	−	0.970841i	\(-0.422943\pi\)
\(17\)			66.2302i			0.944893i	0.881359	+	0.472447i	\(0.156629\pi\)
							−0.881359	+	0.472447i	\(0.843371\pi\)
\(19\)	−12.5299	−	30.2498i	−0.151292	−	0.365251i	0.830004	−	0.557758i	\(-0.188338\pi\)
							−0.981296	+	0.192507i	\(0.938338\pi\)
\(23\)	−63.7309	+	63.7309i	−0.577775	+	0.577775i	−0.934290	−	0.356515i	\(-0.883965\pi\)
							0.356515	+	0.934290i	\(0.383965\pi\)
\(29\)	190.908	−	79.0767i	1.22244	−	0.506351i	0.324255	−	0.945970i	\(-0.394887\pi\)
							0.898184	+	0.439619i	\(0.144887\pi\)
\(31\)	123.811			0.717325			0.358662	−	0.933467i	\(-0.383233\pi\)
							0.358662	+	0.933467i	\(0.383233\pi\)
\(37\)	−46.0901	+	111.271i	−0.204788	+	0.494403i	−0.992588	−	0.121529i	\(-0.961220\pi\)
							0.787799	+	0.615932i	\(0.211220\pi\)
\(41\)	−100.187	+	100.187i	−0.381624	+	0.381624i	−0.871687	−	0.490063i	\(-0.836974\pi\)
							0.490063	+	0.871687i	\(0.336974\pi\)
\(43\)	27.5836	+	11.4255i	0.0978246	+	0.0405203i	0.431059	−	0.902324i	\(-0.358140\pi\)
							−0.333234	+	0.942844i	\(0.608140\pi\)
\(47\)		−	394.293i		−	1.22369i	−0.790976	−	0.611847i	\(-0.790427\pi\)
							0.790976	−	0.611847i	\(-0.209573\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.4.g.a.21.2	✓	44
4.3	odd	2		128.4.g.a.49.9		44
8.3	odd	2		256.4.g.a.97.3		44
8.5	even	2		256.4.g.b.97.9		44
32.3	odd	8		128.4.g.a.81.9		44
32.13	even	8		256.4.g.b.161.9		44
32.19	odd	8		256.4.g.a.161.3		44
32.29	even	8	inner	32.4.g.a.29.2	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.4.g.a.21.2	✓	44	1.1	even	1	trivial
32.4.g.a.29.2	yes	44	32.29	even	8	inner
128.4.g.a.49.9		44	4.3	odd	2
128.4.g.a.81.9		44	32.3	odd	8
256.4.g.a.97.3		44	8.3	odd	2
256.4.g.a.161.3		44	32.19	odd	8
256.4.g.b.97.9		44	8.5	even	2
256.4.g.b.161.9		44	32.13	even	8