Embedded newform 32.4.g.a.5.1

• Label: 32.4.g.a.5.1
• Level: $32$
• Weight: $4$
• Character: 32.5
• 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			5.1
Character	\(\chi\)	\(=\)	32.5
Dual form			32.4.g.a.13.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.82381 - 0.161516i) q^{2} +(-0.477813 - 1.15354i) q^{3} +(7.94783 + 0.912182i) q^{4} +(-16.3468 - 6.77105i) q^{5} +(1.16294 + 3.33456i) q^{6} +(-18.0222 - 18.0222i) q^{7} +(-22.2958 - 3.85953i) q^{8} +(17.9895 - 17.9895i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.82381	−	0.161516i	−0.998368	−	0.0571046i
							\(3\)	−0.477813	−	1.15354i	−0.0919551	−	0.221999i	0.871210	−	0.490911i	\(-0.163336\pi\)
−0.963165	+	0.268912i	\(0.913336\pi\)
\(5\)	−16.3468	−	6.77105i	−1.46210	−	0.605621i	−0.497057	−	0.867718i	\(-0.665586\pi\)
							−0.965042	+	0.262096i	\(0.915586\pi\)
\(7\)	−18.0222	−	18.0222i	−0.973108	−	0.973108i	0.0265394	−	0.999648i	\(-0.491551\pi\)
							−0.999648	+	0.0265394i	\(0.991551\pi\)
\(11\)	−20.1053	+	48.5384i	−0.551088	+	1.33044i	0.365575	+	0.930782i	\(0.380872\pi\)
							−0.916663	+	0.399662i	\(0.869128\pi\)
\(13\)	37.8086	−	15.6609i	0.806633	−	0.334118i	0.0590233	−	0.998257i	\(-0.481201\pi\)
							0.747610	+	0.664138i	\(0.231201\pi\)
\(17\)		−	53.0615i		−	0.757018i	−0.925598	−	0.378509i	\(-0.876437\pi\)
							0.925598	−	0.378509i	\(-0.123563\pi\)
\(19\)	−32.4427	+	13.4382i	−0.391730	+	0.162260i	−0.569850	−	0.821749i	\(-0.692999\pi\)
							0.178120	+	0.984009i	\(0.442999\pi\)
\(23\)	32.1343	−	32.1343i	0.291324	−	0.291324i	−0.546279	−	0.837603i	\(-0.683956\pi\)
							0.837603	+	0.546279i	\(0.183956\pi\)
\(29\)	−52.0634	−	125.692i	−0.333377	−	0.804842i	−0.998320	−	0.0579483i	\(-0.981544\pi\)
							0.664943	−	0.746894i	\(-0.268456\pi\)
\(31\)	−53.3354			−0.309010			−0.154505	−	0.987992i	\(-0.549378\pi\)
							−0.154505	+	0.987992i	\(0.549378\pi\)
\(37\)	−57.3789	−	23.7671i	−0.254947	−	0.105602i	0.251550	−	0.967844i	\(-0.419060\pi\)
							−0.506497	+	0.862242i	\(0.669060\pi\)
\(41\)	240.383	−	240.383i	0.915647	−	0.915647i	−0.0810624	−	0.996709i	\(-0.525831\pi\)
							0.996709	+	0.0810624i	\(0.0258313\pi\)
\(43\)	56.3244	−	135.979i	0.199753	−	0.482247i	−0.791983	−	0.610544i	\(-0.790951\pi\)
							0.991736	+	0.128297i	\(0.0409510\pi\)
\(47\)		−	314.904i		−	0.977309i	−0.872477	−	0.488655i	\(-0.837488\pi\)
							0.872477	−	0.488655i	\(-0.162512\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.4.g.a.5.1	✓	44
4.3	odd	2		128.4.g.a.113.6		44
8.3	odd	2		256.4.g.a.225.6		44
8.5	even	2		256.4.g.b.225.6		44
32.3	odd	8		256.4.g.a.33.6		44
32.13	even	8	inner	32.4.g.a.13.1	yes	44
32.19	odd	8		128.4.g.a.17.6		44
32.29	even	8		256.4.g.b.33.6		44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.4.g.a.5.1	✓	44	1.1	even	1	trivial
32.4.g.a.13.1	yes	44	32.13	even	8	inner
128.4.g.a.17.6		44	32.19	odd	8
128.4.g.a.113.6		44	4.3	odd	2
256.4.g.a.33.6		44	32.3	odd	8
256.4.g.a.225.6		44	8.3	odd	2
256.4.g.b.33.6		44	32.29	even	8
256.4.g.b.225.6		44	8.5	even	2