Embedded newform 32.6.g.a.5.16

• Label: 32.6.g.a.5.16
• Level: $32$
• Weight: $6$
• Character: 32.5
• Analytic conductor: $5.132$
• Analytic rank: $0$
• Dimension: $76$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	32.g (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.13228223402\)
Analytic rank:	\(0\)
Dimension:	\(76\)
Relative dimension:	\(19\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			5.16
Character	\(\chi\)	\(=\)	32.5
Dual form			32.6.g.a.13.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.85839 - 2.89759i) q^{2} +(10.7913 + 26.0525i) q^{3} +(15.2079 - 28.1553i) q^{4} +(-26.5895 - 11.0137i) q^{5} +(127.918 + 95.3043i) q^{6} +(131.646 + 131.646i) q^{7} +(-7.69630 - 180.856i) q^{8} +(-390.452 + 390.452i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.85839	−	2.89759i	0.858850	−	0.512226i
							\(3\)	10.7913	+	26.0525i	0.692261	+	1.67127i	0.740175	+	0.672414i	\(0.234743\pi\)
−0.0479141	+	0.998851i	\(0.515257\pi\)
\(5\)	−26.5895	−	11.0137i	−0.475648	−	0.197020i	0.131963	−	0.991255i	\(-0.457872\pi\)
							−0.607611	+	0.794235i	\(0.707872\pi\)
\(7\)	131.646	+	131.646i	1.01546	+	1.01546i	0.999879	+	0.0155839i	\(0.00496070\pi\)
							0.0155839	+	0.999879i	\(0.495039\pi\)
\(11\)	135.185	−	326.366i	0.336858	−	0.813248i	−0.661155	−	0.750249i	\(-0.729934\pi\)
							0.998014	−	0.0629986i	\(-0.0200664\pi\)
\(13\)	35.1247	−	14.5491i	0.0576440	−	0.0238769i	−0.353675	−	0.935368i	\(-0.615068\pi\)
							0.411319	+	0.911491i	\(0.365068\pi\)
\(17\)		−	2178.41i		−	1.82817i	−0.405522	−	0.914085i	\(-0.632910\pi\)
							0.405522	−	0.914085i	\(-0.367090\pi\)
\(19\)	−852.409	+	353.079i	−0.541706	+	0.224382i	−0.636722	−	0.771094i	\(-0.719710\pi\)
							0.0950153	+	0.995476i	\(0.469710\pi\)
\(23\)	−392.030	+	392.030i	−0.154525	+	0.154525i	−0.780136	−	0.625610i	\(-0.784850\pi\)
							0.625610	+	0.780136i	\(0.284850\pi\)
\(29\)	1340.97	+	3237.39i	0.296090	+	0.714825i	0.999990	+	0.00452531i	\(0.00144045\pi\)
							−0.703900	+	0.710299i	\(0.748560\pi\)
\(31\)	−297.407			−0.0555837			−0.0277918	−	0.999614i	\(-0.508848\pi\)
							−0.0277918	+	0.999614i	\(0.508848\pi\)
\(37\)	8894.30	+	3684.14i	1.06809	+	0.442417i	0.846316	−	0.532681i	\(-0.178816\pi\)
							0.221773	+	0.975098i	\(0.428816\pi\)
\(41\)	550.427	−	550.427i	0.0511376	−	0.0511376i	−0.681076	−	0.732213i	\(-0.738487\pi\)
							0.732213	+	0.681076i	\(0.238487\pi\)
\(43\)	−3661.14	+	8838.76i	−0.301957	+	0.728988i	0.697961	+	0.716136i	\(0.254091\pi\)
							−0.999917	+	0.0128519i	\(0.995909\pi\)
\(47\)		−	18960.2i		−	1.25198i	−0.779830	−	0.625992i	\(-0.784694\pi\)
							0.779830	−	0.625992i	\(-0.215306\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.6.g.a.5.16	✓	76
4.3	odd	2		128.6.g.a.113.1		76
32.13	even	8	inner	32.6.g.a.13.16	yes	76
32.19	odd	8		128.6.g.a.17.1		76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.6.g.a.5.16	✓	76	1.1	even	1	trivial
32.6.g.a.13.16	yes	76	32.13	even	8	inner
128.6.g.a.17.1		76	32.19	odd	8
128.6.g.a.113.1		76	4.3	odd	2