Embedded newform 32.8.g.a.21.27

• Label: 32.8.g.a.21.27
• Level: $32$
• Weight: $8$
• Character: 32.21
• Analytic conductor: $9.996$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(9.99632081549\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(27\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			21.27
Character	\(\chi\)	\(=\)	32.21
Dual form			32.8.g.a.29.27

$q$-expansion

\(f(q)\)	\(=\)	\(q+(11.3101 + 0.284235i) q^{2} +(33.6929 - 13.9561i) q^{3} +(127.838 + 6.42947i) q^{4} +(117.289 - 283.161i) q^{5} +(385.039 - 148.268i) q^{6} +(-590.253 - 590.253i) q^{7} +(1444.04 + 109.054i) q^{8} +(-606.000 + 606.000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	11.3101	+	0.284235i	0.999684	+	0.0251230i
							\(3\)	33.6929	−	13.9561i	0.720468	−	0.298427i	0.00783933	−	0.999969i	\(-0.497505\pi\)
0.712628	+	0.701542i	\(0.247505\pi\)
\(5\)	117.289	−	283.161i	0.419626	−	1.01307i	−0.562830	−	0.826572i	\(-0.690288\pi\)
							0.982456	−	0.186494i	\(-0.0597124\pi\)
\(7\)	−590.253	−	590.253i	−0.650422	−	0.650422i	0.302672	−	0.953095i	\(-0.402121\pi\)
							−0.953095	+	0.302672i	\(0.902121\pi\)
\(11\)	1813.24	+	751.067i	0.410752	+	0.170139i	0.578484	−	0.815693i	\(-0.303644\pi\)
							−0.167732	+	0.985833i	\(0.553644\pi\)
\(13\)	999.831	+	2413.81i	0.126219	+	0.304720i	0.974339	−	0.225084i	\(-0.0722656\pi\)
							−0.848120	+	0.529804i	\(0.822266\pi\)
\(17\)		−	4455.61i		−	0.219956i	−0.993934	−	0.109978i	\(-0.964922\pi\)
							0.993934	−	0.109978i	\(-0.0350780\pi\)
\(19\)	12359.3	+	29838.1i	0.413387	+	0.998006i	0.984222	+	0.176940i	\(0.0566199\pi\)
							−0.570834	+	0.821065i	\(0.693380\pi\)
\(23\)	9861.04	−	9861.04i	0.168996	−	0.168996i	−0.617542	−	0.786538i	\(-0.711872\pi\)
							0.786538	+	0.617542i	\(0.211872\pi\)
\(29\)	46780.8	−	19377.2i	0.356184	−	0.147536i	−0.197414	−	0.980320i	\(-0.563254\pi\)
							0.553598	+	0.832784i	\(0.313254\pi\)
\(31\)	−177683.			−1.07122			−0.535610	−	0.844465i	\(-0.679918\pi\)
							−0.535610	+	0.844465i	\(0.679918\pi\)
\(37\)	−204326.	+	493287.i	−0.663160	+	1.60101i	0.129664	+	0.991558i	\(0.458610\pi\)
							−0.792824	+	0.609451i	\(0.791390\pi\)
\(41\)	−479199.	+	479199.i	−1.08586	+	1.08586i	−0.0899060	+	0.995950i	\(0.528657\pi\)
							−0.995950	+	0.0899060i	\(0.971343\pi\)
\(43\)	−545936.	−	226134.i	−1.04713	−	0.433737i	−0.208265	−	0.978072i	\(-0.566782\pi\)
							−0.838868	+	0.544336i	\(0.816782\pi\)
\(47\)		−	838999.i		−	1.17874i	−0.807863	−	0.589371i	\(-0.799376\pi\)
							0.807863	−	0.589371i	\(-0.200624\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.8.g.a.21.27	✓	108
32.29	even	8	inner	32.8.g.a.29.27	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.8.g.a.21.27	✓	108	1.1	even	1	trivial
32.8.g.a.29.27	yes	108	32.29	even	8	inner