Embedded newform 32.8.g.a.21.13

• Label: 32.8.g.a.21.13
• 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			21.13
Character	\(\chi\)	\(=\)	32.21
Dual form			32.8.g.a.29.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.374822 + 11.3075i) q^{2} +(19.9601 - 8.26773i) q^{3} +(-127.719 + 8.47659i) q^{4} +(-34.6187 + 83.5770i) q^{5} +(100.969 + 222.600i) q^{6} +(-609.571 - 609.571i) q^{7} +(-143.721 - 1441.01i) q^{8} +(-1216.39 + 1216.39i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.374822	+	11.3075i	0.0331299	+	0.999451i
							\(3\)	19.9601	−	8.26773i	0.426813	−	0.176792i	−0.158928	−	0.987290i	\(-0.550804\pi\)
0.585741	+	0.810498i	\(0.300804\pi\)
\(5\)	−34.6187	+	83.5770i	−0.123856	+	0.299014i	−0.973630	−	0.228132i	\(-0.926738\pi\)
							0.849774	+	0.527147i	\(0.176738\pi\)
\(7\)	−609.571	−	609.571i	−0.671709	−	0.671709i	0.286401	−	0.958110i	\(-0.407541\pi\)
							−0.958110	+	0.286401i	\(0.907541\pi\)
\(11\)	−3768.29	−	1560.88i	−0.853631	−	0.353585i	−0.0874173	−	0.996172i	\(-0.527861\pi\)
							−0.766213	+	0.642586i	\(0.777861\pi\)
\(13\)	−450.939	−	1088.66i	−0.0569266	−	0.137433i	0.892857	−	0.450340i	\(-0.148697\pi\)
							−0.949784	+	0.312907i	\(0.898697\pi\)
\(17\)		−	17440.8i		−	0.860983i	−0.902595	−	0.430491i	\(-0.858340\pi\)
							0.902595	−	0.430491i	\(-0.141660\pi\)
\(19\)	−10378.7	−	25056.3i	−0.347140	−	0.838069i	−0.996955	−	0.0779773i	\(-0.975154\pi\)
							0.649815	−	0.760092i	\(-0.274846\pi\)
\(23\)	−23041.5	+	23041.5i	−0.394879	+	0.394879i	−0.876422	−	0.481543i	\(-0.840076\pi\)
							0.481543	+	0.876422i	\(0.340076\pi\)
\(29\)	−163758.	+	67830.7i	−1.24683	+	0.516456i	−0.905844	−	0.423611i	\(-0.860762\pi\)
							−0.340990	+	0.940067i	\(0.610762\pi\)
\(31\)	−73511.9			−0.443192			−0.221596	−	0.975139i	\(-0.571127\pi\)
							−0.221596	+	0.975139i	\(0.571127\pi\)
\(37\)	−65461.4	+	158038.i	−0.212461	+	0.512926i	−0.993800	−	0.111180i	\(-0.964537\pi\)
							0.781339	+	0.624106i	\(0.214537\pi\)
\(41\)	−454708.	+	454708.i	−1.03036	+	1.03036i	−0.0308350	+	0.999524i	\(0.509817\pi\)
							−0.999524	+	0.0308350i	\(0.990183\pi\)
\(43\)	474130.	+	196391.i	0.909406	+	0.376688i	0.787829	−	0.615894i	\(-0.211205\pi\)
							0.121577	+	0.992582i	\(0.461205\pi\)
\(47\)		−	4054.13i		−	0.00569580i	−0.999996	−	0.00284790i	\(-0.999093\pi\)
							0.999996	−	0.00284790i	\(-0.000906516\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.8.g.a.21.13	✓	108
32.29	even	8	inner	32.8.g.a.29.13	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.8.g.a.21.13	✓	108	1.1	even	1	trivial
32.8.g.a.29.13	yes	108	32.29	even	8	inner