Embedded newform 32.4.g.a.13.9

• Label: 32.4.g.a.13.9
• Level: $32$
• Weight: $4$
• Character: 32.13
• 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			13.9
Character	\(\chi\)	\(=\)	32.13
Dual form			32.4.g.a.5.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.01560 + 1.98428i) q^{2} +(-1.20185 + 2.90153i) q^{3} +(0.125267 + 7.99902i) q^{4} +(-3.98512 + 1.65069i) q^{5} +(-8.17991 + 3.46351i) q^{6} +(22.4050 - 22.4050i) q^{7} +(-15.6198 + 16.3714i) q^{8} +(12.1174 + 12.1174i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{7}{8}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.01560	+	1.98428i	0.712621	+	0.701549i
							\(3\)	−1.20185	+	2.90153i	−0.231297	+	0.558400i	−0.996330	−	0.0855902i	\(-0.972722\pi\)
0.765033	+	0.643991i	\(0.222722\pi\)
\(5\)	−3.98512	+	1.65069i	−0.356440	+	0.147642i	−0.553716	−	0.832706i	\(-0.686791\pi\)
							0.197276	+	0.980348i	\(0.436791\pi\)
\(7\)	22.4050	−	22.4050i	1.20976	−	1.20976i	0.238651	−	0.971105i	\(-0.423295\pi\)
							0.971105	−	0.238651i	\(-0.0767051\pi\)
\(11\)	−16.5161	−	39.8735i	−0.452709	−	1.09294i	−0.971288	−	0.237907i	\(-0.923539\pi\)
							0.518578	−	0.855030i	\(-0.326461\pi\)
\(13\)	17.9201	+	7.42277i	0.382320	+	0.158362i	0.565562	−	0.824706i	\(-0.308659\pi\)
							−0.183242	+	0.983068i	\(0.558659\pi\)
\(17\)		−	45.9852i		−	0.656062i	−0.944667	−	0.328031i	\(-0.893615\pi\)
							0.944667	−	0.328031i	\(-0.106385\pi\)
\(19\)	−25.0023	−	10.3563i	−0.301890	−	0.125047i	0.226596	−	0.973989i	\(-0.427240\pi\)
							−0.528486	+	0.848942i	\(0.677240\pi\)
\(23\)	−40.3415	−	40.3415i	−0.365729	−	0.365729i	0.500188	−	0.865917i	\(-0.333264\pi\)
							−0.865917	+	0.500188i	\(0.833264\pi\)
\(29\)	−88.6955	+	214.130i	−0.567943	+	1.37113i	0.335344	+	0.942096i	\(0.391148\pi\)
							−0.903286	+	0.429039i	\(0.858852\pi\)
\(31\)	260.478			1.50913			0.754567	−	0.656223i	\(-0.227847\pi\)
							0.754567	+	0.656223i	\(0.227847\pi\)
\(37\)	70.4470	−	29.1801i	0.313011	−	0.129654i	−0.220647	−	0.975354i	\(-0.570817\pi\)
							0.533658	+	0.845700i	\(0.320817\pi\)
\(41\)	−251.246	−	251.246i	−0.957026	−	0.957026i	0.0420884	−	0.999114i	\(-0.486599\pi\)
							−0.999114	+	0.0420884i	\(0.986599\pi\)
\(43\)	−95.7143	−	231.075i	−0.339449	−	0.819501i	−0.997769	−	0.0667635i	\(-0.978733\pi\)
							0.658320	−	0.752738i	\(-0.271267\pi\)
\(47\)			15.5684i			0.0483167i	0.999708	+	0.0241584i	\(0.00769059\pi\)
							−0.999708	+	0.0241584i	\(0.992309\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.4.g.a.13.9	yes	44
4.3	odd	2		128.4.g.a.17.8		44
8.3	odd	2		256.4.g.a.33.4		44
8.5	even	2		256.4.g.b.33.8		44
32.5	even	8	inner	32.4.g.a.5.9	✓	44
32.11	odd	8		256.4.g.a.225.4		44
32.21	even	8		256.4.g.b.225.8		44
32.27	odd	8		128.4.g.a.113.8		44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.4.g.a.5.9	✓	44	32.5	even	8	inner
32.4.g.a.13.9	yes	44	1.1	even	1	trivial
128.4.g.a.17.8		44	4.3	odd	2
128.4.g.a.113.8		44	32.27	odd	8
256.4.g.a.33.4		44	8.3	odd	2
256.4.g.a.225.4		44	32.11	odd	8
256.4.g.b.33.8		44	8.5	even	2
256.4.g.b.225.8		44	32.21	even	8