Embedded newform 320.2.bd.a.43.17

• Label: 320.2.bd.a.43.17
• Level: $320$
• Weight: $2$
• Character: 320.43
• Analytic conductor: $2.555$
• Analytic rank: $0$
• Dimension: $368$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	320.bd (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.55521286468\)
Analytic rank:	\(0\)
Dimension:	\(368\)
Relative dimension:	\(46\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			43.17
Character	\(\chi\)	\(=\)	320.43
Dual form			320.2.bd.a.67.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.643040 - 1.25956i) q^{2} +(1.22104 + 0.815871i) q^{3} +(-1.17300 + 1.61990i) q^{4} +(2.16782 - 0.548217i) q^{5} +(0.242466 - 2.06261i) q^{6} +(0.908242 + 2.19269i) q^{7} +(2.79465 + 0.435808i) q^{8} +(-0.322763 - 0.779219i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\)	\(191\)	\(257\)	\(261\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{13}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.643040	−	1.25956i	−0.454698	−	0.890646i
							\(3\)	1.22104	+	0.815871i	0.704966	+	0.471044i	0.855661	−	0.517537i	\(-0.173151\pi\)
−0.150694	+	0.988580i	\(0.548151\pi\)
\(5\)	2.16782	−	0.548217i	0.969480	−	0.245170i
							\(7\)	0.908242	+	2.19269i	0.343283	+	0.828759i	0.997379	+	0.0723476i	\(0.0230491\pi\)
−0.654096	+	0.756411i	\(0.726951\pi\)
\(11\)	−1.11132	+	5.58700i	−0.335077	+	1.68454i	0.334972	+	0.942228i	\(0.391273\pi\)
							−0.670049	+	0.742317i	\(0.733727\pi\)
\(13\)	−0.0117508	−	0.0590755i	−0.00325910	−	0.0163846i	0.979122	−	0.203276i	\(-0.0651588\pi\)
							−0.982381	+	0.186891i	\(0.940159\pi\)
\(17\)			1.08110i			0.262204i	0.991369	+	0.131102i	\(0.0418516\pi\)
							−0.991369	+	0.131102i	\(0.958148\pi\)
\(19\)	1.32152	−	1.97780i	0.303178	−	0.453738i	−0.648331	−	0.761359i	\(-0.724533\pi\)
							0.951509	+	0.307620i	\(0.0995327\pi\)
\(23\)	−1.14755	−	0.475330i	−0.239280	−	0.0991131i	0.259821	−	0.965657i	\(-0.416336\pi\)
							−0.499101	+	0.866544i	\(0.666336\pi\)
\(29\)	−0.0558865	−	0.280960i	−0.0103779	−	0.0521730i	0.975249	−	0.221108i	\(-0.0709673\pi\)
							−0.985627	+	0.168935i	\(0.945967\pi\)
\(31\)	9.56200			1.71739			0.858693	−	0.512491i	\(-0.171277\pi\)
							0.858693	+	0.512491i	\(0.171277\pi\)
\(37\)	−3.72557	−	0.741062i	−0.612480	−	0.121830i	−0.120903	−	0.992664i	\(-0.538579\pi\)
							−0.491576	+	0.870835i	\(0.663579\pi\)
\(41\)	−3.67877	−	8.88134i	−0.574527	−	1.38703i	−0.897664	−	0.440680i	\(-0.854738\pi\)
							0.323137	−	0.946352i	\(-0.395262\pi\)
\(43\)	−4.50112	−	6.73640i	−0.686415	−	1.02729i	−0.997049	−	0.0767627i	\(-0.975542\pi\)
							0.310635	−	0.950529i	\(-0.399458\pi\)
\(47\)	−10.2888			−1.50078			−0.750391	−	0.660995i	\(-0.770135\pi\)
							−0.750391	+	0.660995i	\(0.770135\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bd.a.43.17	✓	368
5.2	odd	4		320.2.bj.a.107.40	yes	368
64.3	odd	16		320.2.bj.a.3.40	yes	368
320.67	even	16	inner	320.2.bd.a.67.17	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bd.a.43.17	✓	368	1.1	even	1	trivial
320.2.bd.a.67.17	yes	368	320.67	even	16	inner
320.2.bj.a.3.40	yes	368	64.3	odd	16
320.2.bj.a.107.40	yes	368	5.2	odd	4