Embedded newform 320.2.bd.a.43.11

• Label: 320.2.bd.a.43.11
• 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			43.11
Character	\(\chi\)	\(=\)	320.43
Dual form			320.2.bd.a.67.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.04815 - 0.949416i) q^{2} +(1.57122 + 1.04985i) q^{3} +(0.197220 + 1.99025i) q^{4} +(-1.85697 - 1.24566i) q^{5} +(-0.650118 - 2.59214i) q^{6} +(-1.56668 - 3.78229i) q^{7} +(1.68286 - 2.27332i) q^{8} +(0.218482 + 0.527463i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.04815	−	0.949416i	−0.741151	−	0.671338i
							\(3\)	1.57122	+	1.04985i	0.907143	+	0.606134i	0.919199	−	0.393794i	\(-0.128838\pi\)
−0.0120556	+	0.999927i	\(0.503838\pi\)
\(5\)	−1.85697	−	1.24566i	−0.830462	−	0.557075i
							\(7\)	−1.56668	−	3.78229i	−0.592149	−	1.42957i	−0.881423	−	0.472327i	\(-0.843414\pi\)
0.289275	−	0.957246i	\(-0.406586\pi\)
\(11\)	−0.325819	+	1.63801i	−0.0982383	+	0.493877i	0.900071	+	0.435744i	\(0.143515\pi\)
							−0.998309	+	0.0581330i	\(0.981485\pi\)
\(13\)	−0.953550	−	4.79382i	−0.264467	−	1.32957i	−0.853346	−	0.521345i	\(-0.825431\pi\)
							0.588879	−	0.808221i	\(-0.299569\pi\)
\(17\)			0.700499i			0.169896i	0.996385	+	0.0849480i	\(0.0270724\pi\)
							−0.996385	+	0.0849480i	\(0.972928\pi\)
\(19\)	3.53400	−	5.28900i	0.810755	−	1.21338i	−0.163190	−	0.986595i	\(-0.552178\pi\)
							0.973945	−	0.226786i	\(-0.0728218\pi\)
\(23\)	5.98407	+	2.47868i	1.24776	+	0.516841i	0.906133	−	0.422993i	\(-0.139021\pi\)
							0.341631	+	0.939834i	\(0.389021\pi\)
\(29\)	−0.0727925	−	0.365953i	−0.0135172	−	0.0679557i	0.973438	−	0.228951i	\(-0.0735296\pi\)
							−0.986955	+	0.160995i	\(0.948530\pi\)
\(31\)	−6.15578			−1.10561			−0.552805	−	0.833310i	\(-0.686443\pi\)
							−0.552805	+	0.833310i	\(0.686443\pi\)
\(37\)	−3.84532	−	0.764883i	−0.632167	−	0.125746i	−0.131402	−	0.991329i	\(-0.541948\pi\)
							−0.500765	+	0.865583i	\(0.666948\pi\)
\(41\)	0.984473	+	2.37673i	0.153749	+	0.371182i	0.981921	−	0.189291i	\(-0.0606190\pi\)
							−0.828172	+	0.560474i	\(0.810619\pi\)
\(43\)	−3.67097	−	5.49400i	−0.559818	−	0.837827i	0.438320	−	0.898819i	\(-0.355574\pi\)
							−0.998138	+	0.0609922i	\(0.980574\pi\)
\(47\)	1.53340			0.223669			0.111834	−	0.993727i	\(-0.464327\pi\)
							0.111834	+	0.993727i	\(0.464327\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bd.a.43.11	✓	368
5.2	odd	4		320.2.bj.a.107.34	yes	368
64.3	odd	16		320.2.bj.a.3.34	yes	368
320.67	even	16	inner	320.2.bd.a.67.11	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bd.a.43.11	✓	368	1.1	even	1	trivial
320.2.bd.a.67.11	yes	368	320.67	even	16	inner
320.2.bj.a.3.34	yes	368	64.3	odd	16
320.2.bj.a.107.34	yes	368	5.2	odd	4