Embedded newform 320.2.bd.a.43.12

• Label: 320.2.bd.a.43.12
• 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			43.12
Character	\(\chi\)	\(=\)	320.43
Dual form			320.2.bd.a.67.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00711 - 0.992839i) q^{2} +(-1.39071 - 0.929242i) q^{3} +(0.0285426 + 1.99980i) q^{4} +(0.728044 - 2.11423i) q^{5} +(0.478010 + 2.31660i) q^{6} +(-0.473507 - 1.14315i) q^{7} +(1.95673 - 2.04235i) q^{8} +(-0.0774701 - 0.187029i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00711	−	0.992839i	−0.712135	−	0.702043i
							\(3\)	−1.39071	−	0.929242i	−0.802926	−	0.536498i	0.0850560	−	0.996376i	\(-0.472893\pi\)
−0.887982	+	0.459878i	\(0.847893\pi\)
\(5\)	0.728044	−	2.11423i	0.325591	−	0.945511i
							\(7\)	−0.473507	−	1.14315i	−0.178969	−	0.432069i	0.808782	−	0.588109i	\(-0.200127\pi\)
−0.987751	+	0.156039i	\(0.950127\pi\)
\(11\)	0.0106412	−	0.0534970i	0.00320845	−	0.0161300i	−0.979148	−	0.203149i	\(-0.934883\pi\)
							0.982356	+	0.187019i	\(0.0598825\pi\)
\(13\)	0.141996	+	0.713864i	0.0393827	+	0.197990i	0.995468	−	0.0950988i	\(-0.0303167\pi\)
							−0.956085	+	0.293089i	\(0.905317\pi\)
\(17\)		−	5.02365i		−	1.21841i	−0.793012	−	0.609206i	\(-0.791488\pi\)
							0.793012	−	0.609206i	\(-0.208512\pi\)
\(19\)	−2.64211	+	3.95420i	−0.606142	+	0.907155i	−0.999927	−	0.0120469i	\(-0.996165\pi\)
							0.393786	+	0.919202i	\(0.371165\pi\)
\(23\)	−5.10070	−	2.11278i	−1.06357	−	0.440545i	−0.218852	−	0.975758i	\(-0.570231\pi\)
							−0.844717	+	0.535213i	\(0.820231\pi\)
\(29\)	1.39297	+	7.00293i	0.258668	+	1.30041i	0.863618	+	0.504147i	\(0.168193\pi\)
							−0.604950	+	0.796263i	\(0.706807\pi\)
\(31\)	−6.45695			−1.15970			−0.579851	−	0.814722i	\(-0.696889\pi\)
							−0.579851	+	0.814722i	\(0.696889\pi\)
\(37\)	2.61104	+	0.519368i	0.429252	+	0.0853836i	0.404988	−	0.914322i	\(-0.367276\pi\)
							0.0242642	+	0.999706i	\(0.492276\pi\)
\(41\)	0.428822	+	1.03527i	0.0669708	+	0.161682i	0.953821	−	0.300375i	\(-0.0971117\pi\)
							−0.886850	+	0.462057i	\(0.847112\pi\)
\(43\)	−2.66304	−	3.98552i	−0.406110	−	0.607787i	0.570890	−	0.821027i	\(-0.306598\pi\)
							−0.977000	+	0.213240i	\(0.931598\pi\)
\(47\)	3.89116			0.567585			0.283792	−	0.958886i	\(-0.408407\pi\)
							0.283792	+	0.958886i	\(0.408407\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bd.a.43.12	✓	368
5.2	odd	4		320.2.bj.a.107.35	yes	368
64.3	odd	16		320.2.bj.a.3.35	yes	368
320.67	even	16	inner	320.2.bd.a.67.12	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bd.a.43.12	✓	368	1.1	even	1	trivial
320.2.bd.a.67.12	yes	368	320.67	even	16	inner
320.2.bj.a.3.35	yes	368	64.3	odd	16
320.2.bj.a.107.35	yes	368	5.2	odd	4