Embedded newform 320.2.bd.a.43.40

• Label: 320.2.bd.a.43.40
• 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			43.40
Character	\(\chi\)	\(=\)	320.43
Dual form			320.2.bd.a.67.40

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.30534 + 0.544145i) q^{2} +(-1.17230 - 0.783307i) q^{3} +(1.40781 + 1.42059i) q^{4} +(0.215724 - 2.22564i) q^{5} +(-1.10402 - 1.66038i) q^{6} +(-1.53801 - 3.71308i) q^{7} +(1.06467 + 2.62040i) q^{8} +(-0.387328 - 0.935094i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.30534	+	0.544145i	0.923013	+	0.384768i
							\(3\)	−1.17230	−	0.783307i	−0.676829	−	0.452243i	0.169057	−	0.985606i	\(-0.445928\pi\)
−0.845886	+	0.533364i	\(0.820928\pi\)
\(5\)	0.215724	−	2.22564i	0.0964745	−	0.995335i
							\(7\)	−1.53801	−	3.71308i	−0.581313	−	1.40341i	−0.891624	−	0.452778i	\(-0.850433\pi\)
0.310311	−	0.950635i	\(-0.399567\pi\)
\(11\)	0.657619	−	3.30607i	0.198280	−	0.996819i	−0.745566	−	0.666432i	\(-0.767821\pi\)
							0.943846	−	0.330387i	\(-0.107179\pi\)
\(13\)	0.810210	+	4.07320i	0.224712	+	1.12970i	0.914155	+	0.405364i	\(0.132855\pi\)
							−0.689443	+	0.724339i	\(0.742145\pi\)
\(17\)			4.22745i			1.02531i	0.858596	+	0.512653i	\(0.171337\pi\)
							−0.858596	+	0.512653i	\(0.828663\pi\)
\(19\)	3.18840	−	4.77178i	0.731470	−	1.09472i	−0.260155	−	0.965567i	\(-0.583774\pi\)
							0.991624	−	0.129155i	\(-0.0412264\pi\)
\(23\)	5.70729	+	2.36404i	1.19005	+	0.492936i	0.887772	−	0.460283i	\(-0.152252\pi\)
							0.302280	+	0.953219i	\(0.402252\pi\)
\(29\)	−0.385034	−	1.93570i	−0.0714991	−	0.359450i	0.928428	−	0.371512i	\(-0.121160\pi\)
							−0.999927	+	0.0120619i	\(0.996160\pi\)
\(31\)	−3.07553			−0.552382			−0.276191	−	0.961103i	\(-0.589072\pi\)
							−0.276191	+	0.961103i	\(0.589072\pi\)
\(37\)	11.8567	+	2.35844i	1.94922	+	0.387725i	0.996466	+	0.0839945i	\(0.0267678\pi\)
							0.952758	+	0.303730i	\(0.0982322\pi\)
\(41\)	−1.06277	−	2.56576i	−0.165977	−	0.400705i	0.818905	−	0.573929i	\(-0.194581\pi\)
							−0.984882	+	0.173224i	\(0.944581\pi\)
\(43\)	0.0702829	+	0.105186i	0.0107180	+	0.0160407i	0.836790	−	0.547524i	\(-0.184430\pi\)
							−0.826072	+	0.563565i	\(0.809430\pi\)
\(47\)	−12.0840			−1.76263			−0.881314	−	0.472532i	\(-0.843340\pi\)
							−0.881314	+	0.472532i	\(0.843340\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bd.a.43.40	✓	368
5.2	odd	4		320.2.bj.a.107.19	yes	368
64.3	odd	16		320.2.bj.a.3.19	yes	368
320.67	even	16	inner	320.2.bd.a.67.40	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bd.a.43.40	✓	368	1.1	even	1	trivial
320.2.bd.a.67.40	yes	368	320.67	even	16	inner
320.2.bj.a.3.19	yes	368	64.3	odd	16
320.2.bj.a.107.19	yes	368	5.2	odd	4