Embedded newform 320.2.bj.a.27.44

• Label: 320.2.bj.a.27.44
• Level: $320$
• Weight: $2$
• Character: 320.27
• 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.bj (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			27.44
Character	\(\chi\)	\(=\)	320.27
Dual form			320.2.bj.a.83.44

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.39436 - 0.236162i) q^{2} +(-1.57120 + 0.312532i) q^{3} +(1.88846 - 0.658587i) q^{4} +(1.48382 - 1.67280i) q^{5} +(-2.11701 + 0.806838i) q^{6} +(-1.12936 - 2.72651i) q^{7} +(2.47765 - 1.36428i) q^{8} +(-0.400638 + 0.165950i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(191\)	\(257\)	\(261\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{9}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.39436	−	0.236162i	0.985958	−	0.166991i
							\(3\)	−1.57120	+	0.312532i	−0.907134	+	0.180440i	−0.626543	−	0.779387i	\(-0.715531\pi\)
−0.280591	+	0.959827i	\(0.590531\pi\)
\(5\)	1.48382	−	1.67280i	0.663586	−	0.748100i
							\(7\)	−1.12936	−	2.72651i	−0.426857	−	1.03052i	−0.980278	−	0.197623i	\(-0.936678\pi\)
0.553421	−	0.832902i	\(-0.313322\pi\)
\(11\)	−1.43374	−	2.14574i	−0.432288	−	0.646965i	0.549820	−	0.835283i	\(-0.314696\pi\)
							−0.982108	+	0.188319i	\(0.939696\pi\)
\(13\)	3.45372	+	2.30770i	0.957890	+	0.640042i	0.933087	−	0.359651i	\(-0.117104\pi\)
							0.0248032	+	0.999692i	\(0.492104\pi\)
\(17\)	2.28030			0.553053			0.276527	−	0.961006i	\(-0.410817\pi\)
							0.276527	+	0.961006i	\(0.410817\pi\)
\(19\)	−1.54474	+	0.307267i	−0.354387	+	0.0704919i	−0.369073	−	0.929400i	\(-0.620325\pi\)
							0.0146868	+	0.999892i	\(0.495325\pi\)
\(23\)	5.18425	+	2.14739i	1.08099	+	0.447761i	0.850857	−	0.525398i	\(-0.176084\pi\)
							0.230134	+	0.973159i	\(0.426084\pi\)
\(29\)	−5.10910	+	7.64631i	−0.948737	+	1.41988i	−0.0415594	+	0.999136i	\(0.513233\pi\)
							−0.907177	+	0.420749i	\(0.861767\pi\)
\(31\)	−4.98620			−0.895548			−0.447774	−	0.894147i	\(-0.647783\pi\)
							−0.447774	+	0.894147i	\(0.647783\pi\)
\(37\)	1.74652	+	2.61385i	0.287126	+	0.429715i	0.946793	−	0.321844i	\(-0.104303\pi\)
							−0.659667	+	0.751558i	\(0.729303\pi\)
\(41\)	0.890804	−	0.368983i	0.139120	−	0.0576255i	−0.312037	−	0.950070i	\(-0.601011\pi\)
							0.451157	+	0.892444i	\(0.351011\pi\)
\(43\)	−2.06783	+	10.3957i	−0.315342	+	1.58533i	0.419929	+	0.907557i	\(0.362055\pi\)
							−0.735271	+	0.677773i	\(0.762945\pi\)
\(47\)		−	2.18480i		−	0.318686i	−0.987223	−	0.159343i	\(-0.949062\pi\)
							0.987223	−	0.159343i	\(-0.0509376\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bj.a.27.44	yes	368
5.3	odd	4		320.2.bd.a.283.21	yes	368
64.19	odd	16		320.2.bd.a.147.21	✓	368
320.83	even	16	inner	320.2.bj.a.83.44	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bd.a.147.21	✓	368	64.19	odd	16
320.2.bd.a.283.21	yes	368	5.3	odd	4
320.2.bj.a.27.44	yes	368	1.1	even	1	trivial
320.2.bj.a.83.44	yes	368	320.83	even	16	inner