Embedded newform 320.2.bj.a.27.35

• Label: 320.2.bj.a.27.35
• 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			27.35
Character	\(\chi\)	\(=\)	320.27
Dual form			320.2.bj.a.83.35

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.951179 + 1.04655i) q^{2} +(0.397534 - 0.0790744i) q^{3} +(-0.190517 + 1.99091i) q^{4} +(-1.72090 + 1.42776i) q^{5} +(0.460881 + 0.340824i) q^{6} +(0.0192177 + 0.0463955i) q^{7} +(-2.26479 + 1.69432i) q^{8} +(-2.61986 + 1.08518i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.951179	+	1.04655i	0.672585	+	0.740020i
							\(3\)	0.397534	−	0.0790744i	0.229516	−	0.0456536i	−0.0789921	−	0.996875i	\(-0.525170\pi\)
0.308508	+	0.951222i	\(0.400170\pi\)
\(5\)	−1.72090	+	1.42776i	−0.769610	+	0.638515i
							\(7\)	0.0192177	+	0.0463955i	0.00726359	+	0.0175359i	0.927470	−	0.373898i	\(-0.121979\pi\)
−0.920206	+	0.391434i	\(0.871979\pi\)
\(11\)	1.80142	+	2.69601i	0.543147	+	0.812878i	0.996935	−	0.0782322i	\(-0.0249276\pi\)
							−0.453788	+	0.891110i	\(0.649928\pi\)
\(13\)	−0.557857	−	0.372748i	−0.154722	−	0.103382i	0.475800	−	0.879554i	\(-0.342159\pi\)
							−0.630521	+	0.776172i	\(0.717159\pi\)
\(17\)	6.76682			1.64120			0.820598	−	0.571507i	\(-0.193641\pi\)
							0.820598	+	0.571507i	\(0.193641\pi\)
\(19\)	1.93045	−	0.383991i	0.442876	−	0.0880935i	0.0313851	−	0.999507i	\(-0.490008\pi\)
							0.411491	+	0.911414i	\(0.365008\pi\)
\(23\)	2.67718	+	1.10893i	0.558231	+	0.231227i	0.643917	−	0.765095i	\(-0.277308\pi\)
							−0.0856858	+	0.996322i	\(0.527308\pi\)
\(29\)	2.31797	−	3.46909i	0.430436	−	0.644194i	−0.551330	−	0.834288i	\(-0.685879\pi\)
							0.981766	+	0.190094i	\(0.0608793\pi\)
\(31\)	−4.15267			−0.745842			−0.372921	−	0.927863i	\(-0.621644\pi\)
							−0.372921	+	0.927863i	\(0.621644\pi\)
\(37\)	1.91267	+	2.86251i	0.314441	+	0.470594i	0.954702	−	0.297563i	\(-0.0961738\pi\)
							−0.640261	+	0.768157i	\(0.721174\pi\)
\(41\)	5.65670	−	2.34308i	0.883428	−	0.365928i	0.105602	−	0.994408i	\(-0.466323\pi\)
							0.777825	+	0.628481i	\(0.216323\pi\)
\(43\)	−1.88150	+	9.45894i	−0.286926	+	1.44248i	0.521184	+	0.853444i	\(0.325490\pi\)
							−0.808110	+	0.589031i	\(0.799510\pi\)
\(47\)		−	6.03981i		−	0.880996i	−0.897754	−	0.440498i	\(-0.854802\pi\)
							0.897754	−	0.440498i	\(-0.145198\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bj.a.27.35	yes	368
5.3	odd	4		320.2.bd.a.283.35	yes	368
64.19	odd	16		320.2.bd.a.147.35	✓	368
320.83	even	16	inner	320.2.bj.a.83.35	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bd.a.147.35	✓	368	64.19	odd	16
320.2.bd.a.283.35	yes	368	5.3	odd	4
320.2.bj.a.27.35	yes	368	1.1	even	1	trivial
320.2.bj.a.83.35	yes	368	320.83	even	16	inner