Embedded newform 320.2.bj.a.107.21

• Label: 320.2.bj.a.107.21
• Level: $320$
• Weight: $2$
• Character: 320.107
• 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			107.21
Character	\(\chi\)	\(=\)	320.107
Dual form			320.2.bj.a.3.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.332365 + 1.37460i) q^{2} +(0.678724 - 1.01578i) q^{3} +(-1.77907 - 0.913740i) q^{4} +(1.34691 + 1.78489i) q^{5} +(1.17071 + 1.27059i) q^{6} +(-2.60694 + 1.07983i) q^{7} +(1.84733 - 2.14182i) q^{8} +(0.576903 + 1.39277i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(191\)	\(257\)	\(261\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{13}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.332365	+	1.37460i	−0.235018	+	0.971991i
							\(3\)	0.678724	−	1.01578i	0.391861	−	0.586462i	−0.582116	−	0.813105i	\(-0.697775\pi\)
0.973978	+	0.226643i	\(0.0727752\pi\)
\(5\)	1.34691	+	1.78489i	0.602355	+	0.798228i
							\(7\)	−2.60694	+	1.07983i	−0.985331	+	0.408138i	−0.816398	−	0.577490i	\(-0.804032\pi\)
−0.168933	+	0.985627i	\(0.554032\pi\)
\(11\)	−0.617390	+	3.10383i	−0.186150	+	0.935840i	0.768894	+	0.639376i	\(0.220807\pi\)
							−0.955044	+	0.296464i	\(0.904193\pi\)
\(13\)	0.324926	−	0.0646318i	0.0901183	−	0.0179256i	−0.149825	−	0.988713i	\(-0.547871\pi\)
							0.239943	+	0.970787i	\(0.422871\pi\)
\(17\)	7.28485			1.76684			0.883418	−	0.468586i	\(-0.155237\pi\)
							0.883418	+	0.468586i	\(0.155237\pi\)
\(19\)	−1.21315	+	1.81560i	−0.278315	+	0.416527i	−0.944122	−	0.329597i	\(-0.893087\pi\)
							0.665807	+	0.746124i	\(0.268087\pi\)
\(23\)	−1.57160	+	3.79419i	−0.327702	+	0.791143i	0.671060	+	0.741403i	\(0.265839\pi\)
							−0.998762	+	0.0497398i	\(0.984161\pi\)
\(29\)	−0.750161	−	3.77132i	−0.139301	−	0.700316i	−0.985800	−	0.167925i	\(-0.946294\pi\)
							0.846498	−	0.532391i	\(-0.178706\pi\)
\(31\)	−5.19751			−0.933500			−0.466750	−	0.884389i	\(-0.654575\pi\)
							−0.466750	+	0.884389i	\(0.654575\pi\)
\(37\)	1.44324	−	7.25568i	0.237268	−	1.19283i	−0.659974	−	0.751288i	\(-0.729433\pi\)
							0.897242	−	0.441539i	\(-0.145567\pi\)
\(41\)	−1.84682	−	4.45861i	−0.288425	−	0.696319i	0.711556	−	0.702630i	\(-0.247991\pi\)
							−0.999980	+	0.00631136i	\(0.997991\pi\)
\(43\)	5.87584	−	3.92611i	0.896057	−	0.598726i	−0.0199884	−	0.999800i	\(-0.506363\pi\)
							0.916046	+	0.401074i	\(0.131363\pi\)
\(47\)		−	7.32452i		−	1.06839i	−0.845361	−	0.534196i	\(-0.820615\pi\)
							0.845361	−	0.534196i	\(-0.179385\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bj.a.107.21	yes	368
5.3	odd	4		320.2.bd.a.43.43	✓	368
64.3	odd	16		320.2.bd.a.67.43	yes	368
320.3	even	16	inner	320.2.bj.a.3.21	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bd.a.43.43	✓	368	5.3	odd	4
320.2.bd.a.67.43	yes	368	64.3	odd	16
320.2.bj.a.3.21	yes	368	320.3	even	16	inner
320.2.bj.a.107.21	yes	368	1.1	even	1	trivial