Embedded newform 320.2.bj.a.107.36

• Label: 320.2.bj.a.107.36
• 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			107.36
Character	\(\chi\)	\(=\)	320.107
Dual form			320.2.bj.a.3.36

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.01125 - 0.988622i) q^{2} +(-1.81866 + 2.72182i) q^{3} +(0.0452538 - 1.99949i) q^{4} +(-2.03665 + 0.923066i) q^{5} +(0.851727 + 4.55040i) q^{6} +(-0.386702 + 0.160177i) q^{7} +(-1.93097 - 2.06672i) q^{8} +(-2.95271 - 7.12847i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.01125	−	0.988622i	0.715062	−	0.699061i
							\(3\)	−1.81866	+	2.72182i	−1.05000	+	1.57144i	−0.253000	+	0.967466i	\(0.581417\pi\)
−0.797003	+	0.603975i	\(0.793583\pi\)
\(5\)	−2.03665	+	0.923066i	−0.910818	+	0.412808i
							\(7\)	−0.386702	+	0.160177i	−0.146160	+	0.0605413i	−0.454564	−	0.890714i	\(-0.650205\pi\)
0.308405	+	0.951255i	\(0.400205\pi\)
\(11\)	0.0859762	−	0.432232i	0.0259228	−	0.130323i	−0.965657	−	0.259821i	\(-0.916337\pi\)
							0.991580	+	0.129498i	\(0.0413366\pi\)
\(13\)	−3.24617	+	0.645703i	−0.900325	+	0.179086i	−0.623489	−	0.781832i	\(-0.714286\pi\)
							−0.276835	+	0.960917i	\(0.589286\pi\)
\(17\)	−5.57660			−1.35252			−0.676262	−	0.736661i	\(-0.736401\pi\)
							−0.676262	+	0.736661i	\(0.736401\pi\)
\(19\)	−2.10040	+	3.14347i	−0.481864	+	0.721161i	−0.990147	−	0.140029i	\(-0.955280\pi\)
							0.508283	+	0.861190i	\(0.330280\pi\)
\(23\)	−1.02605	+	2.47710i	−0.213946	+	0.516510i	−0.994023	−	0.109173i	\(-0.965180\pi\)
							0.780077	+	0.625683i	\(0.215180\pi\)
\(29\)	−0.512216	−	2.57508i	−0.0951161	−	0.478181i	−0.998755	−	0.0498855i	\(-0.984114\pi\)
							0.903639	−	0.428295i	\(-0.140886\pi\)
\(31\)	7.29488			1.31020			0.655100	−	0.755542i	\(-0.272627\pi\)
							0.655100	+	0.755542i	\(0.272627\pi\)
\(37\)	−2.03964	+	10.2540i	−0.335315	+	1.68574i	0.333849	+	0.942626i	\(0.391652\pi\)
							−0.669164	+	0.743115i	\(0.733348\pi\)
\(41\)	−2.59529	−	6.26558i	−0.405316	−	0.978519i	−0.986353	−	0.164642i	\(-0.947353\pi\)
							0.581037	−	0.813877i	\(-0.302647\pi\)
\(43\)	1.81597	−	1.21339i	0.276933	−	0.185041i	−0.409346	−	0.912379i	\(-0.634243\pi\)
							0.686279	+	0.727339i	\(0.259243\pi\)
\(47\)			7.69780i			1.12284i	0.827531	+	0.561420i	\(0.189745\pi\)
							−0.827531	+	0.561420i	\(0.810255\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bj.a.107.36	yes	368
5.3	odd	4		320.2.bd.a.43.13	✓	368
64.3	odd	16		320.2.bd.a.67.13	yes	368
320.3	even	16	inner	320.2.bj.a.3.36	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bd.a.43.13	✓	368	5.3	odd	4
320.2.bd.a.67.13	yes	368	64.3	odd	16
320.2.bj.a.3.36	yes	368	320.3	even	16	inner
320.2.bj.a.107.36	yes	368	1.1	even	1	trivial