Embedded newform 320.2.bd.a.43.13

• Label: 320.2.bd.a.43.13
• 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			43.13
Character	\(\chi\)	\(=\)	320.43
Dual form			320.2.bd.a.67.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.988622 - 1.01125i) q^{2} +(-2.72182 - 1.81866i) q^{3} +(-0.0452538 + 1.99949i) q^{4} +(-1.63219 + 1.52838i) q^{5} +(0.851727 + 4.55040i) q^{6} +(0.160177 + 0.386702i) q^{7} +(2.06672 - 1.93097i) q^{8} +(2.95271 + 7.12847i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(191\)	\(257\)	\(261\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{13}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.988622	−	1.01125i	−0.699061	−	0.715062i
							\(3\)	−2.72182	−	1.81866i	−1.57144	−	1.05000i	−0.967466	−	0.253000i	\(-0.918583\pi\)
−0.603975	−	0.797003i	\(-0.706417\pi\)
\(5\)	−1.63219	+	1.52838i	−0.729940	+	0.683512i
							\(7\)	0.160177	+	0.386702i	0.0605413	+	0.146160i	0.951255	−	0.308405i	\(-0.0997950\pi\)
−0.890714	+	0.454564i	\(0.849795\pi\)
\(11\)	0.0859762	−	0.432232i	0.0259228	−	0.130323i	−0.965657	−	0.259821i	\(-0.916337\pi\)
							0.991580	+	0.129498i	\(0.0413366\pi\)
\(13\)	−0.645703	−	3.24617i	−0.179086	−	0.900325i	−0.960917	−	0.276835i	\(-0.910714\pi\)
							0.781832	−	0.623489i	\(-0.214286\pi\)
\(17\)			5.57660i			1.35252i	0.736661	+	0.676262i	\(0.236401\pi\)
							−0.736661	+	0.676262i	\(0.763599\pi\)
\(19\)	2.10040	−	3.14347i	0.481864	−	0.721161i	−0.508283	−	0.861190i	\(-0.669720\pi\)
							0.990147	+	0.140029i	\(0.0447197\pi\)
\(23\)	−2.47710	−	1.02605i	−0.516510	−	0.213946i	0.109173	−	0.994023i	\(-0.465180\pi\)
							−0.625683	+	0.780077i	\(0.715180\pi\)
\(29\)	0.512216	+	2.57508i	0.0951161	+	0.478181i	0.998755	+	0.0498855i	\(0.0158857\pi\)
							−0.903639	+	0.428295i	\(0.859114\pi\)
\(31\)	7.29488			1.31020			0.655100	−	0.755542i	\(-0.272627\pi\)
							0.655100	+	0.755542i	\(0.272627\pi\)
\(37\)	10.2540	+	2.03964i	1.68574	+	0.335315i	0.942626	−	0.333849i	\(-0.108348\pi\)
							0.743115	+	0.669164i	\(0.233348\pi\)
\(41\)	−2.59529	−	6.26558i	−0.405316	−	0.978519i	−0.986353	−	0.164642i	\(-0.947353\pi\)
							0.581037	−	0.813877i	\(-0.302647\pi\)
\(43\)	1.21339	+	1.81597i	0.185041	+	0.276933i	0.912379	−	0.409346i	\(-0.134243\pi\)
							−0.727339	+	0.686279i	\(0.759243\pi\)
\(47\)	7.69780			1.12284			0.561420	−	0.827531i	\(-0.310255\pi\)
							0.561420	+	0.827531i	\(0.310255\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bd.a.43.13	✓	368	1.1	even	1	trivial
320.2.bd.a.67.13	yes	368	320.67	even	16	inner
320.2.bj.a.3.36	yes	368	64.3	odd	16
320.2.bj.a.107.36	yes	368	5.2	odd	4