Embedded newform 320.2.bd.a.203.33

• Label: 320.2.bd.a.203.33
• Level: $320$
• Weight: $2$
• Character: 320.203
• 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			203.33
Character	\(\chi\)	\(=\)	320.203
Dual form			320.2.bd.a.227.33

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.810248 - 1.15909i) q^{2} +(1.10973 - 1.66082i) q^{3} +(-0.686997 - 1.87831i) q^{4} +(0.227422 - 2.22447i) q^{5} +(-1.02590 - 2.63196i) q^{6} +(1.29367 + 3.12319i) q^{7} +(-2.73377 - 0.725599i) q^{8} +(-0.378793 - 0.914486i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(191\)	\(257\)	\(261\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{5}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.810248	−	1.15909i	0.572932	−	0.819603i
							\(3\)	1.10973	−	1.66082i	0.640701	−	0.958877i	−0.358973	−	0.933348i	\(-0.616873\pi\)
0.999674	−	0.0255294i	\(-0.00812713\pi\)
\(5\)	0.227422	−	2.22447i	0.101706	−	0.994814i
							\(7\)	1.29367	+	3.12319i	0.488960	+	1.18045i	0.955244	+	0.295818i	\(0.0955923\pi\)
−0.466284	+	0.884635i	\(0.654408\pi\)
\(11\)	−0.710582	−	0.141343i	−0.214248	−	0.0426167i	0.0867990	−	0.996226i	\(-0.472336\pi\)
							−0.301047	+	0.953609i	\(0.597336\pi\)
\(13\)	1.31285	−	0.261142i	0.364120	−	0.0724279i	−0.00963991	−	0.999954i	\(-0.503069\pi\)
							0.373760	+	0.927526i	\(0.378069\pi\)
\(17\)			3.43364i			0.832780i	0.909186	+	0.416390i	\(0.136705\pi\)
							−0.909186	+	0.416390i	\(0.863295\pi\)
\(19\)	5.56841	+	3.72070i	1.27748	+	0.853586i	0.994418	−	0.105515i	\(-0.0336490\pi\)
							0.283064	+	0.959101i	\(0.408649\pi\)
\(23\)	−6.74722	−	2.79479i	−1.40689	−	0.582754i	−0.455362	−	0.890306i	\(-0.650490\pi\)
							−0.951531	+	0.307552i	\(0.900490\pi\)
\(29\)	−8.41032	+	1.67292i	−1.56176	+	0.310653i	−0.898921	−	0.438111i	\(-0.855648\pi\)
							−0.662836	+	0.748764i	\(0.730648\pi\)
\(31\)	4.57653			0.821970			0.410985	−	0.911642i	\(-0.365185\pi\)
							0.410985	+	0.911642i	\(0.365185\pi\)
\(37\)	0.740055	−	3.72051i	0.121664	−	0.611648i	−0.871054	−	0.491187i	\(-0.836563\pi\)
							0.992718	−	0.120460i	\(-0.0384370\pi\)
\(41\)	−1.33048	−	3.21206i	−0.207786	−	0.501639i	0.785288	−	0.619130i	\(-0.212515\pi\)
							−0.993074	+	0.117491i	\(0.962515\pi\)
\(43\)	7.67050	−	5.12526i	1.16974	−	0.781595i	0.189983	−	0.981787i	\(-0.439157\pi\)
							0.979757	+	0.200192i	\(0.0641567\pi\)
\(47\)	1.49002			0.217341			0.108671	−	0.994078i	\(-0.465341\pi\)
							0.108671	+	0.994078i	\(0.465341\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bd.a.203.33	✓	368
5.2	odd	4		320.2.bj.a.267.38	yes	368
64.35	odd	16		320.2.bj.a.163.38	yes	368
320.227	even	16	inner	320.2.bd.a.227.33	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bd.a.203.33	✓	368	1.1	even	1	trivial
320.2.bd.a.227.33	yes	368	320.227	even	16	inner
320.2.bj.a.163.38	yes	368	64.35	odd	16
320.2.bj.a.267.38	yes	368	5.2	odd	4