Embedded newform 320.2.bf.a.29.30

• Label: 320.2.bf.a.29.30
• Level: $320$
• Weight: $2$
• Character: 320.29
• Analytic conductor: $2.555$
• Analytic rank: $0$
• Dimension: $368$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	320.bf (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			29.30
Character	\(\chi\)	\(=\)	320.29
Dual form			320.2.bf.a.309.30

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.628783 + 1.26674i) q^{2} +(-0.368711 + 1.85364i) q^{3} +(-1.20926 + 1.59301i) q^{4} +(-2.23595 + 0.0229076i) q^{5} +(-2.57992 + 0.698473i) q^{6} +(-1.47604 - 3.56349i) q^{7} +(-2.77829 - 0.530167i) q^{8} +(-0.528383 - 0.218863i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 16 q^{4} - 8 q^{5} - 16 q^{6} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(191\)	\(257\)	\(261\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{11}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.628783	+	1.26674i	0.444617	+	0.895721i
							\(3\)	−0.368711	+	1.85364i	−0.212876	+	1.07020i	0.715517	+	0.698596i	\(0.246191\pi\)
−0.928392	+	0.371602i	\(0.878809\pi\)
\(5\)	−2.23595	+	0.0229076i	−0.999948	+	0.0102446i
							\(7\)	−1.47604	−	3.56349i	−0.557892	−	1.34687i	−0.911432	−	0.411452i	\(-0.865022\pi\)
0.353539	−	0.935420i	\(-0.384978\pi\)
\(11\)	0.755528	+	3.79830i	0.227800	+	1.14523i	0.910174	+	0.414226i	\(0.135948\pi\)
							−0.682374	+	0.731004i	\(0.739052\pi\)
\(13\)	−2.55467	+	1.70697i	−0.708537	+	0.473429i	−0.856886	−	0.515506i	\(-0.827604\pi\)
							0.148349	+	0.988935i	\(0.452604\pi\)
\(17\)	0.420299	−	0.420299i	0.101937	−	0.101937i	−0.654299	−	0.756236i	\(-0.727036\pi\)
							0.756236	+	0.654299i	\(0.227036\pi\)
\(19\)	−6.15466	+	4.11241i	−1.41198	+	0.943452i	−0.412503	+	0.910956i	\(0.635346\pi\)
							−0.999472	+	0.0324957i	\(0.989654\pi\)
\(23\)	4.61793	+	1.91281i	0.962906	+	0.398849i	0.808067	−	0.589091i	\(-0.200514\pi\)
							0.154839	+	0.987940i	\(0.450514\pi\)
\(29\)	1.78738	−	8.98579i	0.331909	−	1.66862i	−0.349669	−	0.936873i	\(-0.613706\pi\)
							0.681578	−	0.731745i	\(-0.261294\pi\)
\(31\)			6.13374i			1.10165i	0.834620	+	0.550826i	\(0.185687\pi\)
							−0.834620	+	0.550826i	\(0.814313\pi\)
\(37\)	−3.16474	+	4.73638i	−0.520281	+	0.778655i	−0.994828	−	0.101575i	\(-0.967612\pi\)
							0.474547	+	0.880230i	\(0.342612\pi\)
\(41\)	0.0274634	−	0.0663026i	0.00428907	−	0.0103547i	−0.921720	−	0.387855i	\(-0.873216\pi\)
							0.926010	+	0.377500i	\(0.123216\pi\)
\(43\)	0.626128	+	3.14776i	0.0954836	+	0.480028i	0.998706	+	0.0508556i	\(0.0161948\pi\)
							−0.903222	+	0.429173i	\(0.858805\pi\)
\(47\)	−9.03628	+	9.03628i	−1.31808	+	1.31808i	−0.402781	+	0.915296i	\(0.631956\pi\)
							−0.915296	+	0.402781i	\(0.868044\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bf.a.29.30	yes	368
5.4	even	2	inner	320.2.bf.a.29.17	✓	368
64.53	even	16	inner	320.2.bf.a.309.17	yes	368
320.309	even	16	inner	320.2.bf.a.309.30	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bf.a.29.17	✓	368	5.4	even	2	inner
320.2.bf.a.29.30	yes	368	1.1	even	1	trivial
320.2.bf.a.309.17	yes	368	64.53	even	16	inner
320.2.bf.a.309.30	yes	368	320.309	even	16	inner