Embedded newform 160.4.ba.a.3.5

• Label: 160.4.ba.a.3.5
• Level: $160$
• Weight: $4$
• Character: 160.3
• Analytic conductor: $9.440$
• Analytic rank: $0$
• Dimension: $280$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	160.ba (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.44030560092\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(70\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			3.5
Character	\(\chi\)	\(=\)	160.3
Dual form			160.4.ba.a.107.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.79487 + 0.434422i) q^{2} +(-0.175127 + 0.0725400i) q^{3} +(7.62256 - 2.42830i) q^{4} +(7.70747 + 8.09907i) q^{5} +(0.457944 - 0.278819i) q^{6} -10.5403 q^{7} +(-20.2491 + 10.0982i) q^{8} +(-19.0665 + 19.0665i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 4 q^{2} - 4 q^{3} - 4 q^{5} - 8 q^{6} - 8 q^{7} - 88 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(97\)	\(101\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{3}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.79487	+	0.434422i	−0.988134	+	0.153591i
							\(3\)	−0.175127	+	0.0725400i	−0.0337032	+	0.0139603i	−0.399471	−	0.916746i	\(-0.630806\pi\)
0.365768	+	0.930706i	\(0.380806\pi\)
\(5\)	7.70747	+	8.09907i	0.689377	+	0.724403i
							\(7\)	−10.5403			−0.569124			−0.284562	−	0.958658i	\(-0.591848\pi\)
−0.284562	+	0.958658i	\(0.591848\pi\)
\(11\)	2.88474	+	6.96439i	0.0790712	+	0.190895i	0.958472	−	0.285188i	\(-0.0920561\pi\)
							−0.879400	+	0.476083i	\(0.842056\pi\)
\(13\)	72.3711	−	29.9771i	1.54401	−	0.639550i	0.561789	−	0.827280i	\(-0.310113\pi\)
							0.982221	+	0.187730i	\(0.0601131\pi\)
\(17\)	−39.0542	−	39.0542i	−0.557179	−	0.557179i	0.371324	−	0.928503i	\(-0.378904\pi\)
							−0.928503	+	0.371324i	\(0.878904\pi\)
\(19\)	−52.4677	+	126.668i	−0.633522	+	1.52946i	0.201644	+	0.979459i	\(0.435372\pi\)
							−0.835166	+	0.549998i	\(0.814628\pi\)
\(23\)	−179.441			−1.62678			−0.813392	−	0.581716i	\(-0.802381\pi\)
							−0.813392	+	0.581716i	\(0.802381\pi\)
\(29\)	−86.9911	+	210.015i	−0.557029	+	1.34479i	0.355078	+	0.934837i	\(0.384454\pi\)
							−0.912107	+	0.409951i	\(0.865546\pi\)
\(31\)		−	17.4165i		−	0.100906i	−0.998726	−	0.0504531i	\(-0.983933\pi\)
							0.998726	−	0.0504531i	\(-0.0160665\pi\)
\(37\)	−41.1988	−	17.0651i	−0.183055	−	0.0758240i	0.289273	−	0.957247i	\(-0.406586\pi\)
							−0.472329	+	0.881423i	\(0.656586\pi\)
\(41\)	192.115	+	192.115i	0.731790	+	0.731790i	0.970974	−	0.239184i	\(-0.0768800\pi\)
							−0.239184	+	0.970974i	\(0.576880\pi\)
\(43\)	30.7302	−	74.1893i	0.108984	−	0.263111i	−0.859973	−	0.510339i	\(-0.829520\pi\)
							0.968957	+	0.247228i	\(0.0795198\pi\)
\(47\)	−394.818	+	394.818i	−1.22532	+	1.22532i	−0.259609	+	0.965714i	\(0.583594\pi\)
							−0.965714	+	0.259609i	\(0.916406\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.4.ba.a.3.5	yes	280
5.2	odd	4		160.4.u.a.67.33	yes	280
32.11	odd	8		160.4.u.a.43.33	✓	280
160.107	even	8	inner	160.4.ba.a.107.5	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.4.u.a.43.33	✓	280	32.11	odd	8
160.4.u.a.67.33	yes	280	5.2	odd	4
160.4.ba.a.3.5	yes	280	1.1	even	1	trivial
160.4.ba.a.107.5	yes	280	160.107	even	8	inner