Embedded newform 320.2.bj.a.27.16

• Label: 320.2.bj.a.27.16
• Level: $320$
• Weight: $2$
• Character: 320.27
• 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			27.16
Character	\(\chi\)	\(=\)	320.27
Dual form			320.2.bj.a.83.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.725691 - 1.21383i) q^{2} +(-1.65741 + 0.329680i) q^{3} +(-0.946744 + 1.76173i) q^{4} +(-1.96369 - 1.06954i) q^{5} +(1.60294 + 1.77256i) q^{6} +(-0.274644 - 0.663049i) q^{7} +(2.82547 - 0.129286i) q^{8} +(-0.133309 + 0.0552186i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(191\)	\(257\)	\(261\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{9}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.725691	−	1.21383i	−0.513141	−	0.858304i
							\(3\)	−1.65741	+	0.329680i	−0.956908	+	0.190341i	−0.648756	−	0.760996i	\(-0.724710\pi\)
−0.308152	+	0.951337i	\(0.599710\pi\)
\(5\)	−1.96369	−	1.06954i	−0.878189	−	0.478314i
							\(7\)	−0.274644	−	0.663049i	−0.103806	−	0.250609i	0.863440	−	0.504452i	\(-0.168305\pi\)
−0.967245	+	0.253843i	\(0.918305\pi\)
\(11\)	2.79927	+	4.18940i	0.844010	+	1.26315i	0.962796	+	0.270231i	\(0.0871000\pi\)
							−0.118785	+	0.992920i	\(0.537900\pi\)
\(13\)	−2.38011	−	1.59034i	−0.660123	−	0.441080i	0.179860	−	0.983692i	\(-0.442435\pi\)
							−0.839983	+	0.542612i	\(0.817435\pi\)
\(17\)	6.03419			1.46351			0.731753	−	0.681570i	\(-0.238702\pi\)
							0.731753	+	0.681570i	\(0.238702\pi\)
\(19\)	3.45429	−	0.687102i	0.792469	−	0.157632i	0.217774	−	0.975999i	\(-0.430120\pi\)
							0.574695	+	0.818367i	\(0.305120\pi\)
\(23\)	2.84797	+	1.17967i	0.593842	+	0.245977i	0.659302	−	0.751878i	\(-0.270852\pi\)
							−0.0654607	+	0.997855i	\(0.520852\pi\)
\(29\)	−4.11620	+	6.16033i	−0.764360	+	1.14395i	0.221301	+	0.975206i	\(0.428970\pi\)
							−0.985661	+	0.168740i	\(0.946030\pi\)
\(31\)	7.87985			1.41526			0.707631	−	0.706582i	\(-0.249764\pi\)
							0.707631	+	0.706582i	\(0.249764\pi\)
\(37\)	−3.69682	−	5.53269i	−0.607754	−	0.909568i	0.392193	−	0.919883i	\(-0.371717\pi\)
							−0.999947	+	0.0103148i	\(0.996717\pi\)
\(41\)	−1.68490	+	0.697907i	−0.263137	+	0.108995i	−0.510351	−	0.859966i	\(-0.670485\pi\)
							0.247215	+	0.968961i	\(0.420485\pi\)
\(43\)	−1.09795	+	5.51979i	−0.167436	+	0.841760i	0.802171	+	0.597094i	\(0.203678\pi\)
							−0.969608	+	0.244666i	\(0.921322\pi\)
\(47\)			7.07259i			1.03164i	0.856696	+	0.515821i	\(0.172513\pi\)
							−0.856696	+	0.515821i	\(0.827487\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bj.a.27.16	yes	368
5.3	odd	4		320.2.bd.a.283.9	yes	368
64.19	odd	16		320.2.bd.a.147.9	✓	368
320.83	even	16	inner	320.2.bj.a.83.16	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bd.a.147.9	✓	368	64.19	odd	16
320.2.bd.a.283.9	yes	368	5.3	odd	4
320.2.bj.a.27.16	yes	368	1.1	even	1	trivial
320.2.bj.a.83.16	yes	368	320.83	even	16	inner