Embedded newform 320.2.bj.a.187.6

• Label: 320.2.bj.a.187.6
• Level: $320$
• Weight: $2$
• Character: 320.187
• 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			187.6
Character	\(\chi\)	\(=\)	320.187
Dual form			320.2.bj.a.243.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.32051 - 0.506205i) q^{2} +(0.625866 + 3.14644i) q^{3} +(1.48751 + 1.33690i) q^{4} +(-2.02429 - 0.949858i) q^{5} +(0.766278 - 4.47173i) q^{6} +(-1.61595 - 3.90126i) q^{7} +(-1.28754 - 2.51838i) q^{8} +(-6.73673 + 2.79045i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.32051	−	0.506205i	−0.933744	−	0.357941i
							\(3\)	0.625866	+	3.14644i	0.361344	+	1.81660i	0.550717	+	0.834692i	\(0.314354\pi\)
−0.189373	+	0.981905i	\(0.560646\pi\)
\(5\)	−2.02429	−	0.949858i	−0.905292	−	0.424790i
							\(7\)	−1.61595	−	3.90126i	−0.610773	−	1.47454i	−0.862153	−	0.506648i	\(-0.830884\pi\)
0.251380	−	0.967889i	\(-0.419116\pi\)
\(11\)	−1.28460	+	0.858345i	−0.387323	+	0.258801i	−0.733952	−	0.679201i	\(-0.762326\pi\)
							0.346629	+	0.938002i	\(0.387326\pi\)
\(13\)	−0.465481	+	0.696642i	−0.129101	+	0.193214i	−0.890389	−	0.455201i	\(-0.849568\pi\)
							0.761288	+	0.648414i	\(0.224568\pi\)
\(17\)	−3.39643			−0.823755			−0.411877	−	0.911239i	\(-0.635127\pi\)
							−0.411877	+	0.911239i	\(0.635127\pi\)
\(19\)	−0.673775	−	3.38730i	−0.154575	−	0.777099i	−0.977826	−	0.209421i	\(-0.932842\pi\)
							0.823251	−	0.567678i	\(-0.192158\pi\)
\(23\)	−6.81123	−	2.82130i	−1.42024	−	0.588282i	−0.465318	−	0.885143i	\(-0.654060\pi\)
							−0.954921	+	0.296861i	\(0.904060\pi\)
\(29\)	5.12683	+	3.42564i	0.952029	+	0.636125i	0.931530	−	0.363665i	\(-0.118475\pi\)
							0.0204989	+	0.999790i	\(0.493475\pi\)
\(31\)	1.40334			0.252048			0.126024	−	0.992027i	\(-0.459778\pi\)
							0.126024	+	0.992027i	\(0.459778\pi\)
\(37\)	−0.828762	+	0.553761i	−0.136248	+	0.0910378i	−0.621826	−	0.783155i	\(-0.713609\pi\)
							0.485579	+	0.874193i	\(0.338609\pi\)
\(41\)	−2.83367	+	1.17375i	−0.442546	+	0.183308i	−0.592818	−	0.805336i	\(-0.701985\pi\)
							0.150273	+	0.988645i	\(0.451985\pi\)
\(43\)	−0.308848	−	0.0614336i	−0.0470988	−	0.00936854i	0.171485	−	0.985187i	\(-0.445144\pi\)
							−0.218584	+	0.975818i	\(0.570144\pi\)
\(47\)			5.50089i			0.802388i	0.915993	+	0.401194i	\(0.131405\pi\)
							−0.915993	+	0.401194i	\(0.868595\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bj.a.187.6	yes	368
5.3	odd	4		320.2.bd.a.123.18	✓	368
64.51	odd	16		320.2.bd.a.307.18	yes	368
320.243	even	16	inner	320.2.bj.a.243.6	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bd.a.123.18	✓	368	5.3	odd	4
320.2.bd.a.307.18	yes	368	64.51	odd	16
320.2.bj.a.187.6	yes	368	1.1	even	1	trivial
320.2.bj.a.243.6	yes	368	320.243	even	16	inner