Embedded newform 320.2.bj.a.3.26

• Label: 320.2.bj.a.3.26
• Level: $320$
• Weight: $2$
• Character: 320.3
• 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			3.26
Character	\(\chi\)	\(=\)	320.3
Dual form			320.2.bj.a.107.26

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.171351 - 1.40379i) q^{2} +(0.198107 + 0.296488i) q^{3} +(-1.94128 - 0.481083i) q^{4} +(-0.936268 + 2.03062i) q^{5} +(0.450154 - 0.227298i) q^{6} +(3.61780 + 1.49854i) q^{7} +(-1.00798 + 2.64272i) q^{8} +(1.09939 - 2.65417i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.171351	−	1.40379i	0.121163	−	0.992633i
							\(3\)	0.198107	+	0.296488i	0.114377	+	0.171177i	0.884241	−	0.467030i	\(-0.154676\pi\)
−0.769864	+	0.638208i	\(0.779676\pi\)
\(5\)	−0.936268	+	2.03062i	−0.418712	+	0.908119i
							\(7\)	3.61780	+	1.49854i	1.36740	+	0.566395i	0.941082	−	0.338178i	\(-0.109811\pi\)
0.426317	+	0.904574i	\(0.359811\pi\)
\(11\)	−0.389836	−	1.95984i	−0.117540	−	0.590913i	−0.993995	−	0.109429i	\(-0.965098\pi\)
							0.876455	−	0.481484i	\(-0.159902\pi\)
\(13\)	4.89338	+	0.973354i	1.35718	+	0.269960i	0.819447	−	0.573156i	\(-0.194281\pi\)
							0.537733	+	0.843115i	\(0.319281\pi\)
\(17\)	0.690091			0.167372			0.0836858	−	0.996492i	\(-0.473331\pi\)
							0.0836858	+	0.996492i	\(0.473331\pi\)
\(19\)	2.43187	+	3.63955i	0.557909	+	0.834970i	0.998015	−	0.0629759i	\(-0.0200591\pi\)
							−0.440106	+	0.897946i	\(0.645059\pi\)
\(23\)	−1.84995	−	4.46618i	−0.385742	−	0.931263i	−0.990831	−	0.135105i	\(-0.956863\pi\)
							0.605090	−	0.796157i	\(-0.293137\pi\)
\(29\)	−1.05714	+	5.31461i	−0.196306	+	0.986899i	0.749460	+	0.662050i	\(0.230313\pi\)
							−0.945766	+	0.324849i	\(0.894687\pi\)
\(31\)	−0.845423			−0.151842			−0.0759212	−	0.997114i	\(-0.524190\pi\)
							−0.0759212	+	0.997114i	\(0.524190\pi\)
\(37\)	1.38403	+	6.95796i	0.227532	+	1.14388i	0.910524	+	0.413456i	\(0.135679\pi\)
							−0.682992	+	0.730426i	\(0.739321\pi\)
\(41\)	−3.34324	+	8.07131i	−0.522127	+	1.26053i	0.414452	+	0.910071i	\(0.363973\pi\)
							−0.936579	+	0.350455i	\(0.886027\pi\)
\(43\)	−10.1318	−	6.76984i	−1.54508	−	1.03239i	−0.977969	−	0.208753i	\(-0.933060\pi\)
							−0.567115	−	0.823639i	\(-0.691940\pi\)
\(47\)		−	12.7257i		−	1.85624i	−0.372280	−	0.928120i	\(-0.621424\pi\)
							0.372280	−	0.928120i	\(-0.378576\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bj.a.3.26	yes	368
5.2	odd	4		320.2.bd.a.67.45	yes	368
64.43	odd	16		320.2.bd.a.43.45	✓	368
320.107	even	16	inner	320.2.bj.a.107.26	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bd.a.43.45	✓	368	64.43	odd	16
320.2.bd.a.67.45	yes	368	5.2	odd	4
320.2.bj.a.3.26	yes	368	1.1	even	1	trivial
320.2.bj.a.107.26	yes	368	320.107	even	16	inner