Embedded newform 320.2.bj.a.107.28

• Label: 320.2.bj.a.107.28
• Level: $320$
• Weight: $2$
• Character: 320.107
• 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			107.28
Character	\(\chi\)	\(=\)	320.107
Dual form			320.2.bj.a.3.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.298862 + 1.38227i) q^{2} +(-0.954542 + 1.42857i) q^{3} +(-1.82136 + 0.826218i) q^{4} +(-0.439674 + 2.19242i) q^{5} +(-2.25996 - 0.892493i) q^{6} +(1.21086 - 0.501553i) q^{7} +(-1.68640 - 2.27070i) q^{8} +(0.0183786 + 0.0443699i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(191\)	\(257\)	\(261\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{13}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.298862	+	1.38227i	0.211327	+	0.977415i
							\(3\)	−0.954542	+	1.42857i	−0.551105	+	0.824787i	−0.997545	−	0.0700262i	\(-0.977692\pi\)
0.446440	+	0.894814i	\(0.352692\pi\)
\(5\)	−0.439674	+	2.19242i	−0.196628	+	0.980478i
							\(7\)	1.21086	−	0.501553i	0.457660	−	0.189569i	−0.141929	−	0.989877i	\(-0.545331\pi\)
0.599590	+	0.800308i	\(0.295331\pi\)
\(11\)	−0.201483	+	1.01293i	−0.0607496	+	0.305409i	−0.999199	−	0.0400069i	\(-0.987262\pi\)
							0.938450	+	0.345416i	\(0.112262\pi\)
\(13\)	−5.13869	+	1.02215i	−1.42521	+	0.283493i	−0.846653	−	0.532146i	\(-0.821386\pi\)
							−0.578562	+	0.815638i	\(0.696386\pi\)
\(17\)	2.47031			0.599138			0.299569	−	0.954075i	\(-0.403157\pi\)
							0.299569	+	0.954075i	\(0.403157\pi\)
\(19\)	2.46536	−	3.68967i	0.565592	−	0.846469i	−0.432895	−	0.901444i	\(-0.642508\pi\)
							0.998488	+	0.0549755i	\(0.0175081\pi\)
\(23\)	0.856259	−	2.06719i	0.178542	−	0.431039i	−0.809119	−	0.587645i	\(-0.800055\pi\)
							0.987661	+	0.156606i	\(0.0500552\pi\)
\(29\)	0.544444	+	2.73710i	0.101101	+	0.508267i	0.997839	+	0.0657063i	\(0.0209300\pi\)
							−0.896738	+	0.442561i	\(0.854070\pi\)
\(31\)	4.57040			0.820868			0.410434	−	0.911890i	\(-0.365377\pi\)
							0.410434	+	0.911890i	\(0.365377\pi\)
\(37\)	−2.14290	+	10.7731i	−0.352291	+	1.77109i	0.245461	+	0.969406i	\(0.421061\pi\)
							−0.597752	+	0.801681i	\(0.703939\pi\)
\(41\)	2.66608	+	6.43649i	0.416372	+	1.00521i	0.983390	+	0.181505i	\(0.0580969\pi\)
							−0.567018	+	0.823705i	\(0.691903\pi\)
\(43\)	−5.46228	+	3.64978i	−0.832990	+	0.556586i	−0.897339	−	0.441341i	\(-0.854503\pi\)
							0.0643492	+	0.997927i	\(0.479503\pi\)
\(47\)			6.45903i			0.942146i	0.882094	+	0.471073i	\(0.156133\pi\)
							−0.882094	+	0.471073i	\(0.843867\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bj.a.107.28	yes	368
5.3	odd	4		320.2.bd.a.43.44	✓	368
64.3	odd	16		320.2.bd.a.67.44	yes	368
320.3	even	16	inner	320.2.bj.a.3.28	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bd.a.43.44	✓	368	5.3	odd	4
320.2.bd.a.67.44	yes	368	64.3	odd	16
320.2.bj.a.3.28	yes	368	320.3	even	16	inner
320.2.bj.a.107.28	yes	368	1.1	even	1	trivial