Embedded newform 320.2.bj.a.27.31

• Label: 320.2.bj.a.27.31
• 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			27.31
Character	\(\chi\)	\(=\)	320.27
Dual form			320.2.bj.a.83.31

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.660171 - 1.25067i) q^{2} +(0.924494 - 0.183893i) q^{3} +(-1.12835 - 1.65131i) q^{4} +(1.84197 - 1.26773i) q^{5} +(0.380334 - 1.27764i) q^{6} +(-1.57130 - 3.79346i) q^{7} +(-2.81015 + 0.321045i) q^{8} +(-1.95077 + 0.808034i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.660171	−	1.25067i	0.466811	−	0.884357i
							\(3\)	0.924494	−	0.183893i	0.533757	−	0.106171i	0.0791494	−	0.996863i	\(-0.474780\pi\)
0.454607	+	0.890692i	\(0.349780\pi\)
\(5\)	1.84197	−	1.26773i	0.823756	−	0.566945i
							\(7\)	−1.57130	−	3.79346i	−0.593896	−	1.43379i	−0.879712	−	0.475508i	\(-0.842264\pi\)
0.285815	−	0.958285i	\(-0.407736\pi\)
\(11\)	3.51182	+	5.25581i	1.05885	+	1.58468i	0.781488	+	0.623920i	\(0.214461\pi\)
							0.277365	+	0.960765i	\(0.410539\pi\)
\(13\)	−0.166085	−	0.110975i	−0.0460638	−	0.0307789i	0.532325	−	0.846540i	\(-0.321318\pi\)
							−0.578389	+	0.815761i	\(0.696318\pi\)
\(17\)	−4.50343			−1.09224			−0.546121	−	0.837706i	\(-0.683896\pi\)
							−0.546121	+	0.837706i	\(0.683896\pi\)
\(19\)	7.57410	−	1.50658i	1.73762	−	0.345633i	0.778284	−	0.627912i	\(-0.216090\pi\)
							0.959332	+	0.282279i	\(0.0910903\pi\)
\(23\)	2.46978	+	1.02302i	0.514984	+	0.213314i	0.625012	−	0.780615i	\(-0.285094\pi\)
							−0.110028	+	0.993929i	\(0.535094\pi\)
\(29\)	2.21500	−	3.31498i	0.411315	−	0.615576i	−0.566747	−	0.823892i	\(-0.691798\pi\)
							0.978062	+	0.208316i	\(0.0667983\pi\)
\(31\)	1.97209			0.354198			0.177099	−	0.984193i	\(-0.443329\pi\)
							0.177099	+	0.984193i	\(0.443329\pi\)
\(37\)	0.230149	+	0.344443i	0.0378363	+	0.0566260i	0.849906	−	0.526935i	\(-0.176659\pi\)
							−0.812069	+	0.583561i	\(0.801659\pi\)
\(41\)	1.84987	−	0.766243i	0.288902	−	0.119667i	−0.233526	−	0.972351i	\(-0.575026\pi\)
							0.522428	+	0.852684i	\(0.325026\pi\)
\(43\)	−0.204946	+	1.03033i	−0.0312539	+	0.157124i	−0.993261	−	0.115901i	\(-0.963024\pi\)
							0.962007	+	0.273026i	\(0.0880244\pi\)
\(47\)			4.20639i			0.613565i	0.951780	+	0.306783i	\(0.0992524\pi\)
							−0.951780	+	0.306783i	\(0.900748\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bj.a.27.31	yes	368
5.3	odd	4		320.2.bd.a.283.8	yes	368
64.19	odd	16		320.2.bd.a.147.8	✓	368
320.83	even	16	inner	320.2.bj.a.83.31	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bd.a.147.8	✓	368	64.19	odd	16
320.2.bd.a.283.8	yes	368	5.3	odd	4
320.2.bj.a.27.31	yes	368	1.1	even	1	trivial
320.2.bj.a.83.31	yes	368	320.83	even	16	inner