Embedded newform 320.2.bf.a.29.13

• Label: 320.2.bf.a.29.13
• Level: $320$
• Weight: $2$
• Character: 320.29
• Analytic conductor: $2.555$
• Analytic rank: $0$
• Dimension: $368$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	320.bf (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			29.13
Character	\(\chi\)	\(=\)	320.29
Dual form			320.2.bf.a.309.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.942250 + 1.05459i) q^{2} +(-0.630723 + 3.17086i) q^{3} +(-0.224328 - 1.98738i) q^{4} +(0.0178005 + 2.23600i) q^{5} +(-2.74966 - 3.65290i) q^{6} +(-1.29895 - 3.13595i) q^{7} +(2.30725 + 1.63603i) q^{8} +(-6.88490 - 2.85182i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 16 q^{4} - 8 q^{5} - 16 q^{6} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(191\)	\(257\)	\(261\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{11}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.942250	+	1.05459i	−0.666272	+	0.745709i
							\(3\)	−0.630723	+	3.17086i	−0.364148	+	1.83070i	0.170172	+	0.985414i	\(0.445568\pi\)
−0.534320	+	0.845282i	\(0.679432\pi\)
\(5\)	0.0178005	+	2.23600i	0.00796061	+	0.999968i
							\(7\)	−1.29895	−	3.13595i	−0.490959	−	1.18528i	−0.954233	−	0.299064i	\(-0.903325\pi\)
0.463274	−	0.886215i	\(-0.346675\pi\)
\(11\)	−0.158217	−	0.795410i	−0.0477042	−	0.239825i	0.949575	−	0.313539i	\(-0.101515\pi\)
							−0.997279	+	0.0737142i	\(0.976515\pi\)
\(13\)	−4.64958	+	3.10675i	−1.28956	+	0.861657i	−0.995558	−	0.0941478i	\(-0.969987\pi\)
							−0.294002	+	0.955805i	\(0.594987\pi\)
\(17\)	−0.606661	+	0.606661i	−0.147137	+	0.147137i	−0.776838	−	0.629701i	\(-0.783177\pi\)
							0.629701	+	0.776838i	\(0.283177\pi\)
\(19\)	3.99547	−	2.66969i	0.916625	−	0.612469i	−0.00523785	−	0.999986i	\(-0.501667\pi\)
							0.921862	+	0.387517i	\(0.126667\pi\)
\(23\)	0.999947	+	0.414192i	0.208503	+	0.0863649i	0.484491	−	0.874796i	\(-0.339005\pi\)
							−0.275988	+	0.961161i	\(0.589005\pi\)
\(29\)	−1.14304	+	5.74646i	−0.212258	+	1.06709i	0.716834	+	0.697243i	\(0.245590\pi\)
							−0.929092	+	0.369848i	\(0.879410\pi\)
\(31\)			3.67898i			0.660765i	0.943847	+	0.330382i	\(0.107178\pi\)
							−0.943847	+	0.330382i	\(0.892822\pi\)
\(37\)	−4.09851	+	6.13385i	−0.673790	+	1.00840i	0.324259	+	0.945968i	\(0.394885\pi\)
							−0.998049	+	0.0624299i	\(0.980115\pi\)
\(41\)	0.598628	−	1.44522i	0.0934900	−	0.225705i	−0.870216	−	0.492670i	\(-0.836021\pi\)
							0.963706	+	0.266965i	\(0.0860209\pi\)
\(43\)	−1.28531	−	6.46167i	−0.196007	−	0.985395i	−0.946054	−	0.324010i	\(-0.894969\pi\)
							0.750046	−	0.661385i	\(-0.230031\pi\)
\(47\)	−0.607762	+	0.607762i	−0.0886512	+	0.0886512i	−0.750042	−	0.661391i	\(-0.769967\pi\)
							0.661391	+	0.750042i	\(0.269967\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bf.a.29.13	✓	368
5.4	even	2	inner	320.2.bf.a.29.34	yes	368
64.53	even	16	inner	320.2.bf.a.309.34	yes	368
320.309	even	16	inner	320.2.bf.a.309.13	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bf.a.29.13	✓	368	1.1	even	1	trivial
320.2.bf.a.29.34	yes	368	5.4	even	2	inner
320.2.bf.a.309.13	yes	368	320.309	even	16	inner
320.2.bf.a.309.34	yes	368	64.53	even	16	inner