Embedded newform 320.2.bf.a.29.2

• Label: 320.2.bf.a.29.2
• 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			29.2
Character	\(\chi\)	\(=\)	320.29
Dual form			320.2.bf.a.309.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40092 - 0.193431i) q^{2} +(-0.612243 + 3.07795i) q^{3} +(1.92517 + 0.541963i) q^{4} +(-2.22253 + 0.245694i) q^{5} +(1.45308 - 4.19355i) q^{6} +(1.41327 + 3.41194i) q^{7} +(-2.59218 - 1.13164i) q^{8} +(-6.32731 - 2.62086i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 16 q^{4} - 8 q^{5} - 16 q^{6} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40092	−	0.193431i	−0.990602	−	0.136776i
							\(3\)	−0.612243	+	3.07795i	−0.353479	+	1.77706i	0.238540	+	0.971133i	\(0.423331\pi\)
−0.592019	+	0.805924i	\(0.701669\pi\)
\(5\)	−2.22253	+	0.245694i	−0.993945	+	0.109878i
							\(7\)	1.41327	+	3.41194i	0.534167	+	1.28959i	0.928741	+	0.370728i	\(0.120892\pi\)
−0.394575	+	0.918864i	\(0.629108\pi\)
\(11\)	0.728826	+	3.66406i	0.219749	+	1.10475i	0.920318	+	0.391170i	\(0.127930\pi\)
							−0.700569	+	0.713584i	\(0.747070\pi\)
\(13\)	0.0680913	−	0.0454972i	0.0188851	−	0.0126186i	−0.546092	−	0.837725i	\(-0.683885\pi\)
							0.564977	+	0.825107i	\(0.308885\pi\)
\(17\)	2.71216	−	2.71216i	0.657797	−	0.657797i	−0.297062	−	0.954858i	\(-0.596007\pi\)
							0.954858	+	0.297062i	\(0.0960067\pi\)
\(19\)	−1.16365	+	0.777526i	−0.266959	+	0.178377i	−0.681844	−	0.731497i	\(-0.738822\pi\)
							0.414885	+	0.909874i	\(0.363822\pi\)
\(23\)	−4.75609	−	1.97004i	−0.991713	−	0.410781i	−0.172961	−	0.984929i	\(-0.555334\pi\)
							−0.818752	+	0.574148i	\(0.805334\pi\)
\(29\)	0.696414	−	3.50111i	0.129321	−	0.650139i	−0.860688	−	0.509133i	\(-0.829966\pi\)
							0.990009	−	0.141006i	\(-0.0450338\pi\)
\(31\)		−	6.61368i		−	1.18785i	−0.804520	−	0.593926i	\(-0.797577\pi\)
							0.804520	−	0.593926i	\(-0.202423\pi\)
\(37\)	−2.65755	+	3.97730i	−0.436898	+	0.653865i	−0.982947	−	0.183889i	\(-0.941131\pi\)
							0.546049	+	0.837753i	\(0.316131\pi\)
\(41\)	−4.39263	+	10.6047i	−0.686013	+	1.65618i	0.0666507	+	0.997776i	\(0.478769\pi\)
							−0.752664	+	0.658405i	\(0.771231\pi\)
\(43\)	0.876872	+	4.40833i	0.133722	+	0.672264i	0.988248	+	0.152857i	\(0.0488473\pi\)
							−0.854527	+	0.519407i	\(0.826153\pi\)
\(47\)	4.04307	−	4.04307i	0.589742	−	0.589742i	−0.347820	−	0.937561i	\(-0.613078\pi\)
							0.937561	+	0.347820i	\(0.113078\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bf.a.29.2	✓	368
5.4	even	2	inner	320.2.bf.a.29.45	yes	368
64.53	even	16	inner	320.2.bf.a.309.45	yes	368
320.309	even	16	inner	320.2.bf.a.309.2	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bf.a.29.2	✓	368	1.1	even	1	trivial
320.2.bf.a.29.45	yes	368	5.4	even	2	inner
320.2.bf.a.309.2	yes	368	320.309	even	16	inner
320.2.bf.a.309.45	yes	368	64.53	even	16	inner