Embedded newform 17.5.e.a.3.4

• Label: 17.5.e.a.3.4
• Level: $17$
• Weight: $5$
• Character: 17.3
• Analytic conductor: $1.757$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 17 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	17.e (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.75728937243\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(5\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			3.4
Character	\(\chi\)	\(=\)	17.3
Dual form			17.5.e.a.6.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.43536 - 1.42297i) q^{2} +(6.42184 + 1.27738i) q^{3} +(-1.53684 + 1.53684i) q^{4} +(-8.81924 - 13.1989i) q^{5} +(23.8790 - 4.74983i) q^{6} +(5.40438 - 8.08823i) q^{7} +(-25.8603 + 62.4323i) q^{8} +(-35.2259 - 14.5911i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/17\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.43536	−	1.42297i	0.858841	−	0.355744i	0.0905867	−	0.995889i	\(-0.471126\pi\)
							0.768254	+	0.640145i	\(0.221126\pi\)
\(3\)	6.42184	+	1.27738i	0.713538	+	0.141931i	0.538494	−	0.842629i	\(-0.318993\pi\)
							0.175043	+	0.984561i	\(0.443993\pi\)
\(5\)	−8.81924	−	13.1989i	−0.352770	−	0.527957i	0.612068	−	0.790806i	\(-0.290338\pi\)
							−0.964837	+	0.262848i	\(0.915338\pi\)
\(7\)	5.40438	−	8.08823i	0.110293	−	0.165066i	−0.772217	−	0.635359i	\(-0.780852\pi\)
							0.882510	+	0.470294i	\(0.155852\pi\)
\(11\)	14.6844	+	73.8233i	0.121358	+	0.610110i	0.992817	+	0.119641i	\(0.0381744\pi\)
							−0.871459	+	0.490469i	\(0.836826\pi\)
\(13\)	−54.9309	−	54.9309i	−0.325035	−	0.325035i	0.525660	−	0.850695i	\(-0.323818\pi\)
							−0.850695	+	0.525660i	\(0.823818\pi\)
\(17\)	272.518	−	96.2017i	0.942970	−	0.332878i
							\(19\)	439.575	−	182.078i	1.21766	−	0.504371i	0.320995	−	0.947081i	\(-0.395983\pi\)
0.896665	+	0.442710i	\(0.145983\pi\)
\(23\)	107.678	−	21.4185i	0.203550	−	0.0404886i	−0.0922621	−	0.995735i	\(-0.529410\pi\)
							0.295812	+	0.955246i	\(0.404410\pi\)
\(29\)	−1143.83	+	764.283i	−1.36008	+	0.908779i	−0.999711	−	0.0240289i	\(-0.992351\pi\)
							−0.360373	+	0.932808i	\(0.617351\pi\)
\(31\)	−152.618	+	767.263i	−0.158812	+	0.798401i	0.816462	+	0.577400i	\(0.195933\pi\)
							−0.975273	+	0.221001i	\(0.929067\pi\)
\(37\)	2165.47	+	430.739i	1.58179	+	0.314638i	0.906269	−	0.422701i	\(-0.138918\pi\)
							0.675523	+	0.737339i	\(0.263918\pi\)
\(41\)	−950.559	+	1422.61i	−0.565472	+	0.846289i	−0.998481	−	0.0551009i	\(-0.982452\pi\)
							0.433009	+	0.901390i	\(0.357452\pi\)
\(43\)	−595.242	−	246.557i	−0.321926	−	0.133346i	0.215868	−	0.976423i	\(-0.430742\pi\)
							−0.537794	+	0.843076i	\(0.680742\pi\)
\(47\)	−2449.02	−	2449.02i	−1.10866	−	1.10866i	−0.993327	−	0.115330i	\(-0.963207\pi\)
							−0.115330	−	0.993327i	\(-0.536793\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.5.e.a.3.4	✓	40
3.2	odd	2		153.5.p.a.37.2		40
17.6	odd	16	inner	17.5.e.a.6.4	yes	40
51.23	even	16		153.5.p.a.91.2		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.5.e.a.3.4	✓	40	1.1	even	1	trivial
17.5.e.a.6.4	yes	40	17.6	odd	16	inner
153.5.p.a.37.2		40	3.2	odd	2
153.5.p.a.91.2		40	51.23	even	16