Embedded newform 17.5.e.a.5.3

• Label: 17.5.e.a.5.3
• Level: $17$
• Weight: $5$
• Character: 17.5
• 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			5.3
Character	\(\chi\)	\(=\)	17.5
Dual form			17.5.e.a.7.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.210299 - 0.507707i) q^{2} +(-5.96248 - 8.92348i) q^{3} +(11.1002 - 11.1002i) q^{4} +(1.57331 - 7.90954i) q^{5} +(-3.27661 + 4.90379i) q^{6} +(7.85546 + 39.4921i) q^{7} +(-16.0933 - 6.66606i) q^{8} +(-13.0800 + 31.5779i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.210299	−	0.507707i	−0.0525748	−	0.126927i	0.895410	−	0.445243i	\(-0.146883\pi\)
							−0.947985	+	0.318316i	\(0.896883\pi\)
\(3\)	−5.96248	−	8.92348i	−0.662498	−	0.991498i	−0.998763	−	0.0497304i	\(-0.984164\pi\)
							0.336265	−	0.941767i	\(-0.390836\pi\)
\(5\)	1.57331	−	7.90954i	0.0629323	−	0.316382i	−0.936478	−	0.350727i	\(-0.885935\pi\)
							0.999410	+	0.0343451i	\(0.0109345\pi\)
\(7\)	7.85546	+	39.4921i	0.160315	+	0.805960i	0.974332	+	0.225118i	\(0.0722767\pi\)
							−0.814016	+	0.580842i	\(0.802723\pi\)
\(11\)	37.6217	+	25.1380i	0.310923	+	0.207752i	0.701234	−	0.712931i	\(-0.252633\pi\)
							−0.390311	+	0.920683i	\(0.627633\pi\)
\(13\)	199.696	+	199.696i	1.18163	+	1.18163i	0.979321	+	0.202314i	\(0.0648461\pi\)
							0.202314	+	0.979321i	\(0.435154\pi\)
\(17\)	14.9096	−	288.615i	0.0515902	−	0.998668i
							\(19\)	−33.4418	−	80.7356i	−0.0926365	−	0.223644i	0.870769	−	0.491692i	\(-0.163621\pi\)
−0.963406	+	0.268048i	\(0.913621\pi\)
\(23\)	−447.780	+	670.150i	−0.846465	+	1.26682i	0.115405	+	0.993319i	\(0.463183\pi\)
							−0.961870	+	0.273506i	\(0.911817\pi\)
\(29\)	−511.346	−	101.713i	−0.608021	−	0.120943i	−0.118529	−	0.992951i	\(-0.537818\pi\)
							−0.489492	+	0.872008i	\(0.662818\pi\)
\(31\)	−365.563	+	244.261i	−0.380398	+	0.254174i	−0.731038	−	0.682336i	\(-0.760964\pi\)
							0.350640	+	0.936510i	\(0.385964\pi\)
\(37\)	971.253	+	1453.58i	0.709462	+	1.06178i	0.994649	+	0.103313i	\(0.0329443\pi\)
							−0.285187	+	0.958472i	\(0.592056\pi\)
\(41\)	−384.510	−	1933.06i	−0.228739	−	1.14995i	−0.908942	−	0.416923i	\(-0.863108\pi\)
							0.680203	−	0.733024i	\(-0.261892\pi\)
\(43\)	−142.187	+	343.271i	−0.0768997	+	0.185652i	−0.957654	−	0.287921i	\(-0.907036\pi\)
							0.880755	+	0.473573i	\(0.157036\pi\)
\(47\)	−2087.62	−	2087.62i	−0.945053	−	0.945053i	0.0535143	−	0.998567i	\(-0.482958\pi\)
							−0.998567	+	0.0535143i	\(0.982958\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.5.e.a.5.3	✓	40
3.2	odd	2		153.5.p.a.73.3		40
17.7	odd	16	inner	17.5.e.a.7.3	yes	40
51.41	even	16		153.5.p.a.109.3		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.5.e.a.5.3	✓	40	1.1	even	1	trivial
17.5.e.a.7.3	yes	40	17.7	odd	16	inner
153.5.p.a.73.3		40	3.2	odd	2
153.5.p.a.109.3		40	51.41	even	16