Embedded newform 17.5.e.a.7.3

• Label: 17.5.e.a.7.3
• Level: $17$
• Weight: $5$
• Character: 17.7
• 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			7.3
Character	\(\chi\)	\(=\)	17.7
Dual form			17.5.e.a.5.3

$q$-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 \(\left(\mathbb{Z}/17\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{11}{16}\right)\)

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.5.e.a.5.3	✓	40	17.5	odd	16	inner
17.5.e.a.7.3	yes	40	1.1	even	1	trivial
153.5.p.a.73.3		40	51.5	even	16
153.5.p.a.109.3		40	3.2	odd	2