Embedded newform 17.5.e.a.3.1

• Label: 17.5.e.a.3.1
• 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			3.1
Character	\(\chi\)	\(=\)	17.3
Dual form			17.5.e.a.6.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.12052 + 2.94942i) q^{2} +(-0.829686 - 0.165035i) q^{3} +(30.6890 - 30.6890i) q^{4} +(-14.8705 - 22.2553i) q^{5} +(6.39455 - 1.27196i) q^{6} +(16.8953 - 25.2857i) q^{7} +(-80.8164 + 195.108i) q^{8} +(-74.1731 - 30.7235i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.12052	+	2.94942i	−1.78013	+	0.737354i	−0.787478	+	0.616342i	\(0.788614\pi\)
							−0.992651	+	0.121011i	\(0.961386\pi\)
\(3\)	−0.829686	−	0.165035i	−0.0921873	−	0.0183372i	0.148781	−	0.988870i	\(-0.452465\pi\)
							−0.240968	+	0.970533i	\(0.577465\pi\)
\(5\)	−14.8705	−	22.2553i	−0.594821	−	0.890213i	0.404887	−	0.914367i	\(-0.367311\pi\)
							−0.999708	+	0.0241536i	\(0.992311\pi\)
\(7\)	16.8953	−	25.2857i	0.344803	−	0.516034i	−0.618022	−	0.786161i	\(-0.712066\pi\)
							0.962825	+	0.270127i	\(0.0870657\pi\)
\(11\)	−28.3501	−	142.526i	−0.234298	−	1.17790i	−0.901419	−	0.432948i	\(-0.857473\pi\)
							0.667121	−	0.744950i	\(-0.267527\pi\)
\(13\)	−15.1728	−	15.1728i	−0.0897796	−	0.0897796i	0.660791	−	0.750570i	\(-0.270221\pi\)
							−0.750570	+	0.660791i	\(0.770221\pi\)
\(17\)	191.261	+	216.657i	0.661804	+	0.749677i
							\(19\)	−190.943	+	79.0912i	−0.528928	+	0.219089i	−0.631134	−	0.775674i	\(-0.717410\pi\)
0.102205	+	0.994763i	\(0.467410\pi\)
\(23\)	−335.795	+	66.7937i	−0.634773	+	0.126264i	−0.501979	−	0.864880i	\(-0.667395\pi\)
							−0.132793	+	0.991144i	\(0.542395\pi\)
\(29\)	923.477	−	617.048i	1.09807	−	0.733707i	0.131810	−	0.991275i	\(-0.457921\pi\)
							0.966260	+	0.257568i	\(0.0829210\pi\)
\(31\)	−7.20195	+	36.2067i	−0.00749423	+	0.0376760i	−0.984351	−	0.176216i	\(-0.943614\pi\)
							0.976857	+	0.213892i	\(0.0686142\pi\)
\(37\)	184.318	+	36.6632i	0.134637	+	0.0267810i	0.261949	−	0.965082i	\(-0.415635\pi\)
							−0.127312	+	0.991863i	\(0.540635\pi\)
\(41\)	926.243	−	1386.22i	0.551007	−	0.824641i	−0.446531	−	0.894768i	\(-0.647341\pi\)
							0.997538	+	0.0701277i	\(0.0223407\pi\)
\(43\)	−1079.53	−	447.157i	−0.583847	−	0.241837i	0.0711542	−	0.997465i	\(-0.477332\pi\)
							−0.655001	+	0.755628i	\(0.727332\pi\)
\(47\)	668.010	+	668.010i	0.302404	+	0.302404i	0.841954	−	0.539550i	\(-0.181406\pi\)
							−0.539550	+	0.841954i	\(0.681406\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.5.e.a.3.1	✓	40
3.2	odd	2		153.5.p.a.37.5		40
17.6	odd	16	inner	17.5.e.a.6.1	yes	40
51.23	even	16		153.5.p.a.91.5		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.5.e.a.3.1	✓	40	1.1	even	1	trivial
17.5.e.a.6.1	yes	40	17.6	odd	16	inner
153.5.p.a.37.5		40	3.2	odd	2
153.5.p.a.91.5		40	51.23	even	16