Embedded newform 17.5.e.a.6.2

• Label: 17.5.e.a.6.2
• Level: $17$
• Weight: $5$
• Character: 17.6
• 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			6.2
Character	\(\chi\)	\(=\)	17.6
Dual form			17.5.e.a.3.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.92749 - 1.21261i) q^{2} +(8.22545 - 1.63614i) q^{3} +(-4.21393 - 4.21393i) q^{4} +(21.2448 - 31.7951i) q^{5} +(-26.0639 - 5.18443i) q^{6} +(5.22138 + 7.81435i) q^{7} +(26.6281 + 64.2859i) q^{8} +(-9.85322 + 4.08134i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{15}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.92749	−	1.21261i	−0.731872	−	0.303151i	−0.0145510	−	0.999894i	\(-0.504632\pi\)
							−0.717321	+	0.696743i	\(0.754632\pi\)
\(3\)	8.22545	−	1.63614i	0.913939	−	0.181794i	0.284348	−	0.958721i	\(-0.408223\pi\)
							0.629590	+	0.776927i	\(0.283223\pi\)
\(5\)	21.2448	−	31.7951i	0.849791	−	1.27180i	−0.110802	−	0.993843i	\(-0.535342\pi\)
							0.960593	−	0.277960i	\(-0.0896582\pi\)
\(7\)	5.22138	+	7.81435i	0.106559	+	0.159477i	0.880917	−	0.473271i	\(-0.156927\pi\)
							−0.774358	+	0.632748i	\(0.781927\pi\)
\(11\)	−30.2570	+	152.112i	−0.250058	+	1.25712i	0.627865	+	0.778322i	\(0.283929\pi\)
							−0.877923	+	0.478802i	\(0.841071\pi\)
\(13\)	144.350	−	144.350i	0.854143	−	0.854143i	−0.136498	−	0.990640i	\(-0.543585\pi\)
							0.990640	+	0.136498i	\(0.0435846\pi\)
\(17\)	−37.1528	+	286.602i	−0.128556	+	0.991702i
							\(19\)	−73.3524	−	30.3835i	−0.203192	−	0.0841649i	0.278766	−	0.960359i	\(-0.410074\pi\)
−0.481959	+	0.876194i	\(0.660074\pi\)
\(23\)	411.759	+	81.9040i	0.778373	+	0.154828i	0.568260	−	0.822849i	\(-0.307617\pi\)
							0.210113	+	0.977677i	\(0.432617\pi\)
\(29\)	696.040	+	465.079i	0.827634	+	0.553007i	0.895692	−	0.444674i	\(-0.146680\pi\)
							−0.0680585	+	0.997681i	\(0.521680\pi\)
\(31\)	−60.5007	−	304.158i	−0.0629560	−	0.316501i	0.936456	−	0.350785i	\(-0.114085\pi\)
							−0.999412	+	0.0342835i	\(0.989085\pi\)
\(37\)	−2146.97	+	427.059i	−1.56827	+	0.311949i	−0.901322	−	0.433150i	\(-0.857402\pi\)
							−0.666953	+	0.745100i	\(0.732402\pi\)
\(41\)	−1181.95	−	1768.92i	−0.703125	−	1.05230i	−0.995385	−	0.0959615i	\(-0.969407\pi\)
							0.292261	−	0.956339i	\(-0.405593\pi\)
\(43\)	745.803	−	308.922i	0.403355	−	0.167075i	−0.171777	−	0.985136i	\(-0.554951\pi\)
							0.575132	+	0.818061i	\(0.304951\pi\)
\(47\)	−1064.52	+	1064.52i	−0.481900	+	0.481900i	−0.905738	−	0.423838i	\(-0.860683\pi\)
							0.423838	+	0.905738i	\(0.360683\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.5.e.a.6.2	yes	40
3.2	odd	2		153.5.p.a.91.4		40
17.3	odd	16	inner	17.5.e.a.3.2	✓	40
51.20	even	16		153.5.p.a.37.4		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.5.e.a.3.2	✓	40	17.3	odd	16	inner
17.5.e.a.6.2	yes	40	1.1	even	1	trivial
153.5.p.a.37.4		40	51.20	even	16
153.5.p.a.91.4		40	3.2	odd	2