Embedded newform 17.10.d.a.9.10

• Label: 17.10.d.a.9.10
• Level: $17$
• Weight: $10$
• Character: 17.9
• Analytic conductor: $8.756$
• Analytic rank: $0$
• Dimension: $52$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 17 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	17.d (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.75560921479\)
Analytic rank:	\(0\)
Dimension:	\(52\)
Relative dimension:	\(13\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			9.10
Character	\(\chi\)	\(=\)	17.9
Dual form			17.10.d.a.2.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(17.3427 - 17.3427i) q^{2} +(-21.9931 - 9.10984i) q^{3} -89.5378i q^{4} +(-1000.76 + 2416.05i) q^{5} +(-539.408 + 223.430i) q^{6} +(1420.57 + 3429.57i) q^{7} +(7326.63 + 7326.63i) q^{8} +(-13517.3 - 13517.3i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 4 q^{2} - 4 q^{3} - 1140 q^{5} + 12308 q^{6} - 4 q^{7} + 2044 q^{8} - 83224 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	17.3427	−	17.3427i	0.766446	−	0.766446i	−0.211033	−	0.977479i	\(-0.567683\pi\)
							0.977479	+	0.211033i	\(0.0676828\pi\)
\(3\)	−21.9931	−	9.10984i	−0.156762	−	0.0649329i	0.302923	−	0.953015i	\(-0.402038\pi\)
							−0.459685	+	0.888082i	\(0.652038\pi\)
\(5\)	−1000.76	+	2416.05i	−0.716086	+	1.72878i	−0.0318850	+	0.999492i	\(0.510151\pi\)
							−0.684201	+	0.729293i	\(0.739849\pi\)
\(7\)	1420.57	+	3429.57i	0.223626	+	0.539881i	0.995377	−	0.0960438i	\(-0.0306189\pi\)
							−0.771751	+	0.635925i	\(0.780619\pi\)
\(11\)	3806.68	−	1576.78i	0.0783934	−	0.0324716i	−0.343142	−	0.939283i	\(-0.611491\pi\)
							0.421536	+	0.906812i	\(0.361491\pi\)
\(13\)		−	18236.6i		−	0.177092i	−0.996072	−	0.0885459i	\(-0.971778\pi\)
							0.996072	−	0.0885459i	\(-0.0282220\pi\)
\(17\)	101311.	+	329126.i	0.294197	+	0.955745i
							\(19\)	−666466.	+	666466.i	−1.17324	+	1.17324i	−0.191806	+	0.981433i	\(0.561435\pi\)
−0.981433	+	0.191806i	\(0.938565\pi\)
\(23\)	1.81374e6	−	751274.i	1.35145	−	0.559787i	0.414751	−	0.909935i	\(-0.363869\pi\)
							0.936695	+	0.350148i	\(0.113869\pi\)
\(29\)	909640.	−	2.19607e6i	0.238824	−	0.576573i	−0.758337	−	0.651862i	\(-0.773988\pi\)
							0.997162	+	0.0752893i	\(0.0239880\pi\)
\(31\)	−2.39565e6	−	992311.i	−0.465903	−	0.192983i	0.137367	−	0.990520i	\(-0.456136\pi\)
							−0.603270	+	0.797537i	\(0.706136\pi\)
\(37\)	1.58254e7	+	6.55511e6i	1.38819	+	0.575005i	0.946657	−	0.322242i	\(-0.104436\pi\)
							0.441529	+	0.897247i	\(0.354436\pi\)
\(41\)	−2.88871e6	−	6.97395e6i	−0.159653	−	0.385435i	0.823730	−	0.566983i	\(-0.191889\pi\)
							−0.983382	+	0.181547i	\(0.941889\pi\)
\(43\)	1.65431e7	+	1.65431e7i	0.737920	+	0.737920i	0.972175	−	0.234255i	\(-0.0752651\pi\)
							−0.234255	+	0.972175i	\(0.575265\pi\)
\(47\)			2.85310e7i			0.852859i	0.904521	+	0.426430i	\(0.140229\pi\)
							−0.904521	+	0.426430i	\(0.859771\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.10.d.a.9.10	yes	52
17.2	even	8	inner	17.10.d.a.2.10	✓	52
17.6	odd	16		289.10.a.i.1.39		52
17.11	odd	16		289.10.a.i.1.40		52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.d.a.2.10	✓	52	17.2	even	8	inner
17.10.d.a.9.10	yes	52	1.1	even	1	trivial
289.10.a.i.1.39		52	17.6	odd	16
289.10.a.i.1.40		52	17.11	odd	16