Embedded newform 17.10.d.a.9.13

• Label: 17.10.d.a.9.13
• 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			9.13
Character	\(\chi\)	\(=\)	17.9
Dual form			17.10.d.a.2.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(31.2113 - 31.2113i) q^{2} +(-114.097 - 47.2607i) q^{3} -1436.29i q^{4} +(205.870 - 497.013i) q^{5} +(-5036.20 + 2086.06i) q^{6} +(3921.77 + 9467.99i) q^{7} +(-28848.3 - 28848.3i) q^{8} +(-3133.33 - 3133.33i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 4 q^{2} - 4 q^{3} - 1140 q^{5} + 12308 q^{6} - 4 q^{7} + 2044 q^{8} - 83224 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	31.2113	−	31.2113i	1.37936	−	1.37936i	0.533657	−	0.845701i	\(-0.320817\pi\)
							0.845701	−	0.533657i	\(-0.179183\pi\)
\(3\)	−114.097	−	47.2607i	−0.813261	−	0.336864i	−0.0630070	−	0.998013i	\(-0.520069\pi\)
							−0.750254	+	0.661149i	\(0.770069\pi\)
\(5\)	205.870	−	497.013i	0.147308	−	0.355634i	−0.832952	−	0.553345i	\(-0.813351\pi\)
							0.980260	+	0.197712i	\(0.0633510\pi\)
\(7\)	3921.77	+	9467.99i	0.617363	+	1.49045i	0.854754	+	0.519033i	\(0.173708\pi\)
							−0.237391	+	0.971414i	\(0.576292\pi\)
\(11\)	5430.59	−	2249.42i	0.111836	−	0.0463238i	−0.326064	−	0.945348i	\(-0.605723\pi\)
							0.437900	+	0.899024i	\(0.355723\pi\)
\(13\)		−	73109.4i		−	0.709951i	−0.934876	−	0.354975i	\(-0.884489\pi\)
							0.934876	−	0.354975i	\(-0.115511\pi\)
\(17\)	252672.	−	233976.i	0.733730	−	0.679441i
							\(19\)	421726.	−	421726.i	0.742402	−	0.742402i	−0.230638	−	0.973040i	\(-0.574081\pi\)
0.973040	+	0.230638i	\(0.0740813\pi\)
\(23\)	−449080.	+	186015.i	−0.334617	+	0.138603i	−0.543665	−	0.839302i	\(-0.682964\pi\)
							0.209047	+	0.977906i	\(0.432964\pi\)
\(29\)	1.80019e6	−	4.34603e6i	0.472636	−	1.14104i	−0.490358	−	0.871521i	\(-0.663134\pi\)
							0.962994	−	0.269523i	\(-0.0868659\pi\)
\(31\)	−2.69479e6	−	1.11622e6i	−0.524080	−	0.217081i	0.104928	−	0.994480i	\(-0.466539\pi\)
							−0.629008	+	0.777399i	\(0.716539\pi\)
\(37\)	3.68495e6	+	1.52636e6i	0.323239	+	0.133890i	0.538402	−	0.842688i	\(-0.319028\pi\)
							−0.215163	+	0.976578i	\(0.569028\pi\)
\(41\)	−1.25691e6	−	3.03445e6i	−0.0694669	−	0.167708i	0.885333	−	0.464958i	\(-0.153931\pi\)
							−0.954800	+	0.297250i	\(0.903931\pi\)
\(43\)	2.61942e7	+	2.61942e7i	1.16842	+	1.16842i	0.982581	+	0.185835i	\(0.0594989\pi\)
							0.185835	+	0.982581i	\(0.440501\pi\)
\(47\)		−	873124.i		−	0.0260997i	−0.999915	−	0.0130499i	\(-0.995846\pi\)
							0.999915	−	0.0130499i	\(-0.00415402\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.10.d.a.9.13	yes	52
17.2	even	8	inner	17.10.d.a.2.13	✓	52
17.6	odd	16		289.10.a.i.1.51		52
17.11	odd	16		289.10.a.i.1.52		52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.d.a.2.13	✓	52	17.2	even	8	inner
17.10.d.a.9.13	yes	52	1.1	even	1	trivial
289.10.a.i.1.51		52	17.6	odd	16
289.10.a.i.1.52		52	17.11	odd	16