Embedded newform 17.10.d.a.2.13

• Label: 17.10.d.a.2.13
• Level: $17$
• Weight: $10$
• Character: 17.2
• 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			2.13
Character	\(\chi\)	\(=\)	17.2
Dual form			17.10.d.a.9.13

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{7}{8}\right)\)

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

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.10.d.a.2.13	✓	52
17.3	odd	16		289.10.a.i.1.51		52
17.9	even	8	inner	17.10.d.a.9.13	yes	52
17.14	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	1.1	even	1	trivial
17.10.d.a.9.13	yes	52	17.9	even	8	inner
289.10.a.i.1.51		52	17.3	odd	16
289.10.a.i.1.52		52	17.14	odd	16