Embedded newform 17.10.d.a.2.9

• Label: 17.10.d.a.2.9
• 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			2.9
Character	\(\chi\)	\(=\)	17.2
Dual form			17.10.d.a.9.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(13.0593 + 13.0593i) q^{2} +(116.759 - 48.3631i) q^{3} -170.907i q^{4} +(663.047 + 1600.74i) q^{5} +(2156.38 + 893.204i) q^{6} +(2945.54 - 7111.17i) q^{7} +(8918.32 - 8918.32i) q^{8} +(-2624.32 + 2624.32i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 4 q^{2} - 4 q^{3} - 1140 q^{5} + 12308 q^{6} - 4 q^{7} + 2044 q^{8} - 83224 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	13.0593	+	13.0593i	0.577147	+	0.577147i	0.934116	−	0.356969i	\(-0.116190\pi\)
							−0.356969	+	0.934116i	\(0.616190\pi\)
\(3\)	116.759	−	48.3631i	0.832232	−	0.344722i	0.0744462	−	0.997225i	\(-0.476281\pi\)
							0.757786	+	0.652503i	\(0.226281\pi\)
\(5\)	663.047	+	1600.74i	0.474438	+	1.14539i	0.962182	+	0.272408i	\(0.0878200\pi\)
							−0.487744	+	0.872987i	\(0.662180\pi\)
\(7\)	2945.54	−	7111.17i	0.463686	−	1.11944i	−0.503186	−	0.864178i	\(-0.667839\pi\)
							0.966872	−	0.255260i	\(-0.0821611\pi\)
\(11\)	57318.2	+	23742.0i	1.18039	+	0.488934i	0.884614	−	0.466323i	\(-0.154422\pi\)
							0.295776	+	0.955257i	\(0.404422\pi\)
\(13\)			70477.0i			0.684388i	0.939629	+	0.342194i	\(0.111170\pi\)
							−0.939629	+	0.342194i	\(0.888830\pi\)
\(17\)	−312389.	−	144917.i	−0.907143	−	0.420822i
							\(19\)	216406.	+	216406.i	0.380959	+	0.380959i	0.871448	−	0.490488i	\(-0.163182\pi\)
−0.490488	+	0.871448i	\(0.663182\pi\)
\(23\)	−1.51666e6	−	628220.i	−1.13009	−	0.468098i	−0.262277	−	0.964993i	\(-0.584473\pi\)
							−0.867811	+	0.496895i	\(0.834473\pi\)
\(29\)	−2.59546e6	−	6.26599e6i	−0.681433	−	1.64512i	−0.761364	−	0.648324i	\(-0.775470\pi\)
							0.0799315	−	0.996800i	\(-0.474530\pi\)
\(31\)	−5.82656e6	+	2.41344e6i	−1.13314	+	0.469363i	−0.868848	−	0.495079i	\(-0.835139\pi\)
							−0.264295	+	0.964442i	\(0.585139\pi\)
\(37\)	6.43151e6	−	2.66402e6i	0.564164	−	0.233684i	−0.0823279	−	0.996605i	\(-0.526235\pi\)
							0.646492	+	0.762921i	\(0.276235\pi\)
\(41\)	−2.52267e6	+	6.09025e6i	−0.139422	+	0.336595i	−0.978132	−	0.207983i	\(-0.933310\pi\)
							0.838710	+	0.544578i	\(0.183310\pi\)
\(43\)	−1.82896e7	+	1.82896e7i	−0.815821	+	0.815821i	−0.985500	−	0.169678i	\(-0.945727\pi\)
							0.169678	+	0.985500i	\(0.445727\pi\)
\(47\)			2.56808e7i			0.767658i	0.923404	+	0.383829i	\(0.125395\pi\)
							−0.923404	+	0.383829i	\(0.874605\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.10.d.a.2.9	✓	52
17.3	odd	16		289.10.a.i.1.36		52
17.9	even	8	inner	17.10.d.a.9.9	yes	52
17.14	odd	16		289.10.a.i.1.35		52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.d.a.2.9	✓	52	1.1	even	1	trivial
17.10.d.a.9.9	yes	52	17.9	even	8	inner
289.10.a.i.1.35		52	17.14	odd	16
289.10.a.i.1.36		52	17.3	odd	16