Embedded newform 17.10.d.a.2.1

• Label: 17.10.d.a.2.1
• 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			2.1
Character	\(\chi\)	\(=\)	17.2
Dual form			17.10.d.a.9.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-29.1798 - 29.1798i) q^{2} +(-162.034 + 67.1166i) q^{3} +1190.92i q^{4} +(-267.860 - 646.671i) q^{5} +(6686.56 + 2769.66i) q^{6} +(-3386.34 + 8175.34i) q^{7} +(19810.7 - 19810.7i) q^{8} +(7832.33 - 7832.33i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 4 q^{2} - 4 q^{3} - 1140 q^{5} + 12308 q^{6} - 4 q^{7} + 2044 q^{8} - 83224 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−29.1798	−	29.1798i	−1.28958	−	1.28958i	−0.935044	−	0.354532i	\(-0.884640\pi\)
							−0.354532	−	0.935044i	\(-0.615360\pi\)
\(3\)	−162.034	+	67.1166i	−1.15494	+	0.478392i	−0.876188	−	0.481970i	\(-0.839921\pi\)
							−0.278754	+	0.960363i	\(0.589921\pi\)
\(5\)	−267.860	−	646.671i	−0.191665	−	0.462720i	0.798609	−	0.601850i	\(-0.205569\pi\)
							−0.990274	+	0.139130i	\(0.955569\pi\)
\(7\)	−3386.34	+	8175.34i	−0.533076	+	1.28696i	0.396401	+	0.918077i	\(0.370259\pi\)
							−0.929477	+	0.368881i	\(0.879741\pi\)
\(11\)	−83690.3	−	34665.6i	−1.72349	−	0.713892i	−0.999716	−	0.0238515i	\(-0.992407\pi\)
							−0.723771	−	0.690040i	\(-0.757593\pi\)
\(13\)			26249.8i			0.254907i	0.991845	+	0.127453i	\(0.0406803\pi\)
							−0.991845	+	0.127453i	\(0.959320\pi\)
\(17\)	209492.	−	273315.i	0.608342	−	0.793675i
							\(19\)	268213.	+	268213.i	0.472160	+	0.472160i	0.902613	−	0.430453i	\(-0.141646\pi\)
−0.430453	+	0.902613i	\(0.641646\pi\)
\(23\)	−184114.	−	76262.4i	−0.137186	−	0.0568244i	0.313034	−	0.949742i	\(-0.398655\pi\)
							−0.450220	+	0.892917i	\(0.648655\pi\)
\(29\)	630380.	+	1.52187e6i	0.165505	+	0.399565i	0.984773	−	0.173847i	\(-0.0556197\pi\)
							−0.819268	+	0.573411i	\(0.805620\pi\)
\(31\)	−5.69663e6	+	2.35962e6i	−1.10787	+	0.458896i	−0.860205	−	0.509949i	\(-0.829664\pi\)
							−0.247669	+	0.968845i	\(0.579664\pi\)
\(37\)	1.85236e7	−	7.67272e6i	1.62486	−	0.673041i	0.630222	−	0.776415i	\(-0.282964\pi\)
							0.994643	+	0.103374i	\(0.0329637\pi\)
\(41\)	−9.72240e6	+	2.34720e7i	−0.537336	+	1.29724i	0.389240	+	0.921137i	\(0.372738\pi\)
							−0.926576	+	0.376108i	\(0.877262\pi\)
\(43\)	−315278.	+	315278.i	−0.0140632	+	0.0140632i	−0.714103	−	0.700040i	\(-0.753165\pi\)
							0.700040	+	0.714103i	\(0.253165\pi\)
\(47\)		−	1.95869e7i		−	0.585498i	−0.956189	−	0.292749i	\(-0.905430\pi\)
							0.956189	−	0.292749i	\(-0.0945701\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.10.d.a.2.1	✓	52
17.3	odd	16		289.10.a.i.1.1		52
17.9	even	8	inner	17.10.d.a.9.1	yes	52
17.14	odd	16		289.10.a.i.1.2		52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.d.a.2.1	✓	52	1.1	even	1	trivial
17.10.d.a.9.1	yes	52	17.9	even	8	inner
289.10.a.i.1.1		52	17.3	odd	16
289.10.a.i.1.2		52	17.14	odd	16