Embedded newform 17.10.d.a.8.6

• Label: 17.10.d.a.8.6
• Level: $17$
• Weight: $10$
• Character: 17.8
• 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			8.6
Character	\(\chi\)	\(=\)	17.8
Dual form			17.10.d.a.15.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.85709 + 4.85709i) q^{2} +(-79.5278 + 191.997i) q^{3} +464.817i q^{4} +(1735.92 + 719.040i) q^{5} +(-546.273 - 1318.82i) q^{6} +(-3022.19 + 1251.83i) q^{7} +(-4744.49 - 4744.49i) q^{8} +(-16620.2 - 16620.2i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 4 q^{2} - 4 q^{3} - 1140 q^{5} + 12308 q^{6} - 4 q^{7} + 2044 q^{8} - 83224 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.85709	+	4.85709i	−0.214655	+	0.214655i	−0.806242	−	0.591586i	\(-0.798502\pi\)
							0.591586	+	0.806242i	\(0.298502\pi\)
\(3\)	−79.5278	+	191.997i	−0.566857	+	1.36851i	0.337334	+	0.941385i	\(0.390475\pi\)
							−0.904191	+	0.427128i	\(0.859525\pi\)
\(5\)	1735.92	+	719.040i	1.24212	+	0.514503i	0.904377	−	0.426735i	\(-0.140336\pi\)
							0.337743	+	0.941238i	\(0.390336\pi\)
\(7\)	−3022.19	+	1251.83i	−0.475753	+	0.197063i	−0.607658	−	0.794199i	\(-0.707891\pi\)
							0.131905	+	0.991262i	\(0.457891\pi\)
\(11\)	23200.0	+	56009.8i	0.477772	+	1.15344i	0.960651	+	0.277757i	\(0.0895912\pi\)
							−0.482879	+	0.875687i	\(0.660409\pi\)
\(13\)		−	74153.5i		−	0.720090i	−0.932935	−	0.360045i	\(-0.882761\pi\)
							0.932935	−	0.360045i	\(-0.117239\pi\)
\(17\)	−5030.70	−	344329.i	−0.0146086	−	0.999893i
							\(19\)	453192.	−	453192.i	0.797795	−	0.797795i	−0.184953	−	0.982747i	\(-0.559213\pi\)
0.982747	+	0.184953i	\(0.0592132\pi\)
\(23\)	608773.	+	1.46971e6i	0.453607	+	1.09510i	0.970941	+	0.239321i	\(0.0769248\pi\)
							−0.517333	+	0.855784i	\(0.673075\pi\)
\(29\)	3.66619e6	+	1.51859e6i	0.962552	+	0.398702i	0.807934	−	0.589273i	\(-0.200586\pi\)
							0.154617	+	0.987974i	\(0.450586\pi\)
\(31\)	−334693.	+	808021.i	−0.0650908	+	0.157143i	−0.953078	−	0.302726i	\(-0.902103\pi\)
							0.887987	+	0.459869i	\(0.152103\pi\)
\(37\)	−3.64596e6	+	8.80213e6i	−0.319819	+	0.772111i	0.679444	+	0.733727i	\(0.262221\pi\)
							−0.999263	+	0.0383840i	\(0.987779\pi\)
\(41\)	−3.17262e7	+	1.31414e7i	−1.75344	+	0.726298i	−0.756016	+	0.654553i	\(0.772857\pi\)
							−0.997423	+	0.0717454i	\(0.977143\pi\)
\(43\)	1.66385e7	+	1.66385e7i	0.742177	+	0.742177i	0.972997	−	0.230820i	\(-0.0741407\pi\)
							−0.230820	+	0.972997i	\(0.574141\pi\)
\(47\)		−	3.35038e7i		−	1.00151i	−0.865590	−	0.500753i	\(-0.833056\pi\)
							0.865590	−	0.500753i	\(-0.166944\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.10.d.a.8.6	✓	52
17.7	odd	16		289.10.a.i.1.30		52
17.10	odd	16		289.10.a.i.1.29		52
17.15	even	8	inner	17.10.d.a.15.6	yes	52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.d.a.8.6	✓	52	1.1	even	1	trivial
17.10.d.a.15.6	yes	52	17.15	even	8	inner
289.10.a.i.1.29		52	17.10	odd	16
289.10.a.i.1.30		52	17.7	odd	16