Embedded newform 17.10.d.a.15.10

• Label: 17.10.d.a.15.10
• Level: $17$
• Weight: $10$
• Character: 17.15
• 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			15.10
Character	\(\chi\)	\(=\)	17.15
Dual form			17.10.d.a.8.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(16.9753 + 16.9753i) q^{2} +(74.3499 + 179.496i) q^{3} +64.3195i q^{4} +(-2350.25 + 973.503i) q^{5} +(-1784.89 + 4309.11i) q^{6} +(1813.43 + 751.147i) q^{7} +(7599.50 - 7599.50i) q^{8} +(-12773.1 + 12773.1i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 4 q^{2} - 4 q^{3} - 1140 q^{5} + 12308 q^{6} - 4 q^{7} + 2044 q^{8} - 83224 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	16.9753	+	16.9753i	0.750208	+	0.750208i	0.974518	−	0.224310i	\(-0.0720128\pi\)
							−0.224310	+	0.974518i	\(0.572013\pi\)
\(3\)	74.3499	+	179.496i	0.529950	+	1.27941i	0.931556	+	0.363599i	\(0.118452\pi\)
							−0.401606	+	0.915813i	\(0.631548\pi\)
\(5\)	−2350.25	+	973.503i	−1.68170	+	0.696582i	−0.999405	−	0.0344957i	\(-0.989018\pi\)
							−0.682294	+	0.731078i	\(0.739018\pi\)
\(7\)	1813.43	+	751.147i	0.285469	+	0.118245i	0.520823	−	0.853664i	\(-0.325625\pi\)
							−0.235354	+	0.971910i	\(0.575625\pi\)
\(11\)	6975.01	−	16839.2i	0.143641	−	0.346780i	−0.835643	−	0.549273i	\(-0.814905\pi\)
							0.979284	+	0.202494i	\(0.0649046\pi\)
\(13\)			165432.i			1.60648i	0.595656	+	0.803240i	\(0.296892\pi\)
							−0.595656	+	0.803240i	\(0.703108\pi\)
\(17\)	−65408.8	+	338097.i	−0.189940	+	0.981796i
							\(19\)	−97283.3	−	97283.3i	−0.171256	−	0.171256i	0.616275	−	0.787531i	\(-0.288641\pi\)
−0.787531	+	0.616275i	\(0.788641\pi\)
\(23\)	−266171.	+	642594.i	−0.198329	+	0.478808i	−0.991487	−	0.130208i	\(-0.958435\pi\)
							0.793158	+	0.609016i	\(0.208435\pi\)
\(29\)	3.90815e6	−	1.61881e6i	1.02608	−	0.425015i	0.194782	−	0.980847i	\(-0.437600\pi\)
							0.831295	+	0.555832i	\(0.187600\pi\)
\(31\)	1.44976e6	+	3.50003e6i	0.281947	+	0.680681i	0.999881	−	0.0154305i	\(-0.00491187\pi\)
							−0.717934	+	0.696112i	\(0.754912\pi\)
\(37\)	4.54160e6	+	1.09644e7i	0.398383	+	0.961782i	0.988050	+	0.154135i	\(0.0492592\pi\)
							−0.589666	+	0.807647i	\(0.700741\pi\)
\(41\)	−9.08663e6	−	3.76380e6i	−0.502198	−	0.208017i	0.117179	−	0.993111i	\(-0.462615\pi\)
							−0.619377	+	0.785093i	\(0.712615\pi\)
\(43\)	2.50560e7	−	2.50560e7i	1.11764	−	1.11764i	0.125557	−	0.992086i	\(-0.459928\pi\)
							0.992086	−	0.125557i	\(-0.0400717\pi\)
\(47\)		−	5.08554e7i		−	1.52018i	−0.649815	−	0.760092i	\(-0.725154\pi\)
							0.649815	−	0.760092i	\(-0.274846\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.10.d.a.15.10	yes	52
17.5	odd	16		289.10.a.i.1.15		52
17.8	even	8	inner	17.10.d.a.8.10	✓	52
17.12	odd	16		289.10.a.i.1.16		52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.d.a.8.10	✓	52	17.8	even	8	inner
17.10.d.a.15.10	yes	52	1.1	even	1	trivial
289.10.a.i.1.15		52	17.5	odd	16
289.10.a.i.1.16		52	17.12	odd	16