Embedded newform 17.10.d.a.2.4

• Label: 17.10.d.a.2.4
• 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			2.4
Character	\(\chi\)	\(=\)	17.2
Dual form			17.10.d.a.9.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-19.0841 - 19.0841i) q^{2} +(-103.159 + 42.7299i) q^{3} +216.409i q^{4} +(813.738 + 1964.54i) q^{5} +(2784.17 + 1153.24i) q^{6} +(3243.32 - 7830.08i) q^{7} +(-5641.10 + 5641.10i) q^{8} +(-5102.02 + 5102.02i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 4 q^{2} - 4 q^{3} - 1140 q^{5} + 12308 q^{6} - 4 q^{7} + 2044 q^{8} - 83224 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−19.0841	−	19.0841i	−0.843408	−	0.843408i	0.145893	−	0.989300i	\(-0.453395\pi\)
							−0.989300	+	0.145893i	\(0.953395\pi\)
\(3\)	−103.159	+	42.7299i	−0.735296	+	0.304570i	−0.718726	−	0.695293i	\(-0.755275\pi\)
							−0.0165696	+	0.999863i	\(0.505275\pi\)
\(5\)	813.738	+	1964.54i	0.582264	+	1.40571i	0.890756	+	0.454482i	\(0.150176\pi\)
							−0.308492	+	0.951227i	\(0.599824\pi\)
\(7\)	3243.32	−	7830.08i	0.510563	−	1.23261i	−0.432994	−	0.901397i	\(-0.642543\pi\)
							0.943557	−	0.331211i	\(-0.107457\pi\)
\(11\)	7177.56	+	2973.04i	0.147812	+	0.0612257i	0.455363	−	0.890306i	\(-0.349509\pi\)
							−0.307551	+	0.951532i	\(0.599509\pi\)
\(13\)		−	188698.i		−	1.83241i	−0.400709	−	0.916205i	\(-0.631236\pi\)
							0.400709	−	0.916205i	\(-0.368764\pi\)
\(17\)	215575.	−	268543.i	0.626005	−	0.779819i
							\(19\)	83394.6	+	83394.6i	0.146807	+	0.146807i	0.776690	−	0.629883i	\(-0.216897\pi\)
−0.629883	+	0.776690i	\(0.716897\pi\)
\(23\)	1.70146e6	+	704768.i	1.26779	+	0.525135i	0.912291	−	0.409542i	\(-0.134311\pi\)
							0.355497	+	0.934677i	\(0.384311\pi\)
\(29\)	−1.98378e6	−	4.78927e6i	−0.520838	−	1.25741i	−0.937384	−	0.348299i	\(-0.886760\pi\)
							0.416546	−	0.909115i	\(-0.363240\pi\)
\(31\)	4.34437e6	−	1.79950e6i	0.844889	−	0.349964i	0.0821094	−	0.996623i	\(-0.473834\pi\)
							0.762779	+	0.646659i	\(0.223834\pi\)
\(37\)	311893.	−	129190.i	0.0273588	−	0.0113324i	−0.368962	−	0.929444i	\(-0.620287\pi\)
							0.396321	+	0.918112i	\(0.370287\pi\)
\(41\)	6.43397e6	−	1.55330e7i	0.355592	−	0.858474i	−0.640317	−	0.768110i	\(-0.721197\pi\)
							0.995909	−	0.0903634i	\(-0.0288028\pi\)
\(43\)	−7.51288e6	+	7.51288e6i	−0.335119	+	0.335119i	−0.854526	−	0.519408i	\(-0.826153\pi\)
							0.519408	+	0.854526i	\(0.326153\pi\)
\(47\)		−	9.90984e6i		−	0.296228i	−0.988970	−	0.148114i	\(-0.952680\pi\)
							0.988970	−	0.148114i	\(-0.0473203\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.10.d.a.2.4	✓	52
17.3	odd	16		289.10.a.i.1.11		52
17.9	even	8	inner	17.10.d.a.9.4	yes	52
17.14	odd	16		289.10.a.i.1.12		52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.d.a.2.4	✓	52	1.1	even	1	trivial
17.10.d.a.9.4	yes	52	17.9	even	8	inner
289.10.a.i.1.11		52	17.3	odd	16
289.10.a.i.1.12		52	17.14	odd	16