Embedded newform 17.10.d.a.9.8

• Label: 17.10.d.a.9.8
• Level: $17$
• Weight: $10$
• Character: 17.9
• 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			9.8
Character	\(\chi\)	\(=\)	17.9
Dual form			17.10.d.a.2.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(11.3845 - 11.3845i) q^{2} +(-244.376 - 101.224i) q^{3} +252.785i q^{4} +(230.356 - 556.130i) q^{5} +(-3934.48 + 1629.72i) q^{6} +(406.478 + 981.326i) q^{7} +(8706.72 + 8706.72i) q^{8} +(35555.2 + 35555.2i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 4 q^{2} - 4 q^{3} - 1140 q^{5} + 12308 q^{6} - 4 q^{7} + 2044 q^{8} - 83224 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	11.3845	−	11.3845i	0.503130	−	0.503130i	−0.409279	−	0.912409i	\(-0.634220\pi\)
							0.912409	+	0.409279i	\(0.134220\pi\)
\(3\)	−244.376	−	101.224i	−1.74186	−	0.721501i	−0.998623	−	0.0524686i	\(-0.983291\pi\)
							−0.743234	−	0.669032i	\(-0.766709\pi\)
\(5\)	230.356	−	556.130i	0.164830	−	0.397934i	−0.819786	−	0.572671i	\(-0.805907\pi\)
							0.984615	+	0.174737i	\(0.0559074\pi\)
\(7\)	406.478	+	981.326i	0.0639877	+	0.154480i	0.952639	−	0.304104i	\(-0.0983569\pi\)
							−0.888651	+	0.458584i	\(0.848357\pi\)
\(11\)	−16492.7	+	6831.49i	−0.339644	+	0.140685i	−0.545985	−	0.837795i	\(-0.683844\pi\)
							0.206341	+	0.978480i	\(0.433844\pi\)
\(13\)			37747.5i			0.366558i	0.983061	+	0.183279i	\(0.0586712\pi\)
							−0.983061	+	0.183279i	\(0.941329\pi\)
\(17\)	38251.7	+	342235.i	0.111079	+	0.993812i
							\(19\)	−121753.	+	121753.i	−0.214333	+	0.214333i	−0.806105	−	0.591772i	\(-0.798429\pi\)
0.591772	+	0.806105i	\(0.298429\pi\)
\(23\)	−2.32978e6	+	965028.i	−1.73596	+	0.719059i	−0.736889	+	0.676014i	\(0.763706\pi\)
							−0.999073	+	0.0430449i	\(0.986294\pi\)
\(29\)	1.21523e6	−	2.93381e6i	0.319055	−	0.770267i	−0.680249	−	0.732981i	\(-0.738129\pi\)
							0.999305	−	0.0372867i	\(-0.0118715\pi\)
\(31\)	6.18488e6	+	2.56186e6i	1.20283	+	0.498227i	0.891911	−	0.452210i	\(-0.149364\pi\)
							0.310916	+	0.950437i	\(0.399364\pi\)
\(37\)	−1.25599e7	−	5.20248e6i	−1.10174	−	0.456355i	−0.243653	−	0.969862i	\(-0.578346\pi\)
							−0.858085	+	0.513507i	\(0.828346\pi\)
\(41\)	9.84536e6	+	2.37688e7i	0.544132	+	1.31365i	0.921784	+	0.387703i	\(0.126731\pi\)
							−0.377652	+	0.925947i	\(0.623269\pi\)
\(43\)	1.15353e7	+	1.15353e7i	0.514541	+	0.514541i	0.915915	−	0.401373i	\(-0.131467\pi\)
							−0.401373	+	0.915915i	\(0.631467\pi\)
\(47\)			4.00867e6i			0.119828i	0.998204	+	0.0599142i	\(0.0190827\pi\)
							−0.998204	+	0.0599142i	\(0.980917\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.10.d.a.9.8	yes	52
17.2	even	8	inner	17.10.d.a.2.8	✓	52
17.6	odd	16		289.10.a.i.1.31		52
17.11	odd	16		289.10.a.i.1.32		52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.d.a.2.8	✓	52	17.2	even	8	inner
17.10.d.a.9.8	yes	52	1.1	even	1	trivial
289.10.a.i.1.31		52	17.6	odd	16
289.10.a.i.1.32		52	17.11	odd	16