Embedded newform 17.10.d.a.2.8

• Label: 17.10.d.a.2.8
• 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			2.8
Character	\(\chi\)	\(=\)	17.2
Dual form			17.10.d.a.9.8

$q$-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 \(\left(\mathbb{Z}/17\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{7}{8}\right)\)

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)

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

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.10.d.a.2.8	✓	52
17.3	odd	16		289.10.a.i.1.31		52
17.9	even	8	inner	17.10.d.a.9.8	yes	52
17.14	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	1.1	even	1	trivial
17.10.d.a.9.8	yes	52	17.9	even	8	inner
289.10.a.i.1.31		52	17.3	odd	16
289.10.a.i.1.32		52	17.14	odd	16