Embedded newform 17.10.d.a.8.1

• Label: 17.10.d.a.8.1
• 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			8.1
Character	\(\chi\)	\(=\)	17.8
Dual form			17.10.d.a.15.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-29.9491 + 29.9491i) q^{2} +(97.1714 - 234.592i) q^{3} -1281.90i q^{4} +(-584.865 - 242.259i) q^{5} +(4115.64 + 9936.04i) q^{6} +(-4412.09 + 1827.55i) q^{7} +(23057.9 + 23057.9i) q^{8} +(-31673.4 - 31673.4i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 4 q^{2} - 4 q^{3} - 1140 q^{5} + 12308 q^{6} - 4 q^{7} + 2044 q^{8} - 83224 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−29.9491	+	29.9491i	−1.32358	+	1.32358i	−0.412720	+	0.910858i	\(0.635421\pi\)
							−0.910858	+	0.412720i	\(0.864579\pi\)
\(3\)	97.1714	−	234.592i	0.692616	−	1.67212i	−0.0468251	−	0.998903i	\(-0.514910\pi\)
							0.739442	−	0.673221i	\(-0.235090\pi\)
\(5\)	−584.865	−	242.259i	−0.418495	−	0.173346i	0.163491	−	0.986545i	\(-0.447724\pi\)
							−0.581987	+	0.813198i	\(0.697724\pi\)
\(7\)	−4412.09	+	1827.55i	−0.694549	+	0.287692i	−0.701894	−	0.712281i	\(-0.747662\pi\)
							0.00734487	+	0.999973i	\(0.497662\pi\)
\(11\)	9130.06	+	22041.9i	0.188021	+	0.453923i	0.989578	−	0.143995i	\(-0.0459949\pi\)
							−0.801557	+	0.597918i	\(0.795995\pi\)
\(13\)			126359.i			1.22705i	0.789675	+	0.613525i	\(0.210249\pi\)
							−0.789675	+	0.613525i	\(0.789751\pi\)
\(17\)	41544.2	−	341851.i	0.120640	−	0.992696i
							\(19\)	−600409.	+	600409.i	−1.05695	+	1.05695i	−0.0586762	+	0.998277i	\(0.518688\pi\)
−0.998277	+	0.0586762i	\(0.981312\pi\)
\(23\)	304305.	+	734657.i	0.226743	+	0.547405i	0.995777	−	0.0918018i	\(-0.0292626\pi\)
							−0.769035	+	0.639207i	\(0.779263\pi\)
\(29\)	−798702.	−	330833.i	−0.209698	−	0.0868597i	0.275362	−	0.961341i	\(-0.411202\pi\)
							−0.485060	+	0.874481i	\(0.661202\pi\)
\(31\)	−194889.	+	470504.i	−0.0379018	+	0.0915031i	−0.941697	−	0.336462i	\(-0.890770\pi\)
							0.903795	+	0.427965i	\(0.140770\pi\)
\(37\)	−3.60357e6	+	8.69979e6i	−0.316100	+	0.763134i	0.683353	+	0.730088i	\(0.260521\pi\)
							−0.999454	+	0.0330462i	\(0.989479\pi\)
\(41\)	1.99946e6	−	828205.i	0.110506	−	0.0457731i	−0.326745	−	0.945112i	\(-0.605952\pi\)
							0.437252	+	0.899339i	\(0.355952\pi\)
\(43\)	−9.48745e6	−	9.48745e6i	−0.423196	−	0.423196i	0.463107	−	0.886303i	\(-0.346735\pi\)
							−0.886303	+	0.463107i	\(0.846735\pi\)
\(47\)		−	3.55358e7i		−	1.06225i	−0.847294	−	0.531124i	\(-0.821770\pi\)
							0.847294	−	0.531124i	\(-0.178230\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.10.d.a.8.1	✓	52
17.7	odd	16		289.10.a.i.1.49		52
17.10	odd	16		289.10.a.i.1.50		52
17.15	even	8	inner	17.10.d.a.15.1	yes	52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.d.a.8.1	✓	52	1.1	even	1	trivial
17.10.d.a.15.1	yes	52	17.15	even	8	inner
289.10.a.i.1.49		52	17.7	odd	16
289.10.a.i.1.50		52	17.10	odd	16