Embedded newform 17.10.d.a.8.7

• Label: 17.10.d.a.8.7
• 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			8.7
Character	\(\chi\)	\(=\)	17.8
Dual form			17.10.d.a.15.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.763172 + 0.763172i) q^{2} +(-66.0099 + 159.362i) q^{3} +510.835i q^{4} +(-1926.04 - 797.791i) q^{5} +(-71.2437 - 171.997i) q^{6} +(7148.81 - 2961.13i) q^{7} +(-780.599 - 780.599i) q^{8} +(-7120.96 - 7120.96i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 4 q^{2} - 4 q^{3} - 1140 q^{5} + 12308 q^{6} - 4 q^{7} + 2044 q^{8} - 83224 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.763172	+	0.763172i	−0.0337277	+	0.0337277i	−0.723770	−	0.690042i	\(-0.757592\pi\)
							0.690042	+	0.723770i	\(0.257592\pi\)
\(3\)	−66.0099	+	159.362i	−0.470504	+	1.13590i	0.493437	+	0.869782i	\(0.335740\pi\)
							−0.963941	+	0.266116i	\(0.914260\pi\)
\(5\)	−1926.04	−	797.791i	−1.37816	−	0.570853i	−0.434172	−	0.900830i	\(-0.642959\pi\)
							−0.943989	+	0.329977i	\(0.892959\pi\)
\(7\)	7148.81	−	2961.13i	1.12536	−	0.466141i	0.259161	−	0.965834i	\(-0.416554\pi\)
							0.866202	+	0.499694i	\(0.166554\pi\)
\(11\)	−33234.4	−	80234.8i	−0.684416	−	1.65233i	−0.755739	−	0.654873i	\(-0.772722\pi\)
							0.0713230	−	0.997453i	\(-0.477278\pi\)
\(13\)			121107.i			1.17605i	0.808844	+	0.588023i	\(0.200094\pi\)
							−0.808844	+	0.588023i	\(0.799906\pi\)
\(17\)	−277938.	−	203318.i	−0.807102	−	0.590412i
							\(19\)	−66596.2	+	66596.2i	−0.117235	+	0.117235i	−0.763291	−	0.646055i	\(-0.776418\pi\)
0.646055	+	0.763291i	\(0.276418\pi\)
\(23\)	283246.	+	683817.i	0.211052	+	0.509524i	0.993585	−	0.113084i	\(-0.0360730\pi\)
							−0.782534	+	0.622608i	\(0.786073\pi\)
\(29\)	−3.63139e6	−	1.50417e6i	−0.953415	−	0.394917i	−0.148901	−	0.988852i	\(-0.547574\pi\)
							−0.804513	+	0.593935i	\(0.797574\pi\)
\(31\)	1.54840e6	−	3.73817e6i	0.301131	−	0.726995i	−0.698801	−	0.715316i	\(-0.746283\pi\)
							0.999932	−	0.0116785i	\(-0.00371746\pi\)
\(37\)	−805034.	+	1.94352e6i	−0.0706166	+	0.170483i	−0.955247	−	0.295809i	\(-0.904411\pi\)
							0.884630	+	0.466293i	\(0.154411\pi\)
\(41\)	−1.08648e7	+	4.50036e6i	−0.600476	+	0.248725i	−0.662150	−	0.749371i	\(-0.730356\pi\)
							0.0616744	+	0.998096i	\(0.480356\pi\)
\(43\)	−2.84382e7	−	2.84382e7i	−1.26851	−	1.26851i	−0.946857	−	0.321655i	\(-0.895761\pi\)
							−0.321655	−	0.946857i	\(-0.604239\pi\)
\(47\)			4.45793e7i			1.33258i	0.745694	+	0.666289i	\(0.232118\pi\)
							−0.745694	+	0.666289i	\(0.767882\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.10.d.a.8.7	✓	52
17.7	odd	16		289.10.a.i.1.28		52
17.10	odd	16		289.10.a.i.1.27		52
17.15	even	8	inner	17.10.d.a.15.7	yes	52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.d.a.8.7	✓	52	1.1	even	1	trivial
17.10.d.a.15.7	yes	52	17.15	even	8	inner
289.10.a.i.1.27		52	17.10	odd	16
289.10.a.i.1.28		52	17.7	odd	16