Embedded newform 17.10.d.a.2.3

• Label: 17.10.d.a.2.3
• 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			2.3
Character	\(\chi\)	\(=\)	17.2
Dual form			17.10.d.a.9.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-22.5808 - 22.5808i) q^{2} +(147.205 - 60.9745i) q^{3} +507.789i q^{4} +(534.784 + 1291.08i) q^{5} +(-4700.88 - 1947.17i) q^{6} +(-3817.95 + 9217.34i) q^{7} +(-95.0916 + 95.0916i) q^{8} +(4033.58 - 4033.58i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 4 q^{2} - 4 q^{3} - 1140 q^{5} + 12308 q^{6} - 4 q^{7} + 2044 q^{8} - 83224 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−22.5808	−	22.5808i	−0.997942	−	0.997942i	0.00205624	−	0.999998i	\(-0.499345\pi\)
							−0.999998	+	0.00205624i	\(0.999345\pi\)
\(3\)	147.205	−	60.9745i	1.04925	−	0.434613i	0.209625	−	0.977782i	\(-0.432776\pi\)
							0.839623	+	0.543169i	\(0.182776\pi\)
\(5\)	534.784	+	1291.08i	0.382660	+	0.923823i	0.991449	+	0.130491i	\(0.0416554\pi\)
							−0.608790	+	0.793332i	\(0.708345\pi\)
\(7\)	−3817.95	+	9217.34i	−0.601020	+	1.45099i	0.271513	+	0.962435i	\(0.412476\pi\)
							−0.872533	+	0.488556i	\(0.837524\pi\)
\(11\)	46761.8	+	19369.4i	0.962994	+	0.398885i	0.808100	−	0.589046i	\(-0.200496\pi\)
							0.154895	+	0.987931i	\(0.450496\pi\)
\(13\)			76005.9i			0.738078i	0.929414	+	0.369039i	\(0.120313\pi\)
							−0.929414	+	0.369039i	\(0.879687\pi\)
\(17\)	−324117.	−	116345.i	−0.941199	−	0.337852i
							\(19\)	−460062.	−	460062.i	−0.809888	−	0.809888i	0.174728	−	0.984617i	\(-0.444095\pi\)
−0.984617	+	0.174728i	\(0.944095\pi\)
\(23\)	547050.	+	226596.i	0.407617	+	0.168840i	0.577064	−	0.816699i	\(-0.304198\pi\)
							−0.169447	+	0.985539i	\(0.554198\pi\)
\(29\)	1.35186e6	+	3.26369e6i	0.354929	+	0.856875i	0.995997	+	0.0893904i	\(0.0284919\pi\)
							−0.641067	+	0.767485i	\(0.721508\pi\)
\(31\)	3.15043e6	−	1.30495e6i	0.612692	−	0.253786i	−0.0546868	−	0.998504i	\(-0.517416\pi\)
							0.667379	+	0.744718i	\(0.267416\pi\)
\(37\)	−5.75330e6	+	2.38309e6i	−0.504672	+	0.209042i	−0.620469	−	0.784231i	\(-0.713058\pi\)
							0.115797	+	0.993273i	\(0.463058\pi\)
\(41\)	−6432.62	+	15529.7i	−0.000355517	+	0.000858294i	−0.924057	−	0.382254i	\(-0.875148\pi\)
							0.923702	+	0.383113i	\(0.125148\pi\)
\(43\)	2.30025e7	−	2.30025e7i	1.02605	−	1.02605i	0.0263957	−	0.999652i	\(-0.491597\pi\)
							0.999652	−	0.0263957i	\(-0.00840299\pi\)
\(47\)			5.02215e7i			1.50124i	0.660736	+	0.750619i	\(0.270244\pi\)
							−0.660736	+	0.750619i	\(0.729756\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.10.d.a.2.3	✓	52
17.3	odd	16		289.10.a.i.1.10		52
17.9	even	8	inner	17.10.d.a.9.3	yes	52
17.14	odd	16		289.10.a.i.1.9		52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.d.a.2.3	✓	52	1.1	even	1	trivial
17.10.d.a.9.3	yes	52	17.9	even	8	inner
289.10.a.i.1.9		52	17.14	odd	16
289.10.a.i.1.10		52	17.3	odd	16