Embedded newform 17.10.d.a.9.2

• Label: 17.10.d.a.9.2
• 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			9.2
Character	\(\chi\)	\(=\)	17.9
Dual form			17.10.d.a.2.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-24.8024 + 24.8024i) q^{2} +(109.545 + 45.3751i) q^{3} -718.321i q^{4} +(-625.855 + 1510.95i) q^{5} +(-3842.40 + 1591.57i) q^{6} +(4127.23 + 9964.00i) q^{7} +(5117.27 + 5117.27i) q^{8} +(-3976.74 - 3976.74i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 4 q^{2} - 4 q^{3} - 1140 q^{5} + 12308 q^{6} - 4 q^{7} + 2044 q^{8} - 83224 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−24.8024	+	24.8024i	−1.09612	+	1.09612i	−0.101263	+	0.994860i	\(0.532288\pi\)
							−0.994860	+	0.101263i	\(0.967712\pi\)
\(3\)	109.545	+	45.3751i	0.780814	+	0.323424i	0.737244	−	0.675627i	\(-0.236127\pi\)
							0.0435700	+	0.999050i	\(0.486127\pi\)
\(5\)	−625.855	+	1510.95i	−0.447825	+	1.08115i	0.525310	+	0.850911i	\(0.323949\pi\)
							−0.973135	+	0.230235i	\(0.926051\pi\)
\(7\)	4127.23	+	9964.00i	0.649706	+	1.56853i	0.813199	+	0.581985i	\(0.197724\pi\)
							−0.163493	+	0.986544i	\(0.552276\pi\)
\(11\)	−19583.8	+	8111.87i	−0.403301	+	0.167053i	−0.575107	−	0.818078i	\(-0.695040\pi\)
							0.171806	+	0.985131i	\(0.445040\pi\)
\(13\)		−	71034.1i		−	0.689798i	−0.938640	−	0.344899i	\(-0.887913\pi\)
							0.938640	−	0.344899i	\(-0.112087\pi\)
\(17\)	44144.3	−	341525.i	0.128190	−	0.991750i
							\(19\)	−169075.	+	169075.i	−0.297638	+	0.297638i	−0.840088	−	0.542450i	\(-0.817497\pi\)
0.542450	+	0.840088i	\(0.317497\pi\)
\(23\)	−2.18834e6	+	906442.i	−1.63057	+	0.675405i	−0.995296	−	0.0968792i	\(-0.969114\pi\)
							−0.635277	+	0.772285i	\(0.719114\pi\)
\(29\)	−527782.	+	1.27418e6i	−0.138568	+	0.334533i	−0.977896	−	0.209093i	\(-0.932949\pi\)
							0.839328	+	0.543626i	\(0.182949\pi\)
\(31\)	4.36823e6	+	1.80938e6i	0.849528	+	0.351886i	0.764603	−	0.644501i	\(-0.222935\pi\)
							0.0849250	+	0.996387i	\(0.472935\pi\)
\(37\)	9.68365e6	+	4.01110e6i	0.849437	+	0.351848i	0.764567	−	0.644544i	\(-0.222953\pi\)
							0.0848697	+	0.996392i	\(0.472953\pi\)
\(41\)	2.76380e6	+	6.67241e6i	0.152749	+	0.368770i	0.981668	−	0.190600i	\(-0.0610433\pi\)
							−0.828918	+	0.559370i	\(0.811043\pi\)
\(43\)	−7.19816e6	−	7.19816e6i	−0.321080	−	0.321080i	0.528101	−	0.849181i	\(-0.322904\pi\)
							−0.849181	+	0.528101i	\(0.822904\pi\)
\(47\)			4.00577e7i			1.19742i	0.800966	+	0.598709i	\(0.204320\pi\)
							−0.800966	+	0.598709i	\(0.795680\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.10.d.a.9.2	yes	52
17.2	even	8	inner	17.10.d.a.2.2	✓	52
17.6	odd	16		289.10.a.i.1.8		52
17.11	odd	16		289.10.a.i.1.7		52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.d.a.2.2	✓	52	17.2	even	8	inner
17.10.d.a.9.2	yes	52	1.1	even	1	trivial
289.10.a.i.1.7		52	17.11	odd	16
289.10.a.i.1.8		52	17.6	odd	16