Embedded newform 17.10.d.a.15.4

• Label: 17.10.d.a.15.4
• Level: $17$
• Weight: $10$
• Character: 17.15
• 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			15.4
Character	\(\chi\)	\(=\)	17.15
Dual form			17.10.d.a.8.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-14.5474 - 14.5474i) q^{2} +(32.3594 + 78.1225i) q^{3} -88.7454i q^{4} +(31.3399 - 12.9814i) q^{5} +(665.735 - 1607.23i) q^{6} +(-4774.06 - 1977.48i) q^{7} +(-8739.29 + 8739.29i) q^{8} +(8861.98 - 8861.98i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 4 q^{2} - 4 q^{3} - 1140 q^{5} + 12308 q^{6} - 4 q^{7} + 2044 q^{8} - 83224 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/17\mathbb{Z}\right)^\times\).

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−14.5474	−	14.5474i	−0.642911	−	0.642911i	0.308359	−	0.951270i	\(-0.400220\pi\)
							−0.951270	+	0.308359i	\(0.900220\pi\)
\(3\)	32.3594	+	78.1225i	0.230651	+	0.556840i	0.996254	−	0.0864725i	\(-0.0275595\pi\)
							−0.765603	+	0.643313i	\(0.777559\pi\)
\(5\)	31.3399	−	12.9814i	0.0224250	−	0.00928875i	−0.371443	−	0.928456i	\(-0.621137\pi\)
							0.393868	+	0.919167i	\(0.371137\pi\)
\(7\)	−4774.06	−	1977.48i	−0.751531	−	0.311294i	−0.0261648	−	0.999658i	\(-0.508329\pi\)
							−0.725366	+	0.688363i	\(0.758329\pi\)
\(11\)	−25659.9	+	61948.5i	−0.528431	+	1.27575i	0.404120	+	0.914706i	\(0.367578\pi\)
							−0.932551	+	0.361039i	\(0.882422\pi\)
\(13\)			194911.i			1.89274i	0.323081	+	0.946371i	\(0.395281\pi\)
							−0.323081	+	0.946371i	\(0.604719\pi\)
\(17\)	−320627.	+	125643.i	−0.931065	+	0.364854i
							\(19\)	−328794.	−	328794.i	−0.578806	−	0.578806i	0.355769	−	0.934574i	\(-0.384219\pi\)
−0.934574	+	0.355769i	\(0.884219\pi\)
\(23\)	144382.	−	348570.i	0.107582	−	0.259725i	−0.860916	−	0.508747i	\(-0.830109\pi\)
							0.968498	+	0.249021i	\(0.0801089\pi\)
\(29\)	−941296.	+	389898.i	−0.247136	+	0.102367i	−0.502813	−	0.864395i	\(-0.667702\pi\)
							0.255677	+	0.966762i	\(0.417702\pi\)
\(31\)	−2.32857e6	−	5.62167e6i	−0.452858	−	1.09330i	−0.971231	−	0.238141i	\(-0.923462\pi\)
							0.518373	−	0.855155i	\(-0.326538\pi\)
\(37\)	−1.67838e6	−	4.05197e6i	−0.147226	−	0.355434i	0.833013	−	0.553254i	\(-0.186614\pi\)
							−0.980238	+	0.197820i	\(0.936614\pi\)
\(41\)	1.46439e7	+	6.06570e6i	0.809336	+	0.335238i	0.748689	−	0.662921i	\(-0.230684\pi\)
							0.0606471	+	0.998159i	\(0.480684\pi\)
\(43\)	−3.45275e6	+	3.45275e6i	−0.154013	+	0.154013i	−0.779908	−	0.625895i	\(-0.784734\pi\)
							0.625895	+	0.779908i	\(0.284734\pi\)
\(47\)			1.42598e7i			0.426258i	0.977024	+	0.213129i	\(0.0683655\pi\)
							−0.977024	+	0.213129i	\(0.931634\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.10.d.a.15.4	yes	52
17.5	odd	16		289.10.a.i.1.37		52
17.8	even	8	inner	17.10.d.a.8.4	✓	52
17.12	odd	16		289.10.a.i.1.38		52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.d.a.8.4	✓	52	17.8	even	8	inner
17.10.d.a.15.4	yes	52	1.1	even	1	trivial
289.10.a.i.1.37		52	17.5	odd	16
289.10.a.i.1.38		52	17.12	odd	16