Embedded newform 17.6.d.a.15.1

• Label: 17.6.d.a.15.1
• Level: $17$
• Weight: $6$
• Character: 17.15
• Analytic conductor: $2.727$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 17 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	17.d (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.72652493682\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(7\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			15.1
Character	\(\chi\)	\(=\)	17.15
Dual form			17.6.d.a.8.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.43061 - 7.43061i) q^{2} +(-9.60499 - 23.1885i) q^{3} +78.4279i q^{4} +(51.4729 - 21.3208i) q^{5} +(-100.934 + 243.676i) q^{6} +(-84.7940 - 35.1228i) q^{7} +(344.988 - 344.988i) q^{8} +(-273.623 + 273.623i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 4 q^{2} - 4 q^{3} + 40 q^{5} - 364 q^{6} - 4 q^{7} + 124 q^{8} - 136 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.43061	−	7.43061i	−1.31356	−	1.31356i	−0.918773	−	0.394785i	\(-0.870819\pi\)
							−0.394785	−	0.918773i	\(-0.629181\pi\)
\(3\)	−9.60499	−	23.1885i	−0.616160	−	1.48754i	−0.856129	−	0.516762i	\(-0.827137\pi\)
							0.239969	−	0.970781i	\(-0.422863\pi\)
\(5\)	51.4729	−	21.3208i	0.920776	−	0.381398i	0.128604	−	0.991696i	\(-0.458950\pi\)
							0.792172	+	0.610298i	\(0.208950\pi\)
\(7\)	−84.7940	−	35.1228i	−0.654064	−	0.270922i	0.0308745	−	0.999523i	\(-0.490171\pi\)
							−0.684938	+	0.728601i	\(0.740171\pi\)
\(11\)	−15.0071	+	36.2302i	−0.0373950	+	0.0902795i	−0.941474	−	0.337087i	\(-0.890558\pi\)
							0.904079	+	0.427366i	\(0.140558\pi\)
\(13\)		−	243.603i		−	0.399783i	−0.979818	−	0.199891i	\(-0.935941\pi\)
							0.979818	−	0.199891i	\(-0.0640589\pi\)
\(17\)	−952.062	+	716.544i	−0.798993	+	0.601341i
							\(19\)	−438.677	−	438.677i	−0.278779	−	0.278779i	0.553842	−	0.832622i	\(-0.313161\pi\)
−0.832622	+	0.553842i	\(0.813161\pi\)
\(23\)	1434.42	−	3462.99i	0.565400	−	1.36500i	−0.339995	−	0.940427i	\(-0.610425\pi\)
							0.905395	−	0.424569i	\(-0.139575\pi\)
\(29\)	4221.23	−	1748.49i	0.932061	−	0.386072i	0.135601	−	0.990764i	\(-0.456703\pi\)
							0.796460	+	0.604691i	\(0.206703\pi\)
\(31\)	−2551.13	−	6158.97i	−0.476791	−	1.15108i	−0.961105	−	0.276182i	\(-0.910931\pi\)
							0.484314	−	0.874894i	\(-0.339069\pi\)
\(37\)	1168.05	+	2819.92i	0.140267	+	0.338635i	0.978365	−	0.206884i	\(-0.0663324\pi\)
							−0.838098	+	0.545519i	\(0.816332\pi\)
\(41\)	6258.10	+	2592.19i	0.581410	+	0.240828i	0.653950	−	0.756537i	\(-0.273111\pi\)
							−0.0725401	+	0.997365i	\(0.523111\pi\)
\(43\)	−11336.6	+	11336.6i	−0.935000	+	0.935000i	−0.998013	−	0.0630125i	\(-0.979929\pi\)
							0.0630125	+	0.998013i	\(0.479929\pi\)
\(47\)		−	21852.3i		−	1.44295i	−0.692440	−	0.721476i	\(-0.743464\pi\)
							0.692440	−	0.721476i	\(-0.256536\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.6.d.a.15.1	yes	28
17.5	odd	16		289.6.a.j.1.28		28
17.8	even	8	inner	17.6.d.a.8.1	✓	28
17.12	odd	16		289.6.a.j.1.27		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.6.d.a.8.1	✓	28	17.8	even	8	inner
17.6.d.a.15.1	yes	28	1.1	even	1	trivial
289.6.a.j.1.27		28	17.12	odd	16
289.6.a.j.1.28		28	17.5	odd	16