Embedded newform 17.6.d.a.15.3

• Label: 17.6.d.a.15.3
• Level: $17$
• Weight: $6$
• Character: 17.15
• Analytic conductor: $2.727$
• Analytic rank: $0$
• Dimension: $28$
• 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			15.3
Character	\(\chi\)	\(=\)	17.15
Dual form			17.6.d.a.8.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.68811 - 3.68811i) q^{2} +(-0.537605 - 1.29789i) q^{3} -4.79575i q^{4} +(-80.0401 + 33.1537i) q^{5} +(-2.80402 + 6.76951i) q^{6} +(-92.4561 - 38.2966i) q^{7} +(-135.707 + 135.707i) q^{8} +(170.431 - 170.431i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 4 q^{2} - 4 q^{3} + 40 q^{5} - 364 q^{6} - 4 q^{7} + 124 q^{8} - 136 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.68811	−	3.68811i	−0.651971	−	0.651971i	0.301496	−	0.953467i	\(-0.402514\pi\)
							−0.953467	+	0.301496i	\(0.902514\pi\)
\(3\)	−0.537605	−	1.29789i	−0.0344874	−	0.0832599i	0.905697	−	0.423925i	\(-0.139348\pi\)
							−0.940185	+	0.340665i	\(0.889348\pi\)
\(5\)	−80.0401	+	33.1537i	−1.43180	+	0.593071i	−0.957795	−	0.287451i	\(-0.907192\pi\)
							−0.474005	+	0.880522i	\(0.657192\pi\)
\(7\)	−92.4561	−	38.2966i	−0.713166	−	0.295403i	−0.00355187	−	0.999994i	\(-0.501131\pi\)
							−0.709614	+	0.704591i	\(0.751131\pi\)
\(11\)	122.677	−	296.167i	0.305689	−	0.737999i	−0.694146	−	0.719834i	\(-0.744218\pi\)
							0.999835	−	0.0181645i	\(-0.00578225\pi\)
\(13\)		−	440.455i		−	0.722841i	−0.932403	−	0.361420i	\(-0.882292\pi\)
							0.932403	−	0.361420i	\(-0.117708\pi\)
\(17\)	26.8685	+	1191.27i	0.0225487	+	0.999746i
							\(19\)	−1690.57	−	1690.57i	−1.07436	−	1.07436i	−0.997003	−	0.0773574i	\(-0.975352\pi\)
−0.0773574	−	0.997003i	\(-0.524648\pi\)
\(23\)	144.456	−	348.746i	0.0569396	−	0.137464i	−0.892850	−	0.450355i	\(-0.851297\pi\)
							0.949789	+	0.312891i	\(0.101297\pi\)
\(29\)	−1815.32	+	751.932i	−0.400829	+	0.166029i	−0.573985	−	0.818866i	\(-0.694603\pi\)
							0.173156	+	0.984894i	\(0.444603\pi\)
\(31\)	1354.29	+	3269.54i	0.253109	+	0.611058i	0.998452	−	0.0556227i	\(-0.0177144\pi\)
							−0.745343	+	0.666681i	\(0.767714\pi\)
\(37\)	547.883	+	1322.71i	0.0657936	+	0.158840i	0.953356	−	0.301847i	\(-0.0976032\pi\)
							−0.887563	+	0.460687i	\(0.847603\pi\)
\(41\)	−18494.5	−	7660.67i	−1.71823	−	0.711716i	−0.999871	−	0.0160484i	\(-0.994891\pi\)
							−0.718364	−	0.695668i	\(-0.755109\pi\)
\(43\)	9783.49	−	9783.49i	0.806906	−	0.806906i	−0.177259	−	0.984164i	\(-0.556723\pi\)
							0.984164	+	0.177259i	\(0.0567229\pi\)
\(47\)			16051.0i			1.05988i	0.848035	+	0.529940i	\(0.177786\pi\)
							−0.848035	+	0.529940i	\(0.822214\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.6.d.a.15.3	yes	28
17.5	odd	16		289.6.a.j.1.20		28
17.8	even	8	inner	17.6.d.a.8.3	✓	28
17.12	odd	16		289.6.a.j.1.19		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.6.d.a.8.3	✓	28	17.8	even	8	inner
17.6.d.a.15.3	yes	28	1.1	even	1	trivial
289.6.a.j.1.19		28	17.12	odd	16
289.6.a.j.1.20		28	17.5	odd	16