Embedded newform 63.4.s.a.47.18

• Label: 63.4.s.a.47.18
• Level: $63$
• Weight: $4$
• Character: 63.47
• Analytic conductor: $3.717$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	63.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.71712033036\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(22\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			47.18
Character	\(\chi\)	\(=\)	63.47
Dual form			63.4.s.a.59.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.22215 - 1.86031i) q^{2} +(-1.86223 - 4.85099i) q^{3} +(2.92152 - 5.06021i) q^{4} +13.6667 q^{5} +(-15.0248 - 12.1663i) q^{6} +(-10.0995 - 15.5242i) q^{7} +8.02526i q^{8} +(-20.0642 + 18.0673i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 3 q^{2} - 3 q^{3} + 81 q^{4} - 6 q^{5} - 24 q^{6} + 5 q^{7} - 3 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\)	\(10\)	\(29\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.22215	−	1.86031i	1.13920	−	0.657719i	0.192970	−	0.981205i	\(-0.438188\pi\)
							0.946233	+	0.323485i	\(0.104855\pi\)
\(3\)	−1.86223	−	4.85099i	−0.358387	−	0.933573i
							\(5\)	13.6667			1.22239			0.611194	−	0.791481i	\(-0.290690\pi\)
0.611194	+	0.791481i	\(0.290690\pi\)
\(7\)	−10.0995	−	15.5242i	−0.545320	−	0.838228i
							\(11\)		−	24.6001i		−	0.674291i	−0.941453	−	0.337145i	\(-0.890539\pi\)
0.941453	−	0.337145i	\(-0.109461\pi\)
\(13\)	−2.03747	+	1.17634i	−0.0434688	+	0.0250967i	−0.521577	−	0.853204i	\(-0.674656\pi\)
							0.478108	+	0.878301i	\(0.341323\pi\)
\(17\)	63.8439	+	110.581i	0.910848	+	1.57764i	0.812868	+	0.582447i	\(0.197905\pi\)
							0.0979801	+	0.995188i	\(0.468762\pi\)
\(19\)	87.2223	+	50.3578i	1.05317	+	0.608046i	0.923534	−	0.383516i	\(-0.125287\pi\)
							0.129633	+	0.991562i	\(0.458620\pi\)
\(23\)		−	48.5293i		−	0.439959i	−0.975504	−	0.219980i	\(-0.929401\pi\)
							0.975504	−	0.219980i	\(-0.0705991\pi\)
\(29\)	−171.644	−	99.0987i	−1.09909	−	0.634557i	−0.163105	−	0.986609i	\(-0.552151\pi\)
							−0.935981	+	0.352051i	\(0.885484\pi\)
\(31\)	−37.2350	−	21.4976i	−0.215729	−	0.124551i	0.388242	−	0.921557i	\(-0.373082\pi\)
							−0.603971	+	0.797006i	\(0.706416\pi\)
\(37\)	−42.0011	+	72.7481i	−0.186620	+	0.323235i	−0.944121	−	0.329598i	\(-0.893087\pi\)
							0.757501	+	0.652834i	\(0.226420\pi\)
\(41\)	−92.8172	−	160.764i	−0.353551	−	0.612369i	0.633318	−	0.773892i	\(-0.281693\pi\)
							−0.986869	+	0.161523i	\(0.948359\pi\)
\(43\)	−185.452	+	321.213i	−0.657702	+	1.13917i	0.323507	+	0.946226i	\(0.395138\pi\)
							−0.981209	+	0.192948i	\(0.938195\pi\)
\(47\)	−252.364	−	437.108i	−0.783215	−	1.35657i	−0.930059	−	0.367409i	\(-0.880245\pi\)
							0.146844	−	0.989160i	\(-0.453089\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.s.a.47.18	yes	44
3.2	odd	2		189.4.s.a.89.5		44
7.3	odd	6		63.4.i.a.38.18	yes	44
9.4	even	3		189.4.i.a.152.18		44
9.5	odd	6		63.4.i.a.5.5	✓	44
21.17	even	6		189.4.i.a.143.5		44
63.31	odd	6		189.4.s.a.17.5		44
63.59	even	6	inner	63.4.s.a.59.18	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.i.a.5.5	✓	44	9.5	odd	6
63.4.i.a.38.18	yes	44	7.3	odd	6
63.4.s.a.47.18	yes	44	1.1	even	1	trivial
63.4.s.a.59.18	yes	44	63.59	even	6	inner
189.4.i.a.143.5		44	21.17	even	6
189.4.i.a.152.18		44	9.4	even	3
189.4.s.a.17.5		44	63.31	odd	6
189.4.s.a.89.5		44	3.2	odd	2