Embedded newform 63.4.g.a.4.19

• Label: 63.4.g.a.4.19
• Level: $63$
• Weight: $4$
• Character: 63.4
• 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.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.71712033036\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(22\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			4.19
Character	\(\chi\)	\(=\)	63.4
Dual form			63.4.g.a.16.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.15287 - 3.72888i) q^{2} +(-4.60682 + 2.40358i) q^{3} +(-5.26968 - 9.12735i) q^{4} -15.9966 q^{5} +(-0.955246 + 22.3529i) q^{6} +(-16.8821 - 7.61550i) q^{7} -10.9338 q^{8} +(15.4456 - 22.1457i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + q^{2} - q^{3} - 79 q^{4} + 38 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} - 31 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(10\)	\(29\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.15287	−	3.72888i	0.761154	−	1.31836i	−0.181103	−	0.983464i	\(-0.557967\pi\)
							0.942256	−	0.334892i	\(-0.108700\pi\)
\(3\)	−4.60682	+	2.40358i	−0.886584	+	0.462568i
							\(5\)	−15.9966			−1.43078			−0.715389	−	0.698726i	\(-0.753751\pi\)
−0.715389	+	0.698726i	\(0.753751\pi\)
\(7\)	−16.8821	−	7.61550i	−0.911546	−	0.411198i
							\(11\)	−12.3484			−0.338470			−0.169235	−	0.985576i	\(-0.554130\pi\)
−0.169235	+	0.985576i	\(0.554130\pi\)
\(13\)	35.8182	−	62.0390i	0.764168	−	1.32358i	−0.176517	−	0.984298i	\(-0.556483\pi\)
							0.940685	−	0.339281i	\(-0.110184\pi\)
\(17\)	−42.2423	+	73.1658i	−0.602663	+	1.04384i	0.389753	+	0.920919i	\(0.372560\pi\)
							−0.992416	+	0.122923i	\(0.960773\pi\)
\(19\)	13.6505	+	23.6434i	0.164823	+	0.285482i	0.936592	−	0.350421i	\(-0.113961\pi\)
							−0.771769	+	0.635903i	\(0.780628\pi\)
\(23\)	−40.5398			−0.367527			−0.183764	−	0.982970i	\(-0.558828\pi\)
							−0.183764	+	0.982970i	\(0.558828\pi\)
\(29\)	−99.2642	−	171.931i	−0.635617	−	1.10092i	−0.986384	−	0.164458i	\(-0.947412\pi\)
							0.350767	−	0.936463i	\(-0.385921\pi\)
\(31\)	−146.116	−	253.080i	−0.846555	−	1.46628i	−0.884264	−	0.466988i	\(-0.845339\pi\)
							0.0377089	−	0.999289i	\(-0.487994\pi\)
\(37\)	58.7266	+	101.717i	0.260935	+	0.451952i	0.966491	−	0.256702i	\(-0.0826359\pi\)
							−0.705556	+	0.708654i	\(0.749303\pi\)
\(41\)	−18.6085	+	32.2308i	−0.0708818	+	0.122771i	−0.899288	−	0.437357i	\(-0.855915\pi\)
							0.828406	+	0.560128i	\(0.189248\pi\)
\(43\)	122.917	+	212.898i	0.435921	+	0.755037i	0.997370	−	0.0724738i	\(-0.0230894\pi\)
							−0.561449	+	0.827511i	\(0.689756\pi\)
\(47\)	−45.8404	+	79.3979i	−0.142266	+	0.246412i	−0.928350	−	0.371708i	\(-0.878772\pi\)
							0.786084	+	0.618120i	\(0.212106\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.g.a.4.19	✓	44
3.2	odd	2		189.4.g.a.172.4		44
7.2	even	3		63.4.h.a.58.4	yes	44
9.2	odd	6		189.4.h.a.46.19		44
9.7	even	3		63.4.h.a.25.4	yes	44
21.2	odd	6		189.4.h.a.37.19		44
63.2	odd	6		189.4.g.a.100.4		44
63.16	even	3	inner	63.4.g.a.16.19	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.19	✓	44	1.1	even	1	trivial
63.4.g.a.16.19	yes	44	63.16	even	3	inner
63.4.h.a.25.4	yes	44	9.7	even	3
63.4.h.a.58.4	yes	44	7.2	even	3
189.4.g.a.100.4		44	63.2	odd	6
189.4.g.a.172.4		44	3.2	odd	2
189.4.h.a.37.19		44	21.2	odd	6
189.4.h.a.46.19		44	9.2	odd	6