Embedded newform 63.4.h.a.58.4

• Label: 63.4.h.a.58.4
• Level: $63$
• Weight: $4$
• Character: 63.58
• 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.h (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			58.4
Character	\(\chi\)	\(=\)	63.58
Dual form			63.4.h.a.25.4

$q$-expansion

\(f(q)\)	\(=\)	\(q-4.30573 q^{2} +(0.221854 - 5.19141i) q^{3} +10.5394 q^{4} +(7.99829 + 13.8535i) q^{5} +(-0.955246 + 22.3529i) q^{6} +(1.84582 - 18.4280i) q^{7} -10.9338 q^{8} +(-26.9016 - 2.30348i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 2 q^{2} - q^{3} + 158 q^{4} - 19 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} + 11 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.30573			−1.52231			−0.761154	−	0.648572i	\(-0.775367\pi\)
							−0.761154	+	0.648572i	\(0.775367\pi\)
\(3\)	0.221854	−	5.19141i	0.0426959	−	0.999088i
							\(5\)	7.99829	+	13.8535i	0.715389	+	1.23909i	0.962809	+	0.270182i	\(0.0870839\pi\)
−0.247420	+	0.968908i	\(0.579583\pi\)
\(7\)	1.84582	−	18.4280i	0.0996649	−	0.995021i
							\(11\)	6.17419	−	10.6940i	0.169235	−	0.293124i	−0.768916	−	0.639350i	\(-0.779204\pi\)
0.938151	+	0.346226i	\(0.112537\pi\)
\(13\)	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	−0.992416	−	0.122923i	\(-0.960773\pi\)
							0.389753	−	0.920919i	\(-0.372560\pi\)
\(19\)	13.6505	−	23.6434i	0.164823	−	0.285482i	−0.771769	−	0.635903i	\(-0.780628\pi\)
							0.936592	+	0.350421i	\(0.113961\pi\)
\(23\)	20.2699	+	35.1085i	0.183764	+	0.318288i	0.943159	−	0.332341i	\(-0.107839\pi\)
							−0.759396	+	0.650629i	\(0.774505\pi\)
\(29\)	−99.2642	−	171.931i	−0.635617	−	1.10092i	−0.986384	−	0.164458i	\(-0.947412\pi\)
							0.350767	−	0.936463i	\(-0.385921\pi\)
\(31\)	292.232			1.69311			0.846555	−	0.532301i	\(-0.178673\pi\)
							0.846555	+	0.532301i	\(0.178673\pi\)
\(37\)	58.7266	−	101.717i	0.260935	−	0.451952i	−0.705556	−	0.708654i	\(-0.749303\pi\)
							0.966491	+	0.256702i	\(0.0826359\pi\)
\(41\)	−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	\(0.689756\pi\)
\(47\)	91.6808			0.284532			0.142266	−	0.989828i	\(-0.454561\pi\)
							0.142266	+	0.989828i	\(0.454561\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.19	✓	44	7.4	even	3
63.4.g.a.16.19	yes	44	9.7	even	3
63.4.h.a.25.4	yes	44	63.25	even	3	inner
63.4.h.a.58.4	yes	44	1.1	even	1	trivial
189.4.g.a.100.4		44	9.2	odd	6
189.4.g.a.172.4		44	21.11	odd	6
189.4.h.a.37.19		44	3.2	odd	2
189.4.h.a.46.19		44	63.11	odd	6