Embedded newform 63.4.h.a.25.6

• Label: 63.4.h.a.25.6
• Level: $63$
• Weight: $4$
• Character: 63.25
• 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			25.6
Character	\(\chi\)	\(=\)	63.25
Dual form			63.4.h.a.58.6

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.93457 q^{2} +(-3.89946 - 3.43427i) q^{3} +0.611720 q^{4} +(1.84855 - 3.20179i) q^{5} +(11.4432 + 10.0781i) q^{6} +(-18.1273 - 3.79476i) q^{7} +21.6814 q^{8} +(3.41152 + 26.7836i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 2 q^{2} - q^{3} + 158 q^{4} - 19 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} + 11 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.93457			−1.03753			−0.518764	−	0.854917i	\(-0.673608\pi\)
							−0.518764	+	0.854917i	\(0.673608\pi\)
\(3\)	−3.89946	−	3.43427i	−0.750451	−	0.660926i
							\(5\)	1.84855	−	3.20179i	0.165340	−	0.286377i	−0.771436	−	0.636307i	\(-0.780461\pi\)
0.936776	+	0.349930i	\(0.113795\pi\)
\(7\)	−18.1273	−	3.79476i	−0.978783	−	0.204898i
							\(11\)	32.8537	+	56.9043i	0.900524	+	1.55975i	0.826816	+	0.562473i	\(0.190150\pi\)
0.0737076	+	0.997280i	\(0.476517\pi\)
\(13\)	2.73952	+	4.74498i	0.0584465	+	0.101232i	0.893768	−	0.448529i	\(-0.148052\pi\)
							−0.835322	+	0.549761i	\(0.814719\pi\)
\(17\)	−25.1343	+	43.5340i	−0.358587	+	0.621090i	−0.987725	−	0.156203i	\(-0.950075\pi\)
							0.629138	+	0.777293i	\(0.283408\pi\)
\(19\)	0.769884	+	1.33348i	0.00929597	+	0.0161011i	0.870636	−	0.491928i	\(-0.163708\pi\)
							−0.861340	+	0.508029i	\(0.830374\pi\)
\(23\)	60.0078	−	103.936i	0.544021	−	0.942272i	−0.454647	−	0.890672i	\(-0.650235\pi\)
							0.998668	−	0.0516002i	\(-0.0164322\pi\)
\(29\)	−39.2967	+	68.0640i	−0.251628	+	0.435833i	−0.963974	−	0.265995i	\(-0.914299\pi\)
							0.712346	+	0.701829i	\(0.247633\pi\)
\(31\)	−303.332			−1.75742			−0.878709	−	0.477358i	\(-0.841595\pi\)
							−0.878709	+	0.477358i	\(0.841595\pi\)
\(37\)	96.6335	+	167.374i	0.429363	+	0.743679i	0.996817	−	0.0797263i	\(-0.0254046\pi\)
							−0.567453	+	0.823406i	\(0.692071\pi\)
\(41\)	196.671	+	340.645i	0.749144	+	1.29756i	0.948233	+	0.317574i	\(0.102868\pi\)
							−0.199090	+	0.979981i	\(0.563798\pi\)
\(43\)	−138.067	+	239.139i	−0.489651	+	0.848101i	−0.999929	−	0.0119087i	\(-0.996209\pi\)
							0.510278	+	0.860010i	\(0.329543\pi\)
\(47\)	−252.760			−0.784443			−0.392221	−	0.919871i	\(-0.628293\pi\)
							−0.392221	+	0.919871i	\(0.628293\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.h.a.25.6	yes	44
3.2	odd	2		189.4.h.a.46.17		44
7.2	even	3		63.4.g.a.16.17	yes	44
9.4	even	3		63.4.g.a.4.17	✓	44
9.5	odd	6		189.4.g.a.172.6		44
21.2	odd	6		189.4.g.a.100.6		44
63.23	odd	6		189.4.h.a.37.17		44
63.58	even	3	inner	63.4.h.a.58.6	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.17	✓	44	9.4	even	3
63.4.g.a.16.17	yes	44	7.2	even	3
63.4.h.a.25.6	yes	44	1.1	even	1	trivial
63.4.h.a.58.6	yes	44	63.58	even	3	inner
189.4.g.a.100.6		44	21.2	odd	6
189.4.g.a.172.6		44	9.5	odd	6
189.4.h.a.37.17		44	63.23	odd	6
189.4.h.a.46.17		44	3.2	odd	2