Embedded newform 63.4.h.a.25.12

• Label: 63.4.h.a.25.12
• 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			25.12
Character	\(\chi\)	\(=\)	63.25
Dual form			63.4.h.a.58.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+0.534259 q^{2} +(1.18489 - 5.05925i) q^{3} -7.71457 q^{4} +(0.696621 - 1.20658i) q^{5} +(0.633036 - 2.70295i) q^{6} +(-2.10659 - 18.4001i) q^{7} -8.39564 q^{8} +(-24.1921 - 11.9893i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 2 q^{2} - q^{3} + 158 q^{4} - 19 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} + 11 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.534259			0.188889			0.0944445	−	0.995530i	\(-0.469893\pi\)
							0.0944445	+	0.995530i	\(0.469893\pi\)
\(3\)	1.18489	−	5.05925i	0.228031	−	0.973654i
							\(5\)	0.696621	−	1.20658i	0.0623077	−	0.107920i	−0.833189	−	0.552989i	\(-0.813487\pi\)
0.895497	+	0.445068i	\(0.146821\pi\)
\(7\)	−2.10659	−	18.4001i	−0.113745	−	0.993510i
							\(11\)	−9.69762	−	16.7968i	−0.265813	−	0.460402i	0.701963	−	0.712213i	\(-0.252307\pi\)
−0.967776	+	0.251811i	\(0.918974\pi\)
\(13\)	4.05999	+	7.03211i	0.0866184	+	0.150027i	0.906080	−	0.423107i	\(-0.139061\pi\)
							−0.819461	+	0.573134i	\(0.805727\pi\)
\(17\)	28.8598	−	49.9867i	0.411737	−	0.713150i	−0.583342	−	0.812226i	\(-0.698255\pi\)
							0.995080	+	0.0990761i	\(0.0315887\pi\)
\(19\)	35.7871	+	61.9851i	0.432112	+	0.748440i	0.997055	−	0.0766902i	\(-0.0244352\pi\)
							−0.564943	+	0.825130i	\(0.691102\pi\)
\(23\)	105.165	−	182.151i	0.953411	−	1.65136i	0.215449	−	0.976515i	\(-0.430878\pi\)
							0.737962	−	0.674842i	\(-0.235788\pi\)
\(29\)	−78.1687	+	135.392i	−0.500537	+	0.866955i	0.499463	+	0.866335i	\(0.333531\pi\)
							−1.00000		0.000619685i	\(0.999803\pi\)
\(31\)	111.132			0.643870			0.321935	−	0.946762i	\(-0.395667\pi\)
							0.321935	+	0.946762i	\(0.395667\pi\)
\(37\)	26.2246	+	45.4224i	0.116522	+	0.201821i	0.918387	−	0.395683i	\(-0.129492\pi\)
							−0.801865	+	0.597505i	\(0.796159\pi\)
\(41\)	−201.952	−	349.790i	−0.769257	−	1.33239i	−0.937966	−	0.346726i	\(-0.887293\pi\)
							0.168709	−	0.985666i	\(-0.446040\pi\)
\(43\)	89.7250	−	155.408i	0.318208	−	0.551152i	−0.661906	−	0.749586i	\(-0.730252\pi\)
							0.980114	+	0.198435i	\(0.0635857\pi\)
\(47\)	−92.8201			−0.288068			−0.144034	−	0.989573i	\(-0.546007\pi\)
							−0.144034	+	0.989573i	\(0.546007\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.h.a.25.12	yes	44
3.2	odd	2		189.4.h.a.46.11		44
7.2	even	3		63.4.g.a.16.11	yes	44
9.4	even	3		63.4.g.a.4.11	✓	44
9.5	odd	6		189.4.g.a.172.12		44
21.2	odd	6		189.4.g.a.100.12		44
63.23	odd	6		189.4.h.a.37.11		44
63.58	even	3	inner	63.4.h.a.58.12	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.11	✓	44	9.4	even	3
63.4.g.a.16.11	yes	44	7.2	even	3
63.4.h.a.25.12	yes	44	1.1	even	1	trivial
63.4.h.a.58.12	yes	44	63.58	even	3	inner
189.4.g.a.100.12		44	21.2	odd	6
189.4.g.a.172.12		44	9.5	odd	6
189.4.h.a.37.11		44	63.23	odd	6
189.4.h.a.46.11		44	3.2	odd	2