Embedded newform 63.4.h.a.25.17

• Label: 63.4.h.a.25.17
• 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			25.17
Character	\(\chi\)	\(=\)	63.25
Dual form			63.4.h.a.58.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+3.35315 q^{2} +(3.94919 - 3.37697i) q^{3} +3.24362 q^{4} +(4.35326 - 7.54007i) q^{5} +(13.2422 - 11.3235i) q^{6} +(-2.88142 + 18.2947i) q^{7} -15.9489 q^{8} +(4.19213 - 26.6726i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 2 q^{2} - q^{3} + 158 q^{4} - 19 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} + 11 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.35315			1.18552			0.592759	−	0.805380i	\(-0.298039\pi\)
							0.592759	+	0.805380i	\(0.298039\pi\)
\(3\)	3.94919	−	3.37697i	0.760021	−	0.649898i
							\(5\)	4.35326	−	7.54007i	0.389367	−	0.674404i	−0.602997	−	0.797743i	\(-0.706027\pi\)
0.992365	+	0.123339i	\(0.0393603\pi\)
\(7\)	−2.88142	+	18.2947i	−0.155582	+	0.987823i
							\(11\)	5.89868	+	10.2168i	0.161684	+	0.280044i	0.935473	−	0.353399i	\(-0.114974\pi\)
−0.773789	+	0.633443i	\(0.781641\pi\)
\(13\)	26.5809	+	46.0395i	0.567094	+	0.982235i	0.996852	+	0.0792910i	\(0.0252656\pi\)
							−0.429758	+	0.902944i	\(0.641401\pi\)
\(17\)	−22.5463	+	39.0513i	−0.321663	+	0.557137i	−0.980831	−	0.194858i	\(-0.937575\pi\)
							0.659168	+	0.751996i	\(0.270909\pi\)
\(19\)	−13.0906	−	22.6736i	−0.158062	−	0.273772i	0.776107	−	0.630601i	\(-0.217191\pi\)
							−0.934170	+	0.356828i	\(0.883858\pi\)
\(23\)	29.0940	−	50.3923i	0.263762	−	0.456848i	−0.703477	−	0.710718i	\(-0.748370\pi\)
							0.967238	+	0.253870i	\(0.0817035\pi\)
\(29\)	36.7015	−	63.5688i	0.235010	−	0.407049i	−0.724266	−	0.689521i	\(-0.757821\pi\)
							0.959276	+	0.282472i	\(0.0911544\pi\)
\(31\)	−314.805			−1.82389			−0.911947	−	0.410308i	\(-0.865421\pi\)
							−0.911947	+	0.410308i	\(0.865421\pi\)
\(37\)	−189.337	−	327.941i	−0.841263	−	1.45711i	−0.888827	−	0.458242i	\(-0.848479\pi\)
							0.0475640	−	0.998868i	\(-0.484854\pi\)
\(41\)	181.180	+	313.812i	0.690134	+	1.19535i	0.971794	+	0.235833i	\(0.0757819\pi\)
							−0.281659	+	0.959515i	\(0.590885\pi\)
\(43\)	58.7824	−	101.814i	0.208470	−	0.361081i	−0.742762	−	0.669555i	\(-0.766485\pi\)
							0.951233	+	0.308474i	\(0.0998182\pi\)
\(47\)	228.220			0.708284			0.354142	−	0.935192i	\(-0.384773\pi\)
							0.354142	+	0.935192i	\(0.384773\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.h.a.25.17	yes	44
3.2	odd	2		189.4.h.a.46.6		44
7.2	even	3		63.4.g.a.16.6	yes	44
9.4	even	3		63.4.g.a.4.6	✓	44
9.5	odd	6		189.4.g.a.172.17		44
21.2	odd	6		189.4.g.a.100.17		44
63.23	odd	6		189.4.h.a.37.6		44
63.58	even	3	inner	63.4.h.a.58.17	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.6	✓	44	9.4	even	3
63.4.g.a.16.6	yes	44	7.2	even	3
63.4.h.a.25.17	yes	44	1.1	even	1	trivial
63.4.h.a.58.17	yes	44	63.58	even	3	inner
189.4.g.a.100.17		44	21.2	odd	6
189.4.g.a.172.17		44	9.5	odd	6
189.4.h.a.37.6		44	63.23	odd	6
189.4.h.a.46.6		44	3.2	odd	2