Embedded newform 63.4.h.a.25.9

• Label: 63.4.h.a.25.9
• 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			25.9
Character	\(\chi\)	\(=\)	63.25
Dual form			63.4.h.a.58.9

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.80909 q^{2} +(-5.15261 + 0.671239i) q^{3} -4.72719 q^{4} +(1.04890 - 1.81674i) q^{5} +(9.32155 - 1.21433i) q^{6} +(18.3541 + 2.47507i) q^{7} +23.0246 q^{8} +(26.0989 - 6.91727i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 2 q^{2} - q^{3} + 158 q^{4} - 19 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} + 11 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.80909			−0.639610			−0.319805	−	0.947483i	\(-0.603617\pi\)
							−0.319805	+	0.947483i	\(0.603617\pi\)
\(3\)	−5.15261	+	0.671239i	−0.991621	+	0.129180i
							\(5\)	1.04890	−	1.81674i	0.0938161	−	0.162494i	−0.815298	−	0.579042i	\(-0.803427\pi\)
0.909114	+	0.416548i	\(0.136760\pi\)
\(7\)	18.3541	+	2.47507i	0.991030	+	0.133641i
							\(11\)	−11.7410	−	20.3360i	−0.321822	−	0.557412i	0.659042	−	0.752106i	\(-0.270962\pi\)
−0.980864	+	0.194694i	\(0.937629\pi\)
\(13\)	27.8713	+	48.2745i	0.594623	+	1.02992i	0.993600	+	0.112957i	\(0.0360321\pi\)
							−0.398977	+	0.916961i	\(0.630635\pi\)
\(17\)	55.3837	−	95.9274i	0.790148	−	1.36858i	−0.135726	−	0.990746i	\(-0.543337\pi\)
							0.925875	−	0.377831i	\(-0.123330\pi\)
\(19\)	9.75396	+	16.8944i	0.117774	+	0.203991i	0.918885	−	0.394525i	\(-0.129091\pi\)
							−0.801111	+	0.598516i	\(0.795757\pi\)
\(23\)	6.87475	−	11.9074i	0.0623254	−	0.107951i	−0.833179	−	0.553003i	\(-0.813482\pi\)
							0.895504	+	0.445053i	\(0.146815\pi\)
\(29\)	59.9165	−	103.778i	0.383663	−	0.664523i	−0.607920	−	0.793998i	\(-0.707996\pi\)
							0.991583	+	0.129475i	\(0.0413292\pi\)
\(31\)	158.986			0.921121			0.460561	−	0.887628i	\(-0.347648\pi\)
							0.460561	+	0.887628i	\(0.347648\pi\)
\(37\)	−79.4438	−	137.601i	−0.352986	−	0.611389i	0.633785	−	0.773509i	\(-0.281500\pi\)
							−0.986771	+	0.162120i	\(0.948167\pi\)
\(41\)	208.432	+	361.015i	0.793942	+	1.37515i	0.923509	+	0.383578i	\(0.125308\pi\)
							−0.129566	+	0.991571i	\(0.541358\pi\)
\(43\)	131.264	−	227.355i	0.465524	−	0.806311i	−0.533701	−	0.845673i	\(-0.679199\pi\)
							0.999225	+	0.0393623i	\(0.0125327\pi\)
\(47\)	190.228			0.590376			0.295188	−	0.955439i	\(-0.404618\pi\)
							0.295188	+	0.955439i	\(0.404618\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.h.a.25.9	yes	44
3.2	odd	2		189.4.h.a.46.14		44
7.2	even	3		63.4.g.a.16.14	yes	44
9.4	even	3		63.4.g.a.4.14	✓	44
9.5	odd	6		189.4.g.a.172.9		44
21.2	odd	6		189.4.g.a.100.9		44
63.23	odd	6		189.4.h.a.37.14		44
63.58	even	3	inner	63.4.h.a.58.9	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.14	✓	44	9.4	even	3
63.4.g.a.16.14	yes	44	7.2	even	3
63.4.h.a.25.9	yes	44	1.1	even	1	trivial
63.4.h.a.58.9	yes	44	63.58	even	3	inner
189.4.g.a.100.9		44	21.2	odd	6
189.4.g.a.172.9		44	9.5	odd	6
189.4.h.a.37.14		44	63.23	odd	6
189.4.h.a.46.14		44	3.2	odd	2