Embedded newform 63.4.s.a.59.4

• Label: 63.4.s.a.59.4
• Level: $63$
• Weight: $4$
• Character: 63.59
• 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.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.71712033036\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(22\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			59.4
Character	\(\chi\)	\(=\)	63.59
Dual form			63.4.s.a.47.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.22249 - 1.86051i) q^{2} +(4.92551 + 1.65511i) q^{3} +(2.92296 + 5.06272i) q^{4} +2.66012 q^{5} +(-12.7931 - 14.4975i) q^{6} +(9.53109 + 15.8795i) q^{7} +8.01534i q^{8} +(21.5212 + 16.3045i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 3 q^{2} - 3 q^{3} + 81 q^{4} - 6 q^{5} - 24 q^{6} + 5 q^{7} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(10\)	\(29\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.22249	−	1.86051i	−1.13932	−	0.657788i	−0.193060	−	0.981187i	\(-0.561841\pi\)
							−0.946263	+	0.323399i	\(0.895174\pi\)
\(3\)	4.92551	+	1.65511i	0.947914	+	0.318526i
							\(5\)	2.66012			0.237928			0.118964	−	0.992899i	\(-0.462043\pi\)
0.118964	+	0.992899i	\(0.462043\pi\)
\(7\)	9.53109	+	15.8795i	0.514630	+	0.857412i
							\(11\)		−	67.9830i		−	1.86342i	−0.363200	−	0.931711i	\(-0.618316\pi\)
0.363200	−	0.931711i	\(-0.381684\pi\)
\(13\)	69.6961	+	40.2391i	1.48694	+	0.858485i	0.999889	−	0.0148882i	\(-0.00473924\pi\)
							0.487051	+	0.873374i	\(0.338073\pi\)
\(17\)	29.6144	−	51.2937i	0.422503	−	0.731797i	−0.573681	−	0.819079i	\(-0.694485\pi\)
							0.996184	+	0.0872825i	\(0.0278183\pi\)
\(19\)	−5.86312	+	3.38507i	−0.0707943	+	0.0408731i	−0.534979	−	0.844865i	\(-0.679681\pi\)
							0.464185	+	0.885738i	\(0.346347\pi\)
\(23\)			126.401i			1.14593i	0.819579	+	0.572967i	\(0.194208\pi\)
							−0.819579	+	0.572967i	\(0.805792\pi\)
\(29\)	−114.327	+	66.0067i	−0.732068	+	0.422660i	−0.819178	−	0.573539i	\(-0.805570\pi\)
							0.0871101	+	0.996199i	\(0.472237\pi\)
\(31\)	62.8514	−	36.2873i	0.364143	−	0.210238i	−0.306753	−	0.951789i	\(-0.599243\pi\)
							0.670897	+	0.741551i	\(0.265909\pi\)
\(37\)	−63.8671	−	110.621i	−0.283775	−	0.491513i	0.688536	−	0.725202i	\(-0.258254\pi\)
							−0.972311	+	0.233689i	\(0.924920\pi\)
\(41\)	4.93523	−	8.54808i	0.0187989	−	0.0325606i	−0.856473	−	0.516192i	\(-0.827349\pi\)
							0.875272	+	0.483631i	\(0.160682\pi\)
\(43\)	−108.717	−	188.303i	−0.385561	−	0.667812i	0.606286	−	0.795247i	\(-0.292659\pi\)
							−0.991847	+	0.127435i	\(0.959326\pi\)
\(47\)	−92.9465	+	160.988i	−0.288460	+	0.499628i	−0.973442	−	0.228932i	\(-0.926477\pi\)
							0.684982	+	0.728560i	\(0.259810\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.s.a.59.4	yes	44
3.2	odd	2		189.4.s.a.17.19		44
7.5	odd	6		63.4.i.a.5.19	✓	44
9.2	odd	6		63.4.i.a.38.4	yes	44
9.7	even	3		189.4.i.a.143.19		44
21.5	even	6		189.4.i.a.152.4		44
63.47	even	6	inner	63.4.s.a.47.4	yes	44
63.61	odd	6		189.4.s.a.89.19		44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.i.a.5.19	✓	44	7.5	odd	6
63.4.i.a.38.4	yes	44	9.2	odd	6
63.4.s.a.47.4	yes	44	63.47	even	6	inner
63.4.s.a.59.4	yes	44	1.1	even	1	trivial
189.4.i.a.143.19		44	9.7	even	3
189.4.i.a.152.4		44	21.5	even	6
189.4.s.a.17.19		44	3.2	odd	2
189.4.s.a.89.19		44	63.61	odd	6