Embedded newform 63.4.i.a.5.21

• Label: 63.4.i.a.5.21
• Level: $63$
• Weight: $4$
• Character: 63.5
• 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.i (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			5.21
Character	\(\chi\)	\(=\)	63.5
Dual form			63.4.i.a.38.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+5.00965i q^{2} +(-2.93718 + 4.28637i) q^{3} -17.0966 q^{4} +(9.28418 + 16.0807i) q^{5} +(-21.4732 - 14.7142i) q^{6} +(12.9757 - 13.2148i) q^{7} -45.5709i q^{8} +(-9.74599 - 25.1797i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 3 q^{3} - 162 q^{4} - 3 q^{5} + 24 q^{6} + 5 q^{7} - 45 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)			5.00965i			1.77118i	0.464468	+	0.885590i	\(0.346245\pi\)
							−0.464468	+	0.885590i	\(0.653755\pi\)
\(3\)	−2.93718	+	4.28637i	−0.565260	+	0.824913i
							\(5\)	9.28418	+	16.0807i	0.830402	+	1.43830i	0.897719	+	0.440568i	\(0.145223\pi\)
−0.0673170	+	0.997732i	\(0.521444\pi\)
\(7\)	12.9757	−	13.2148i	0.700620	−	0.713535i
							\(11\)	30.4270	+	17.5670i	0.834008	+	0.481515i	0.855223	−	0.518260i	\(-0.173420\pi\)
−0.0212150	+	0.999775i	\(0.506753\pi\)
\(13\)	−3.94292	−	2.27645i	−0.0841207	−	0.0485671i	0.457350	−	0.889287i	\(-0.348799\pi\)
							−0.541470	+	0.840720i	\(0.682132\pi\)
\(17\)	−20.7867	−	36.0037i	−0.296560	−	0.513658i	0.678786	−	0.734336i	\(-0.262506\pi\)
							−0.975347	+	0.220678i	\(0.929173\pi\)
\(19\)	64.8837	+	37.4606i	0.783439	+	0.452319i	0.837648	−	0.546211i	\(-0.183930\pi\)
							−0.0542084	+	0.998530i	\(0.517264\pi\)
\(23\)	−132.242	+	76.3499i	−1.19888	+	0.692176i	−0.960307	−	0.278947i	\(-0.910015\pi\)
							−0.238578	+	0.971123i	\(0.576681\pi\)
\(29\)	42.5645	−	24.5746i	0.272552	−	0.157358i	−0.357495	−	0.933915i	\(-0.616369\pi\)
							0.630047	+	0.776557i	\(0.283036\pi\)
\(31\)		−	101.212i		−	0.586396i	−0.956052	−	0.293198i	\(-0.905280\pi\)
							0.956052	−	0.293198i	\(-0.0947195\pi\)
\(37\)	2.86552	−	4.96323i	0.0127321	−	0.0220527i	−0.859589	−	0.510986i	\(-0.829280\pi\)
							0.872321	+	0.488933i	\(0.162614\pi\)
\(41\)	6.35071	−	10.9998i	0.0241906	−	0.0418993i	−0.853677	−	0.520803i	\(-0.825632\pi\)
							0.877867	+	0.478904i	\(0.158966\pi\)
\(43\)	69.1907	+	119.842i	0.245383	+	0.425016i	0.962239	−	0.272205i	\(-0.0877528\pi\)
							−0.716856	+	0.697221i	\(0.754420\pi\)
\(47\)	538.178			1.67024			0.835121	−	0.550067i	\(-0.185398\pi\)
							0.835121	+	0.550067i	\(0.185398\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.i.a.5.21	✓	44
3.2	odd	2		189.4.i.a.152.2		44
7.3	odd	6		63.4.s.a.59.2	yes	44
9.2	odd	6		63.4.s.a.47.2	yes	44
9.7	even	3		189.4.s.a.89.21		44
21.17	even	6		189.4.s.a.17.21		44
63.38	even	6	inner	63.4.i.a.38.2	yes	44
63.52	odd	6		189.4.i.a.143.21		44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.i.a.5.21	✓	44	1.1	even	1	trivial
63.4.i.a.38.2	yes	44	63.38	even	6	inner
63.4.s.a.47.2	yes	44	9.2	odd	6
63.4.s.a.59.2	yes	44	7.3	odd	6
189.4.i.a.143.21		44	63.52	odd	6
189.4.i.a.152.2		44	3.2	odd	2
189.4.s.a.17.21		44	21.17	even	6
189.4.s.a.89.21		44	9.7	even	3