Embedded newform 63.4.o.a.20.19

• Label: 63.4.o.a.20.19
• Level: $63$
• Weight: $4$
• Character: 63.20
• Analytic conductor: $3.717$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	63.o (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			20.19
Character	\(\chi\)	\(=\)	63.20
Dual form			63.4.o.a.41.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.92583 + 2.26658i) q^{2} +(-3.93727 + 3.39086i) q^{3} +(6.27474 + 10.8682i) q^{4} +(0.0687529 + 0.119084i) q^{5} +(-23.1427 + 4.38780i) q^{6} +(2.53789 + 18.3455i) q^{7} +20.6235i q^{8} +(4.00416 - 26.7014i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 6 q^{2} + 78 q^{4} + 5 q^{7}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.92583	+	2.26658i	1.38799	+	0.801356i	0.993089	−	0.117368i	\(-0.0374456\pi\)
							0.394901	+	0.918724i	\(0.370779\pi\)
\(3\)	−3.93727	+	3.39086i	−0.757728	+	0.652571i
							\(5\)	0.0687529	+	0.119084i	0.00614945	+	0.0106512i	0.869084	−	0.494665i	\(-0.164709\pi\)
−0.862934	+	0.505316i	\(0.831376\pi\)
\(7\)	2.53789	+	18.3455i	0.137033	+	0.990566i
							\(11\)	8.78077	+	5.06958i	0.240682	+	0.138958i	0.615490	−	0.788145i	\(-0.288958\pi\)
−0.374808	+	0.927102i	\(0.622291\pi\)
\(13\)	13.1146	−	7.57170i	0.279794	−	0.161539i	−0.353536	−	0.935421i	\(-0.615021\pi\)
							0.633330	+	0.773882i	\(0.281687\pi\)
\(17\)	88.2223			1.25865			0.629325	−	0.777142i	\(-0.283331\pi\)
							0.629325	+	0.777142i	\(0.283331\pi\)
\(19\)		−	66.1014i		−	0.798142i	−0.916920	−	0.399071i	\(-0.869333\pi\)
							0.916920	−	0.399071i	\(-0.130667\pi\)
\(23\)	−49.7310	+	28.7122i	−0.450854	+	0.260300i	−0.708191	−	0.706021i	\(-0.750488\pi\)
							0.257337	+	0.966322i	\(0.417155\pi\)
\(29\)	102.744	+	59.3195i	0.657902	+	0.379840i	0.791477	−	0.611199i	\(-0.209312\pi\)
							−0.133575	+	0.991039i	\(0.542646\pi\)
\(31\)	−266.140	+	153.656i	−1.54194	+	0.890238i	−0.543221	+	0.839590i	\(0.682796\pi\)
							−0.998717	+	0.0506486i	\(0.983871\pi\)
\(37\)	−337.159			−1.49807			−0.749035	−	0.662531i	\(-0.769482\pi\)
							−0.749035	+	0.662531i	\(0.769482\pi\)
\(41\)	−220.622	−	382.129i	−0.840376	−	1.45557i	−0.889577	−	0.456785i	\(-0.849001\pi\)
							0.0492008	−	0.998789i	\(-0.484333\pi\)
\(43\)	46.7113	−	80.9064i	0.165661	−	0.286933i	−0.771229	−	0.636558i	\(-0.780358\pi\)
							0.936890	+	0.349625i	\(0.113691\pi\)
\(47\)	155.927	−	270.073i	0.483921	−	0.838176i	−0.515909	−	0.856644i	\(-0.672546\pi\)
							0.999829	+	0.0184681i	\(0.00587892\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.o.a.20.19	✓	44
3.2	odd	2		189.4.o.a.62.3		44
7.6	odd	2	inner	63.4.o.a.20.20	yes	44
9.2	odd	6		567.4.c.c.566.18		44
9.4	even	3		189.4.o.a.125.4		44
9.5	odd	6	inner	63.4.o.a.41.20	yes	44
9.7	even	3		567.4.c.c.566.27		44
21.20	even	2		189.4.o.a.62.4		44
63.13	odd	6		189.4.o.a.125.3		44
63.20	even	6		567.4.c.c.566.28		44
63.34	odd	6		567.4.c.c.566.17		44
63.41	even	6	inner	63.4.o.a.41.19	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.o.a.20.19	✓	44	1.1	even	1	trivial
63.4.o.a.20.20	yes	44	7.6	odd	2	inner
63.4.o.a.41.19	yes	44	63.41	even	6	inner
63.4.o.a.41.20	yes	44	9.5	odd	6	inner
189.4.o.a.62.3		44	3.2	odd	2
189.4.o.a.62.4		44	21.20	even	2
189.4.o.a.125.3		44	63.13	odd	6
189.4.o.a.125.4		44	9.4	even	3
567.4.c.c.566.17		44	63.34	odd	6
567.4.c.c.566.18		44	9.2	odd	6
567.4.c.c.566.27		44	9.7	even	3
567.4.c.c.566.28		44	63.20	even	6