Embedded newform 63.4.o.a.41.16

• Label: 63.4.o.a.41.16
• Level: $63$
• Weight: $4$
• Character: 63.41
• 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			41.16
Character	\(\chi\)	\(=\)	63.41
Dual form			63.4.o.a.20.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.93743 - 1.11857i) q^{2} +(5.00204 - 1.40697i) q^{3} +(-1.49759 + 2.59390i) q^{4} +(2.75758 - 4.77627i) q^{5} +(8.11730 - 8.32104i) q^{6} +(13.1049 - 13.0867i) q^{7} +24.5978i q^{8} +(23.0409 - 14.0754i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 6 q^{2} + 78 q^{4} + 5 q^{7}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.93743	−	1.11857i	0.684983	−	0.395475i	−0.116747	−	0.993162i	\(-0.537247\pi\)
							0.801730	+	0.597686i	\(0.203913\pi\)
\(3\)	5.00204	−	1.40697i	0.962644	−	0.270771i
							\(5\)	2.75758	−	4.77627i	0.246645	−	0.427202i	−0.715948	−	0.698154i	\(-0.754005\pi\)
0.962593	+	0.270952i	\(0.0873384\pi\)
\(7\)	13.1049	−	13.0867i	0.707599	−	0.706614i
							\(11\)	−30.2441	+	17.4614i	−0.828995	+	0.478620i	−0.853508	−	0.521079i	\(-0.825530\pi\)
0.0245137	+	0.999699i	\(0.492196\pi\)
\(13\)	−23.1532	−	13.3675i	−0.493965	−	0.285191i	0.232253	−	0.972656i	\(-0.425390\pi\)
							−0.726218	+	0.687464i	\(0.758724\pi\)
\(17\)	−35.9093			−0.512311			−0.256156	−	0.966636i	\(-0.582456\pi\)
							−0.256156	+	0.966636i	\(0.582456\pi\)
\(19\)			78.0198i			0.942051i	0.882120	+	0.471025i	\(0.156116\pi\)
							−0.882120	+	0.471025i	\(0.843884\pi\)
\(23\)	−50.1284	−	28.9416i	−0.454456	−	0.262380i	0.255254	−	0.966874i	\(-0.417841\pi\)
							−0.709710	+	0.704494i	\(0.751174\pi\)
\(29\)	−192.066	+	110.889i	−1.22985	+	0.710056i	−0.966999	−	0.254779i	\(-0.917997\pi\)
							−0.262854	+	0.964836i	\(0.584664\pi\)
\(31\)	−126.002	−	72.7471i	−0.730018	−	0.421476i	0.0884105	−	0.996084i	\(-0.471821\pi\)
							−0.818429	+	0.574608i	\(0.805155\pi\)
\(37\)	290.956			1.29278			0.646390	−	0.763008i	\(-0.276278\pi\)
							0.646390	+	0.763008i	\(0.276278\pi\)
\(41\)	54.0606	−	93.6357i	0.205923	−	0.356669i	−0.744503	−	0.667619i	\(-0.767314\pi\)
							0.950426	+	0.310949i	\(0.100647\pi\)
\(43\)	174.061	+	301.482i	0.617303	+	1.06920i	0.989976	+	0.141238i	\(0.0451082\pi\)
							−0.372673	+	0.927963i	\(0.621559\pi\)
\(47\)	−125.545	−	217.450i	−0.389630	−	0.674858i	0.602770	−	0.797915i	\(-0.294064\pi\)
							−0.992400	+	0.123057i	\(0.960730\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.o.a.41.16	yes	44
3.2	odd	2		189.4.o.a.125.7		44
7.6	odd	2	inner	63.4.o.a.41.15	yes	44
9.2	odd	6	inner	63.4.o.a.20.15	✓	44
9.4	even	3		567.4.c.c.566.30		44
9.5	odd	6		567.4.c.c.566.15		44
9.7	even	3		189.4.o.a.62.8		44
21.20	even	2		189.4.o.a.125.8		44
63.13	odd	6		567.4.c.c.566.16		44
63.20	even	6	inner	63.4.o.a.20.16	yes	44
63.34	odd	6		189.4.o.a.62.7		44
63.41	even	6		567.4.c.c.566.29		44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.o.a.20.15	✓	44	9.2	odd	6	inner
63.4.o.a.20.16	yes	44	63.20	even	6	inner
63.4.o.a.41.15	yes	44	7.6	odd	2	inner
63.4.o.a.41.16	yes	44	1.1	even	1	trivial
189.4.o.a.62.7		44	63.34	odd	6
189.4.o.a.62.8		44	9.7	even	3
189.4.o.a.125.7		44	3.2	odd	2
189.4.o.a.125.8		44	21.20	even	2
567.4.c.c.566.15		44	9.5	odd	6
567.4.c.c.566.16		44	63.13	odd	6
567.4.c.c.566.29		44	63.41	even	6
567.4.c.c.566.30		44	9.4	even	3