Embedded newform 63.4.i.a.5.12

• Label: 63.4.i.a.5.12
• 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			5.12
Character	\(\chi\)	\(=\)	63.5
Dual form			63.4.i.a.38.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+0.257625i q^{2} +(1.70621 + 4.90804i) q^{3} +7.93363 q^{4} +(3.19386 + 5.53193i) q^{5} +(-1.26443 + 0.439563i) q^{6} +(-15.2112 - 10.5650i) q^{7} +4.10490i q^{8} +(-21.1777 + 16.7483i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 3 q^{3} - 162 q^{4} - 3 q^{5} + 24 q^{6} + 5 q^{7} - 45 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)			0.257625i			0.0910841i	0.998962	+	0.0455421i	\(0.0145015\pi\)
							−0.998962	+	0.0455421i	\(0.985498\pi\)
\(3\)	1.70621	+	4.90804i	0.328361	+	0.944552i
							\(5\)	3.19386	+	5.53193i	0.285668	+	0.494791i	0.972771	−	0.231769i	\(-0.0744513\pi\)
−0.687103	+	0.726560i	\(0.741118\pi\)
\(7\)	−15.2112	−	10.5650i	−0.821328	−	0.570456i
							\(11\)	52.8984	+	30.5409i	1.44995	+	0.837130i	0.998478	−	0.0551546i	\(-0.0175652\pi\)
0.451474	+	0.892284i	\(0.350898\pi\)
\(13\)	−8.78677	−	5.07304i	−0.187462	−	0.108231i	0.403332	−	0.915054i	\(-0.367852\pi\)
							−0.590794	+	0.806822i	\(0.701185\pi\)
\(17\)	−22.5082	−	38.9853i	−0.321120	−	0.556196i	0.659599	−	0.751617i	\(-0.270726\pi\)
							−0.980719	+	0.195421i	\(0.937393\pi\)
\(19\)	−69.6373	−	40.2051i	−0.840837	−	0.485457i	0.0167118	−	0.999860i	\(-0.494680\pi\)
							−0.857549	+	0.514403i	\(0.828014\pi\)
\(23\)	23.9716	−	13.8400i	0.217323	−	0.125471i	−0.387387	−	0.921917i	\(-0.626622\pi\)
							0.604710	+	0.796446i	\(0.293289\pi\)
\(29\)	48.9383	−	28.2545i	0.313366	−	0.180922i	−0.335066	−	0.942195i	\(-0.608759\pi\)
							0.648432	+	0.761273i	\(0.275425\pi\)
\(31\)		−	106.409i		−	0.616502i	−0.951305	−	0.308251i	\(-0.900256\pi\)
							0.951305	−	0.308251i	\(-0.0997436\pi\)
\(37\)	95.6403	−	165.654i	0.424951	−	0.736036i	−0.571465	−	0.820626i	\(-0.693625\pi\)
							0.996416	+	0.0845902i	\(0.0269581\pi\)
\(41\)	15.0856	−	26.1289i	0.0574626	−	0.0995282i	−0.835863	−	0.548938i	\(-0.815032\pi\)
							0.893326	+	0.449410i	\(0.148366\pi\)
\(43\)	−185.062	−	320.536i	−0.656317	−	1.13678i	−0.981562	−	0.191145i	\(-0.938780\pi\)
							0.325244	−	0.945630i	\(-0.394553\pi\)
\(47\)	496.082			1.53960			0.769798	−	0.638288i	\(-0.220357\pi\)
							0.769798	+	0.638288i	\(0.220357\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.i.a.5.12	✓	44
3.2	odd	2		189.4.i.a.152.11		44
7.3	odd	6		63.4.s.a.59.11	yes	44
9.2	odd	6		63.4.s.a.47.11	yes	44
9.7	even	3		189.4.s.a.89.12		44
21.17	even	6		189.4.s.a.17.12		44
63.38	even	6	inner	63.4.i.a.38.11	yes	44
63.52	odd	6		189.4.i.a.143.12		44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.i.a.5.12	✓	44	1.1	even	1	trivial
63.4.i.a.38.11	yes	44	63.38	even	6	inner
63.4.s.a.47.11	yes	44	9.2	odd	6
63.4.s.a.59.11	yes	44	7.3	odd	6
189.4.i.a.143.12		44	63.52	odd	6
189.4.i.a.152.11		44	3.2	odd	2
189.4.s.a.17.12		44	21.17	even	6
189.4.s.a.89.12		44	9.7	even	3