Embedded newform 63.3.n.b.2.11

• Label: 63.3.n.b.2.11
• Level: $63$
• Weight: $3$
• Character: 63.2
• Analytic conductor: $1.717$
• Analytic rank: $0$
• Dimension: $22$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	63.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.71662566547\)
Analytic rank:	\(0\)
Dimension:	\(22\)
Relative dimension:	\(11\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2.11
Character	\(\chi\)	\(=\)	63.2
Dual form			63.3.n.b.32.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.48702 - 1.43588i) q^{2} +(0.950543 - 2.84543i) q^{3} +(2.12350 - 3.67801i) q^{4} +7.54889i q^{5} +(-1.72168 - 8.44150i) q^{6} +(-6.82274 + 1.56534i) q^{7} -0.709334i q^{8} +(-7.19294 - 5.40941i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 6 q^{2} + 8 q^{3} + 12 q^{4} - 8 q^{6} + 3 q^{7} + 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(10\)	\(29\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.48702	−	1.43588i	1.24351	−	0.717940i	0.273702	−	0.961815i	\(-0.411752\pi\)
							0.969807	+	0.243875i	\(0.0784186\pi\)
\(3\)	0.950543	−	2.84543i	0.316848	−	0.948476i
							\(5\)			7.54889i			1.50978i	0.655853	+	0.754889i	\(0.272309\pi\)
−0.655853	+	0.754889i	\(0.727691\pi\)
\(7\)	−6.82274	+	1.56534i	−0.974677	+	0.223619i
							\(11\)		−	15.1994i		−	1.38176i	−0.722970	−	0.690880i	\(-0.757223\pi\)
0.722970	−	0.690880i	\(-0.242777\pi\)
\(13\)	4.30409	+	7.45491i	0.331084	+	0.573455i	0.982725	−	0.185073i	\(-0.0592522\pi\)
							−0.651641	+	0.758528i	\(0.725919\pi\)
\(17\)	4.60986	−	2.66151i	0.271168	−	0.156559i	−0.358250	−	0.933626i	\(-0.616626\pi\)
							0.629419	+	0.777066i	\(0.283293\pi\)
\(19\)	−0.417241	+	0.722683i	−0.0219601	+	0.0380360i	−0.876797	−	0.480861i	\(-0.840324\pi\)
							0.854837	+	0.518897i	\(0.173657\pi\)
\(23\)		−	39.1297i		−	1.70129i	−0.525740	−	0.850646i	\(-0.676211\pi\)
							0.525740	−	0.850646i	\(-0.323789\pi\)
\(29\)	12.5660	+	7.25497i	0.433309	+	0.250171i	0.700755	−	0.713402i	\(-0.252846\pi\)
							−0.267446	+	0.963573i	\(0.586180\pi\)
\(31\)	−6.37095	+	11.0348i	−0.205515	+	0.355962i	−0.950297	−	0.311346i	\(-0.899220\pi\)
							0.744782	+	0.667308i	\(0.232553\pi\)
\(37\)	−11.7490	+	20.3498i	−0.317540	+	0.549995i	−0.979974	−	0.199125i	\(-0.936190\pi\)
							0.662434	+	0.749120i	\(0.269523\pi\)
\(41\)	−13.4288	+	7.75311i	−0.327531	+	0.189100i	−0.654744	−	0.755850i	\(-0.727224\pi\)
							0.327213	+	0.944951i	\(0.393890\pi\)
\(43\)	0.448287	−	0.776457i	0.0104253	−	0.0180571i	−0.860766	−	0.509001i	\(-0.830015\pi\)
							0.871191	+	0.490944i	\(0.163348\pi\)
\(47\)	2.03864	−	1.17701i	0.0433754	−	0.0250428i	−0.478155	−	0.878275i	\(-0.658694\pi\)
							0.521531	+	0.853232i	\(0.325361\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.3.n.b.2.11	yes	22
3.2	odd	2		189.3.n.b.170.1		22
7.2	even	3		441.3.r.g.344.1		22
7.3	odd	6		441.3.j.f.263.11		22
7.4	even	3		63.3.j.b.11.11	✓	22
7.5	odd	6		441.3.r.f.344.1		22
7.6	odd	2		441.3.n.f.128.11		22
9.4	even	3		189.3.j.b.44.11		22
9.5	odd	6		63.3.j.b.23.1	yes	22
21.11	odd	6		189.3.j.b.116.1		22
63.4	even	3		189.3.n.b.179.1		22
63.5	even	6		441.3.r.f.50.1		22
63.23	odd	6		441.3.r.g.50.1		22
63.32	odd	6	inner	63.3.n.b.32.11	yes	22
63.41	even	6		441.3.j.f.275.1		22
63.59	even	6		441.3.n.f.410.11		22

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.3.j.b.11.11	✓	22	7.4	even	3
63.3.j.b.23.1	yes	22	9.5	odd	6
63.3.n.b.2.11	yes	22	1.1	even	1	trivial
63.3.n.b.32.11	yes	22	63.32	odd	6	inner
189.3.j.b.44.11		22	9.4	even	3
189.3.j.b.116.1		22	21.11	odd	6
189.3.n.b.170.1		22	3.2	odd	2
189.3.n.b.179.1		22	63.4	even	3
441.3.j.f.263.11		22	7.3	odd	6
441.3.j.f.275.1		22	63.41	even	6
441.3.n.f.128.11		22	7.6	odd	2
441.3.n.f.410.11		22	63.59	even	6
441.3.r.f.50.1		22	63.5	even	6
441.3.r.f.344.1		22	7.5	odd	6
441.3.r.g.50.1		22	63.23	odd	6
441.3.r.g.344.1		22	7.2	even	3