Embedded newform 63.6.s.a.47.7

• Label: 63.6.s.a.47.7
• Level: $63$
• Weight: $6$
• Character: 63.47
• Analytic conductor: $10.104$
• Analytic rank: $0$
• Dimension: $76$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(10.1041806482\)
Analytic rank:	\(0\)
Dimension:	\(76\)
Relative dimension:	\(38\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			47.7
Character	\(\chi\)	\(=\)	63.47
Dual form			63.6.s.a.59.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.18813 + 4.15007i) q^{2} +(15.2650 + 3.15899i) q^{3} +(18.4461 - 31.9497i) q^{4} -13.4718 q^{5} +(-122.837 + 40.6437i) q^{6} +(121.599 + 44.9527i) q^{7} +40.6067i q^{8} +(223.042 + 96.4440i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 3 q^{2} - 3 q^{3} + 577 q^{4} - 6 q^{5} - 96 q^{6} - 30 q^{7} - 81 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.18813	+	4.15007i	−1.27069	+	0.733636i	−0.975119	−	0.221684i	\(-0.928845\pi\)
							−0.295575	+	0.955319i	\(0.595511\pi\)
\(3\)	15.2650	+	3.15899i	0.979251	+	0.202649i
							\(5\)	−13.4718			−0.240991			−0.120495	−	0.992714i	\(-0.538448\pi\)
−0.120495	+	0.992714i	\(0.538448\pi\)
\(7\)	121.599	+	44.9527i	0.937959	+	0.346746i
							\(11\)		−	605.551i		−	1.50893i	−0.656340	−	0.754465i	\(-0.727896\pi\)
0.656340	−	0.754465i	\(-0.272104\pi\)
\(13\)	362.675	−	209.390i	0.595195	−	0.343636i	−0.171954	−	0.985105i	\(-0.555008\pi\)
							0.767149	+	0.641469i	\(0.221675\pi\)
\(17\)	165.403	+	286.486i	0.138810	+	0.240426i	0.927046	−	0.374947i	\(-0.122339\pi\)
							−0.788237	+	0.615372i	\(0.789006\pi\)
\(19\)	1722.48	+	994.474i	1.09464	+	0.631989i	0.934807	−	0.355156i	\(-0.115572\pi\)
							0.159830	+	0.987145i	\(0.448906\pi\)
\(23\)			3117.84i			1.22895i	0.788936	+	0.614475i	\(0.210632\pi\)
							−0.788936	+	0.614475i	\(0.789368\pi\)
\(29\)	−83.1858	−	48.0273i	−0.0183677	−	0.0106046i	0.490788	−	0.871279i	\(-0.336709\pi\)
							−0.509156	+	0.860674i	\(0.670042\pi\)
\(31\)	4153.05	+	2397.77i	0.776181	+	0.448128i	0.835075	−	0.550136i	\(-0.185424\pi\)
							−0.0588939	+	0.998264i	\(0.518757\pi\)
\(37\)	58.2940	−	100.968i	0.00700034	−	0.0121249i	−0.862504	−	0.506050i	\(-0.831105\pi\)
							0.869504	+	0.493925i	\(0.164438\pi\)
\(41\)	7297.17	+	12639.1i	0.677945	+	1.17424i	0.975599	+	0.219562i	\(0.0704627\pi\)
							−0.297653	+	0.954674i	\(0.596204\pi\)
\(43\)	10129.6	−	17544.9i	0.835449	−	1.44704i	−0.0582155	−	0.998304i	\(-0.518541\pi\)
							0.893664	−	0.448736i	\(-0.148126\pi\)
\(47\)	−6872.22	−	11903.0i	−0.453787	−	0.785983i	0.544830	−	0.838546i	\(-0.316594\pi\)
							−0.998618	+	0.0525637i	\(0.983261\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.6.s.a.47.7	yes	76
3.2	odd	2		189.6.s.a.89.32		76
7.3	odd	6		63.6.i.a.38.7	yes	76
9.4	even	3		189.6.i.a.152.7		76
9.5	odd	6		63.6.i.a.5.32	✓	76
21.17	even	6		189.6.i.a.143.32		76
63.31	odd	6		189.6.s.a.17.32		76
63.59	even	6	inner	63.6.s.a.59.7	yes	76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.6.i.a.5.32	✓	76	9.5	odd	6
63.6.i.a.38.7	yes	76	7.3	odd	6
63.6.s.a.47.7	yes	76	1.1	even	1	trivial
63.6.s.a.59.7	yes	76	63.59	even	6	inner
189.6.i.a.143.32		76	21.17	even	6
189.6.i.a.152.7		76	9.4	even	3
189.6.s.a.17.32		76	63.31	odd	6
189.6.s.a.89.32		76	3.2	odd	2