Embedded newform 63.4.i.a.5.2

• Label: 63.4.i.a.5.2
• 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			5.2
Character	\(\chi\)	\(=\)	63.5
Dual form			63.4.i.a.38.21

$q$-expansion

\(f(q)\)	\(=\)	\(q-5.07706i q^{2} +(-4.42649 - 2.72143i) q^{3} -17.7766 q^{4} +(4.21335 + 7.29774i) q^{5} +(-13.8169 + 22.4736i) q^{6} +(-4.29909 - 18.0144i) q^{7} +49.6363i q^{8} +(12.1877 + 24.0928i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 3 q^{3} - 162 q^{4} - 3 q^{5} + 24 q^{6} + 5 q^{7} - 45 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	5.07706i		−	1.79501i	−0.441001	−	0.897506i	\(-0.645377\pi\)
							0.441001	−	0.897506i	\(-0.354623\pi\)
\(3\)	−4.42649	−	2.72143i	−0.851879	−	0.523739i
							\(5\)	4.21335	+	7.29774i	0.376854	+	0.652730i	0.990603	−	0.136772i	\(-0.0436726\pi\)
−0.613749	+	0.789501i	\(0.710339\pi\)
\(7\)	−4.29909	−	18.0144i	−0.232129	−	0.972685i
							\(11\)	−39.7187	−	22.9316i	−1.08869	−	0.628557i	−0.155465	−	0.987841i	\(-0.549688\pi\)
−0.933228	+	0.359284i	\(0.883021\pi\)
\(13\)	18.3978	+	10.6219i	0.392509	+	0.226615i	0.683247	−	0.730188i	\(-0.260567\pi\)
							−0.290738	+	0.956803i	\(0.593901\pi\)
\(17\)	−8.26439	−	14.3143i	−0.117906	−	0.204220i	0.801031	−	0.598622i	\(-0.204285\pi\)
							−0.918938	+	0.394402i	\(0.870952\pi\)
\(19\)	−49.7708	−	28.7352i	−0.600958	−	0.346963i	0.168461	−	0.985708i	\(-0.446120\pi\)
							−0.769418	+	0.638745i	\(0.779454\pi\)
\(23\)	−167.733	+	96.8406i	−1.52064	+	0.877942i	−0.520936	+	0.853596i	\(0.674417\pi\)
							−0.999704	+	0.0243457i	\(0.992250\pi\)
\(29\)	−47.1761	+	27.2371i	−0.302082	+	0.174407i	−0.643378	−	0.765549i	\(-0.722468\pi\)
							0.341296	+	0.939956i	\(0.389134\pi\)
\(31\)		−	294.064i		−	1.70372i	−0.523766	−	0.851862i	\(-0.675473\pi\)
							0.523766	−	0.851862i	\(-0.324527\pi\)
\(37\)	185.255	−	320.871i	0.823128	−	1.42570i	−0.0802142	−	0.996778i	\(-0.525560\pi\)
							0.903342	−	0.428921i	\(-0.141106\pi\)
\(41\)	166.132	−	287.749i	0.632815	−	1.09607i	−0.354158	−	0.935186i	\(-0.615233\pi\)
							0.986974	−	0.160883i	\(-0.0514341\pi\)
\(43\)	104.372	+	180.777i	0.370152	+	0.641121i	0.989589	−	0.143925i	\(-0.0459724\pi\)
							−0.619437	+	0.785046i	\(0.712639\pi\)
\(47\)	419.964			1.30336			0.651681	−	0.758493i	\(-0.274064\pi\)
							0.651681	+	0.758493i	\(0.274064\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.i.a.5.2	✓	44
3.2	odd	2		189.4.i.a.152.21		44
7.3	odd	6		63.4.s.a.59.21	yes	44
9.2	odd	6		63.4.s.a.47.21	yes	44
9.7	even	3		189.4.s.a.89.2		44
21.17	even	6		189.4.s.a.17.2		44
63.38	even	6	inner	63.4.i.a.38.21	yes	44
63.52	odd	6		189.4.i.a.143.2		44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.i.a.5.2	✓	44	1.1	even	1	trivial
63.4.i.a.38.21	yes	44	63.38	even	6	inner
63.4.s.a.47.21	yes	44	9.2	odd	6
63.4.s.a.59.21	yes	44	7.3	odd	6
189.4.i.a.143.2		44	63.52	odd	6
189.4.i.a.152.21		44	3.2	odd	2
189.4.s.a.17.2		44	21.17	even	6
189.4.s.a.89.2		44	9.7	even	3