Embedded newform 63.4.g.a.4.5

• Label: 63.4.g.a.4.5
• Level: $63$
• Weight: $4$
• Character: 63.4
• 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.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.71712033036\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(22\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			4.5
Character	\(\chi\)	\(=\)	63.4
Dual form			63.4.g.a.16.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.83489 + 3.17813i) q^{2} +(-2.73769 - 4.41645i) q^{3} +(-2.73366 - 4.73484i) q^{4} +14.7742 q^{5} +(19.0594 - 0.597013i) q^{6} +(0.242329 + 18.5187i) q^{7} -9.29438 q^{8} +(-12.0101 + 24.1818i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + q^{2} - q^{3} - 79 q^{4} + 38 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} - 31 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(10\)	\(29\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.83489	+	3.17813i	−0.648732	+	1.12364i	0.334693	+	0.942327i	\(0.391367\pi\)
							−0.983426	+	0.181311i	\(0.941966\pi\)
\(3\)	−2.73769	−	4.41645i	−0.526869	−	0.849947i
							\(5\)	14.7742			1.32144			0.660721	−	0.750632i	\(-0.270251\pi\)
0.660721	+	0.750632i	\(0.270251\pi\)
\(7\)	0.242329	+	18.5187i	0.0130845	+	0.999914i
							\(11\)	48.5907			1.33188			0.665939	−	0.746006i	\(-0.268031\pi\)
0.665939	+	0.746006i	\(0.268031\pi\)
\(13\)	−33.6012	+	58.1989i	−0.716868	+	1.24165i	0.245367	+	0.969430i	\(0.421092\pi\)
							−0.962235	+	0.272221i	\(0.912242\pi\)
\(17\)	−5.40836	+	9.36755i	−0.0771599	+	0.133645i	−0.902024	−	0.431687i	\(-0.857919\pi\)
							0.824864	+	0.565332i	\(0.191252\pi\)
\(19\)	67.2344	+	116.453i	0.811822	+	1.40612i	0.911587	+	0.411106i	\(0.134858\pi\)
							−0.0997650	+	0.995011i	\(0.531809\pi\)
\(23\)	84.3293			0.764516			0.382258	−	0.924056i	\(-0.375147\pi\)
							0.382258	+	0.924056i	\(0.375147\pi\)
\(29\)	−55.1123	−	95.4573i	−0.352900	−	0.611241i	0.633856	−	0.773451i	\(-0.281471\pi\)
							−0.986756	+	0.162210i	\(0.948138\pi\)
\(31\)	−75.5983	−	130.940i	−0.437995	−	0.758630i	0.559539	−	0.828804i	\(-0.310978\pi\)
							−0.997535	+	0.0701734i	\(0.977645\pi\)
\(37\)	−152.177	−	263.578i	−0.676154	−	1.17113i	−0.976130	−	0.217187i	\(-0.930312\pi\)
							0.299976	−	0.953947i	\(-0.403021\pi\)
\(41\)	127.117	−	220.173i	0.484203	−	0.838665i	−0.515632	−	0.856810i	\(-0.672443\pi\)
							0.999835	+	0.0181454i	\(0.00577619\pi\)
\(43\)	41.3056	+	71.5433i	0.146489	+	0.253727i	0.929928	−	0.367743i	\(-0.119869\pi\)
							−0.783438	+	0.621470i	\(0.786536\pi\)
\(47\)	−23.0188	+	39.8697i	−0.0714390	+	0.123736i	−0.899532	−	0.436855i	\(-0.856092\pi\)
							0.828093	+	0.560590i	\(0.189426\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.g.a.4.5	✓	44
3.2	odd	2		189.4.g.a.172.18		44
7.2	even	3		63.4.h.a.58.18	yes	44
9.2	odd	6		189.4.h.a.46.5		44
9.7	even	3		63.4.h.a.25.18	yes	44
21.2	odd	6		189.4.h.a.37.5		44
63.2	odd	6		189.4.g.a.100.18		44
63.16	even	3	inner	63.4.g.a.16.5	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.5	✓	44	1.1	even	1	trivial
63.4.g.a.16.5	yes	44	63.16	even	3	inner
63.4.h.a.25.18	yes	44	9.7	even	3
63.4.h.a.58.18	yes	44	7.2	even	3
189.4.g.a.100.18		44	63.2	odd	6
189.4.g.a.172.18		44	3.2	odd	2
189.4.h.a.37.5		44	21.2	odd	6
189.4.h.a.46.5		44	9.2	odd	6