Embedded newform 63.4.h.a.58.18

• Label: 63.4.h.a.58.18
• Level: $63$
• Weight: $4$
• Character: 63.58
• 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.h (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			58.18
Character	\(\chi\)	\(=\)	63.58
Dual form			63.4.h.a.25.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+3.66978 q^{2} +(5.19361 - 0.162683i) q^{3} +5.46732 q^{4} +(-7.38708 - 12.7948i) q^{5} +(19.0594 - 0.597013i) q^{6} +(15.9165 + 9.46920i) q^{7} -9.29438 q^{8} +(26.9471 - 1.68983i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 2 q^{2} - q^{3} + 158 q^{4} - 19 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} + 11 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.66978			1.29746			0.648732	−	0.761017i	\(-0.275299\pi\)
							0.648732	+	0.761017i	\(0.275299\pi\)
\(3\)	5.19361	−	0.162683i	0.999510	−	0.0313084i
							\(5\)	−7.38708	−	12.7948i	−0.660721	−	1.14440i	−0.980427	−	0.196885i	\(-0.936917\pi\)
0.319706	−	0.947517i	\(-0.396416\pi\)
\(7\)	15.9165	+	9.46920i	0.859409	+	0.511289i
							\(11\)	−24.2954	+	42.0808i	−0.665939	+	1.15344i	0.313091	+	0.949723i	\(0.398636\pi\)
−0.979030	+	0.203717i	\(0.934698\pi\)
\(13\)	−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	0.824864	−	0.565332i	\(-0.191252\pi\)
							−0.902024	+	0.431687i	\(0.857919\pi\)
\(19\)	67.2344	−	116.453i	0.811822	−	1.40612i	−0.0997650	−	0.995011i	\(-0.531809\pi\)
							0.911587	−	0.411106i	\(-0.134858\pi\)
\(23\)	−42.1647	−	73.0313i	−0.382258	−	0.662091i	0.609127	−	0.793073i	\(-0.291520\pi\)
							−0.991385	+	0.130983i	\(0.958187\pi\)
\(29\)	−55.1123	−	95.4573i	−0.352900	−	0.611241i	0.633856	−	0.773451i	\(-0.281471\pi\)
							−0.986756	+	0.162210i	\(0.948138\pi\)
\(31\)	151.197			0.875991			0.437995	−	0.898977i	\(-0.355689\pi\)
							0.437995	+	0.898977i	\(0.355689\pi\)
\(37\)	−152.177	+	263.578i	−0.676154	+	1.17113i	0.299976	+	0.953947i	\(0.403021\pi\)
							−0.976130	+	0.217187i	\(0.930312\pi\)
\(41\)	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	\(0.786536\pi\)
\(47\)	46.0375			0.142878			0.0714390	−	0.997445i	\(-0.477241\pi\)
							0.0714390	+	0.997445i	\(0.477241\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.5	✓	44	7.4	even	3
63.4.g.a.16.5	yes	44	9.7	even	3
63.4.h.a.25.18	yes	44	63.25	even	3	inner
63.4.h.a.58.18	yes	44	1.1	even	1	trivial
189.4.g.a.100.18		44	9.2	odd	6
189.4.g.a.172.18		44	21.11	odd	6
189.4.h.a.37.5		44	3.2	odd	2
189.4.h.a.46.5		44	63.11	odd	6