Embedded newform 63.4.h.a.25.5

• Label: 63.4.h.a.25.5
• Level: $63$
• Weight: $4$
• Character: 63.25
• 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			25.5
Character	\(\chi\)	\(=\)	63.25
Dual form			63.4.h.a.58.5

$q$-expansion

\(f(q)\)	\(=\)	\(q-3.37342 q^{2} +(4.47377 - 2.64298i) q^{3} +3.37993 q^{4} +(4.87266 - 8.43970i) q^{5} +(-15.0919 + 8.91589i) q^{6} +(-16.1176 + 9.12268i) q^{7} +15.5854 q^{8} +(13.0293 - 23.6482i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 2 q^{2} - q^{3} + 158 q^{4} - 19 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} + 11 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.37342			−1.19268			−0.596341	−	0.802731i	\(-0.703379\pi\)
							−0.596341	+	0.802731i	\(0.703379\pi\)
\(3\)	4.47377	−	2.64298i	0.860978	−	0.508643i
							\(5\)	4.87266	−	8.43970i	0.435824	−	0.754869i	−0.561538	−	0.827451i	\(-0.689790\pi\)
0.997363	+	0.0725813i	\(0.0231237\pi\)
\(7\)	−16.1176	+	9.12268i	−0.870268	+	0.492578i
							\(11\)	−19.4562	−	33.6990i	−0.533296	−	0.923695i	−0.999244	−	0.0388830i	\(-0.987620\pi\)
0.465948	−	0.884812i	\(-0.345713\pi\)
\(13\)	−31.2817	−	54.1816i	−0.667384	−	1.15594i	−0.978633	−	0.205615i	\(-0.934081\pi\)
							0.311249	−	0.950328i	\(-0.399253\pi\)
\(17\)	63.3614	−	109.745i	0.903964	−	1.56571i	0.0816621	−	0.996660i	\(-0.473977\pi\)
							0.822302	−	0.569051i	\(-0.192689\pi\)
\(19\)	22.7332	+	39.3751i	0.274493	+	0.475435i	0.970007	−	0.243077i	\(-0.0781568\pi\)
							−0.695514	+	0.718512i	\(0.744823\pi\)
\(23\)	−76.9324	+	133.251i	−0.697457	+	1.20803i	0.271889	+	0.962329i	\(0.412352\pi\)
							−0.969345	+	0.245702i	\(0.920982\pi\)
\(29\)	27.4655	−	47.5716i	0.175869	−	0.304615i	−0.764592	−	0.644514i	\(-0.777060\pi\)
							0.940462	+	0.339899i	\(0.110393\pi\)
\(31\)	109.702			0.635581			0.317791	−	0.948161i	\(-0.397059\pi\)
							0.317791	+	0.948161i	\(0.397059\pi\)
\(37\)	144.482	+	250.249i	0.641962	+	1.11191i	0.984994	+	0.172588i	\(0.0552129\pi\)
							−0.343032	+	0.939324i	\(0.611454\pi\)
\(41\)	−11.3009	−	19.5738i	−0.0430465	−	0.0745587i	0.843699	−	0.536816i	\(-0.180373\pi\)
							−0.886746	+	0.462257i	\(0.847040\pi\)
\(43\)	23.9450	−	41.4740i	0.0849204	−	0.147086i	−0.820437	−	0.571737i	\(-0.806270\pi\)
							0.905357	+	0.424651i	\(0.139603\pi\)
\(47\)	386.770			1.20035			0.600173	−	0.799870i	\(-0.295098\pi\)
							0.600173	+	0.799870i	\(0.295098\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.h.a.25.5	yes	44
3.2	odd	2		189.4.h.a.46.18		44
7.2	even	3		63.4.g.a.16.18	yes	44
9.4	even	3		63.4.g.a.4.18	✓	44
9.5	odd	6		189.4.g.a.172.5		44
21.2	odd	6		189.4.g.a.100.5		44
63.23	odd	6		189.4.h.a.37.18		44
63.58	even	3	inner	63.4.h.a.58.5	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.g.a.4.18	✓	44	9.4	even	3
63.4.g.a.16.18	yes	44	7.2	even	3
63.4.h.a.25.5	yes	44	1.1	even	1	trivial
63.4.h.a.58.5	yes	44	63.58	even	3	inner
189.4.g.a.100.5		44	21.2	odd	6
189.4.g.a.172.5		44	9.5	odd	6
189.4.h.a.37.18		44	63.23	odd	6
189.4.h.a.46.18		44	3.2	odd	2