Embedded newform 177.3.h.a.104.20

• Label: 177.3.h.a.104.20
• Level: $177$
• Weight: $3$
• Character: 177.104
• Analytic conductor: $4.823$
• Analytic rank: $0$
• Dimension: $1064$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	177.h (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.82290067918\)
Analytic rank:	\(0\)
Dimension:	\(1064\)
Relative dimension:	\(38\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			104.20
Character	\(\chi\)	\(=\)	177.104
Dual form			177.3.h.a.80.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0752804 + 0.125117i) q^{2} +(-2.94815 + 0.555335i) q^{3} +(1.86365 - 3.51521i) q^{4} +(0.165637 + 1.52301i) q^{5} +(-0.291420 - 0.327058i) q^{6} +(-1.36154 + 0.458757i) q^{7} +(1.16333 - 0.0630737i) q^{8} +(8.38321 - 3.27442i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/177\mathbb{Z}\right)^\times\).

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{24}{29}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0752804	+	0.125117i	0.0376402	+	0.0625585i	0.875106	−	0.483931i	\(-0.160791\pi\)
							−0.837466	+	0.546489i	\(0.815964\pi\)
\(3\)	−2.94815	+	0.555335i	−0.982717	+	0.185112i
							\(5\)	0.165637	+	1.52301i	0.0331274	+	0.304601i	0.998966	+	0.0454578i	\(0.0144747\pi\)
−0.965839	+	0.259143i	\(0.916560\pi\)
\(7\)	−1.36154	+	0.458757i	−0.194506	+	0.0655368i	−0.414869	−	0.909881i	\(-0.636173\pi\)
							0.220363	+	0.975418i	\(0.429276\pi\)
\(11\)	−9.09777	−	6.16844i	−0.827070	−	0.560767i	0.0725886	−	0.997362i	\(-0.476874\pi\)
							−0.899658	+	0.436595i	\(0.856184\pi\)
\(13\)	−10.4527	−	9.90132i	−0.804053	−	0.761640i	0.170605	−	0.985340i	\(-0.445428\pi\)
							−0.974658	+	0.223700i	\(0.928186\pi\)
\(17\)	10.3712	−	30.7806i	0.610070	−	1.81062i	0.0249318	−	0.999689i	\(-0.492063\pi\)
							0.585138	−	0.810934i	\(-0.301040\pi\)
\(19\)	0.113546	+	0.692602i	0.00597613	+	0.0364528i	0.989646	−	0.143528i	\(-0.0458447\pi\)
							−0.983670	+	0.179981i	\(0.942396\pi\)
\(23\)	−20.7348	+	5.75700i	−0.901515	+	0.250304i	−0.687207	−	0.726462i	\(-0.741163\pi\)
							−0.214308	+	0.976766i	\(0.568750\pi\)
\(29\)	1.90867	−	3.17223i	0.0658161	−	0.109387i	−0.822202	−	0.569196i	\(-0.807255\pi\)
							0.888018	+	0.459809i	\(0.152082\pi\)
\(31\)	6.31440	−	38.5161i	0.203690	−	1.24246i	−0.664903	−	0.746930i	\(-0.731527\pi\)
							0.868593	−	0.495526i	\(-0.165025\pi\)
\(37\)	−2.40465	+	44.3513i	−0.0649907	+	1.19868i	0.766165	+	0.642643i	\(0.222162\pi\)
							−0.831156	+	0.556039i	\(0.812320\pi\)
\(41\)	−41.4338	−	11.5040i	−1.01058	−	0.280587i	−0.277499	−	0.960726i	\(-0.589505\pi\)
							−0.733083	+	0.680140i	\(0.761919\pi\)
\(43\)	32.1372	+	47.3988i	0.747376	+	1.10230i	0.991186	+	0.132474i	\(0.0422921\pi\)
							−0.243811	+	0.969823i	\(0.578398\pi\)
\(47\)	−1.01352	+	9.31912i	−0.0215642	+	0.198279i	−0.999964	−	0.00844043i	\(-0.997313\pi\)
							0.978400	+	0.206720i	\(0.0662788\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.104.20	yes	1064
3.2	odd	2	inner	177.3.h.a.104.19	yes	1064
59.21	even	29	inner	177.3.h.a.80.19	✓	1064
177.80	odd	58	inner	177.3.h.a.80.20	yes	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.80.19	✓	1064	59.21	even	29	inner
177.3.h.a.80.20	yes	1064	177.80	odd	58	inner
177.3.h.a.104.19	yes	1064	3.2	odd	2	inner
177.3.h.a.104.20	yes	1064	1.1	even	1	trivial