Embedded newform 177.3.h.a.104.3

• Label: 177.3.h.a.104.3
• 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			104.3
Character	\(\chi\)	\(=\)	177.104
Dual form			177.3.h.a.80.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.88346 - 3.13033i) q^{2} +(-1.65608 + 2.50148i) q^{3} +(-4.37793 + 8.25764i) q^{4} +(0.175772 + 1.61620i) q^{5} +(10.9496 + 0.472652i) q^{6} +(0.807096 - 0.271942i) q^{7} +(19.5031 - 1.05743i) q^{8} +(-3.51479 - 8.28530i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.88346	−	3.13033i	−0.941729	−	1.56517i	−0.819818	−	0.572624i	\(-0.805925\pi\)
							−0.121911	−	0.992541i	\(-0.538902\pi\)
\(3\)	−1.65608	+	2.50148i	−0.552027	+	0.833826i
							\(5\)	0.175772	+	1.61620i	0.0351544	+	0.323239i	0.998490	+	0.0549390i	\(0.0174964\pi\)
−0.963335	+	0.268300i	\(0.913538\pi\)
\(7\)	0.807096	−	0.271942i	0.115299	−	0.0388489i	−0.261060	−	0.965322i	\(-0.584072\pi\)
							0.376360	+	0.926474i	\(0.377176\pi\)
\(11\)	−0.113447	−	0.0769192i	−0.0103134	−	0.00699266i	0.556019	−	0.831169i	\(-0.312328\pi\)
							−0.566333	+	0.824177i	\(0.691638\pi\)
\(13\)	−4.49293	−	4.25593i	−0.345610	−	0.327380i	0.495129	−	0.868819i	\(-0.335121\pi\)
							−0.840739	+	0.541440i	\(0.817879\pi\)
\(17\)	2.77402	−	8.23301i	0.163178	−	0.484295i	−0.834605	−	0.550849i	\(-0.814304\pi\)
							0.997783	+	0.0665544i	\(0.0212006\pi\)
\(19\)	−5.30438	−	32.3553i	−0.279178	−	1.70291i	−0.638439	−	0.769673i	\(-0.720420\pi\)
							0.359261	−	0.933237i	\(-0.383029\pi\)
\(23\)	−21.1069	+	5.86031i	−0.917692	+	0.254796i	−0.694093	−	0.719885i	\(-0.744194\pi\)
							−0.223599	+	0.974681i	\(0.571781\pi\)
\(29\)	3.98070	−	6.61597i	0.137265	−	0.228137i	−0.780598	−	0.625034i	\(-0.785085\pi\)
							0.917863	+	0.396897i	\(0.129913\pi\)
\(31\)	−3.27544	+	19.9793i	−0.105660	+	0.644494i	0.879899	+	0.475160i	\(0.157610\pi\)
							−0.985559	+	0.169334i	\(0.945838\pi\)
\(37\)	3.29145	−	60.7072i	0.0889581	−	1.64074i	−0.524788	−	0.851233i	\(-0.675856\pi\)
							0.613747	−	0.789503i	\(-0.289662\pi\)
\(41\)	−3.82607	−	1.06230i	−0.0933187	−	0.0259098i	0.220555	−	0.975375i	\(-0.429213\pi\)
							−0.313874	+	0.949465i	\(0.601627\pi\)
\(43\)	−32.6292	−	48.1244i	−0.758818	−	1.11917i	−0.989271	−	0.146089i	\(-0.953331\pi\)
							0.230454	−	0.973083i	\(-0.425979\pi\)
\(47\)	9.16178	−	84.2412i	0.194932	−	1.79237i	−0.332844	−	0.942982i	\(-0.608008\pi\)
							0.527775	−	0.849384i	\(-0.323026\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.104.3	yes	1064
3.2	odd	2	inner	177.3.h.a.104.36	yes	1064
59.21	even	29	inner	177.3.h.a.80.36	yes	1064
177.80	odd	58	inner	177.3.h.a.80.3	✓	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.80.3	✓	1064	177.80	odd	58	inner
177.3.h.a.80.36	yes	1064	59.21	even	29	inner
177.3.h.a.104.3	yes	1064	1.1	even	1	trivial
177.3.h.a.104.36	yes	1064	3.2	odd	2	inner