Embedded newform 177.2.e.b.4.5

• Label: 177.2.e.b.4.5
• Level: $177$
• Weight: $2$
• Character: 177.4
• Analytic conductor: $1.413$
• Analytic rank: $0$
• Dimension: $140$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.e (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.41335211578\)
Analytic rank:	\(0\)
Dimension:	\(140\)
Relative dimension:	\(5\) over \(\Q(\zeta_{29})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{29}]$

Embedding invariants

Embedding label			4.5
Character	\(\chi\)	\(=\)	177.4
Dual form			177.2.e.b.133.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.71009 + 0.294740i) q^{2} +(-0.647386 + 0.762162i) q^{3} +(5.30448 + 1.16760i) q^{4} +(-2.37039 - 1.80192i) q^{5} +(-1.97912 + 1.87472i) q^{6} +(-0.223242 + 0.560295i) q^{7} +(8.86474 + 2.98688i) q^{8} +(-0.161782 - 0.986827i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q + q^{2} + 5 q^{3} - q^{4} + 2 q^{5} - q^{6} - 2 q^{7} - 3 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.71009	+	0.294740i	1.91632	+	0.208413i	0.986985	−	0.160815i	\(-0.0514123\pi\)
							0.929339	+	0.369228i	\(0.120378\pi\)
\(3\)	−0.647386	+	0.762162i	−0.373769	+	0.440034i
							\(5\)	−2.37039	−	1.80192i	−1.06007	−	0.805844i	−0.0784930	−	0.996915i	\(-0.525011\pi\)
−0.981576	+	0.191071i	\(0.938804\pi\)
\(7\)	−0.223242	+	0.560295i	−0.0843775	+	0.211772i	−0.965111	−	0.261840i	\(-0.915671\pi\)
							0.880734	+	0.473612i	\(0.157050\pi\)
\(11\)	−5.57154	+	2.57767i	−1.67988	+	0.777197i	−0.680888	+	0.732387i	\(0.738406\pi\)
							−0.998996	+	0.0448094i	\(0.985732\pi\)
\(13\)	0.929830	−	5.67171i	0.257888	−	1.57305i	−0.467047	−	0.884233i	\(-0.654682\pi\)
							0.724935	−	0.688817i	\(-0.241870\pi\)
\(17\)	1.13804	+	2.85626i	0.276015	+	0.692745i	0.999996	+	0.00292495i	\(0.000931041\pi\)
							−0.723981	+	0.689820i	\(0.757690\pi\)
\(19\)	−1.07725	−	1.58883i	−0.247138	−	0.364502i	0.683805	−	0.729664i	\(-0.260324\pi\)
							−0.930944	+	0.365163i	\(0.881013\pi\)
\(23\)	−0.0863465	+	1.59257i	−0.0180045	+	0.332073i	0.975630	+	0.219422i	\(0.0704171\pi\)
							−0.993635	+	0.112651i	\(0.964066\pi\)
\(29\)	1.31807	−	0.143348i	0.244759	−	0.0266191i	0.0150826	−	0.999886i	\(-0.495199\pi\)
							0.229676	+	0.973267i	\(0.426233\pi\)
\(31\)	−1.55765	+	2.29736i	−0.279761	+	0.412617i	−0.941445	−	0.337166i	\(-0.890532\pi\)
							0.661684	+	0.749783i	\(0.269842\pi\)
\(37\)	4.85008	−	1.63418i	0.797348	−	0.268658i	0.109003	−	0.994041i	\(-0.465234\pi\)
							0.688345	+	0.725384i	\(0.258338\pi\)
\(41\)	−0.113269	−	2.08913i	−0.0176897	−	0.326267i	−0.993962	−	0.109722i	\(-0.965004\pi\)
							0.976273	−	0.216545i	\(-0.0694788\pi\)
\(43\)	0.406516	+	0.188074i	0.0619931	+	0.0286811i	0.450641	−	0.892705i	\(-0.351196\pi\)
							−0.388648	+	0.921386i	\(0.627058\pi\)
\(47\)	0.142877	−	0.108612i	0.0208407	−	0.0158427i	−0.594702	−	0.803946i	\(-0.702730\pi\)
							0.615543	+	0.788104i	\(0.288937\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.b.4.5	✓	140
3.2	odd	2		531.2.i.b.181.1		140
59.15	even	29	inner	177.2.e.b.133.5	yes	140
177.74	odd	58		531.2.i.b.487.1		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.b.4.5	✓	140	1.1	even	1	trivial
177.2.e.b.133.5	yes	140	59.15	even	29	inner
531.2.i.b.181.1		140	3.2	odd	2
531.2.i.b.487.1		140	177.74	odd	58