Embedded newform 177.2.e.b.139.3

• Label: 177.2.e.b.139.3
• Level: $177$
• Weight: $2$
• Character: 177.139
• 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			139.3
Character	\(\chi\)	\(=\)	177.139
Dual form			177.2.e.b.163.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.183344 + 0.110315i) q^{2} +(0.370138 - 0.928977i) q^{3} +(-0.915371 - 1.72657i) q^{4} +(-4.31685 - 0.469486i) q^{5} +(0.170342 - 0.129491i) q^{6} +(2.04860 + 0.690253i) q^{7} +(0.0458067 - 0.844855i) q^{8} +(-0.725995 - 0.687699i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q + q^{2} + 5 q^{3} - q^{4} + 2 q^{5} - q^{6} - 2 q^{7} - 3 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{5}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.183344	+	0.110315i	0.129644	+	0.0780042i	0.578898	−	0.815400i	\(-0.303483\pi\)
							−0.449254	+	0.893404i	\(0.648310\pi\)
\(3\)	0.370138	−	0.928977i	0.213699	−	0.536345i
							\(5\)	−4.31685	−	0.469486i	−1.93055	−	0.209960i	−0.939414	−	0.342784i	\(-0.888630\pi\)
−0.991140	+	0.132824i	\(0.957595\pi\)
\(7\)	2.04860	+	0.690253i	0.774298	+	0.260891i	0.678589	−	0.734518i	\(-0.262592\pi\)
							0.0957087	+	0.995409i	\(0.469488\pi\)
\(11\)	−2.75585	−	4.06457i	−0.830919	−	1.22551i	−0.971875	−	0.235498i	\(-0.924328\pi\)
							0.140956	−	0.990016i	\(-0.454982\pi\)
\(13\)	0.125570	−	0.118947i	0.0348270	−	0.0329899i	−0.670088	−	0.742282i	\(-0.733744\pi\)
							0.704915	+	0.709292i	\(0.250985\pi\)
\(17\)	6.32648	−	2.13164i	1.53440	−	0.516999i	0.579765	−	0.814784i	\(-0.303144\pi\)
							0.954633	+	0.297785i	\(0.0962479\pi\)
\(19\)	0.309375	−	1.88710i	0.0709754	−	0.432931i	−0.927305	−	0.374307i	\(-0.877881\pi\)
							0.998280	−	0.0586235i	\(-0.0186712\pi\)
\(23\)	−0.787649	+	2.83686i	−0.164236	+	0.591525i	0.834917	+	0.550376i	\(0.185516\pi\)
							−0.999153	+	0.0411492i	\(0.986898\pi\)
\(29\)	2.05174	−	1.23449i	0.380998	−	0.229239i	−0.312152	−	0.950032i	\(-0.601050\pi\)
							0.693150	+	0.720793i	\(0.256222\pi\)
\(31\)	−0.933639	−	5.69495i	−0.167687	−	1.02284i	−0.927807	−	0.373061i	\(-0.878308\pi\)
							0.760120	−	0.649782i	\(-0.225140\pi\)
\(37\)	0.237912	+	4.38804i	0.0391126	+	0.721389i	0.950668	+	0.310211i	\(0.100400\pi\)
							−0.911555	+	0.411178i	\(0.865118\pi\)
\(41\)	−0.159931	−	0.576019i	−0.0249770	−	0.0899591i	0.950011	−	0.312216i	\(-0.101071\pi\)
							−0.974988	+	0.222257i	\(0.928658\pi\)
\(43\)	2.72864	−	4.02444i	0.416114	−	0.613722i	−0.560647	−	0.828055i	\(-0.689448\pi\)
							0.976761	+	0.214333i	\(0.0687579\pi\)
\(47\)	10.0745	−	1.09567i	1.46952	−	0.159820i	0.661929	−	0.749567i	\(-0.269738\pi\)
							0.807588	+	0.589747i	\(0.200773\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.b.139.3	✓	140
3.2	odd	2		531.2.i.b.316.3		140
59.45	even	29	inner	177.2.e.b.163.3	yes	140
177.104	odd	58		531.2.i.b.163.3		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.b.139.3	✓	140	1.1	even	1	trivial
177.2.e.b.163.3	yes	140	59.45	even	29	inner
531.2.i.b.163.3		140	177.104	odd	58
531.2.i.b.316.3		140	3.2	odd	2