Embedded newform 177.2.e.a.7.2

• Label: 177.2.e.a.7.2
• Level: $177$
• Weight: $2$
• Character: 177.7
• 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			7.2
Character	\(\chi\)	\(=\)	177.7
Dual form			177.2.e.a.76.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.408490 - 0.602477i) q^{2} +(0.0541389 - 0.998533i) q^{3} +(0.544161 - 1.36574i) q^{4} +(-2.21596 + 1.02521i) q^{5} +(-0.623709 + 0.375273i) q^{6} +(0.762154 - 2.74503i) q^{7} +(-2.46688 + 0.543002i) q^{8} +(-0.994138 - 0.108119i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - q^{2} - 5 q^{3} - 9 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} - 9 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{9}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.408490	−	0.602477i	−0.288846	−	0.426016i	0.655388	−	0.755293i	\(-0.272505\pi\)
							−0.944233	+	0.329277i	\(0.893195\pi\)
\(3\)	0.0541389	−	0.998533i	0.0312571	−	0.576504i
							\(5\)	−2.21596	+	1.02521i	−0.991006	+	0.458488i	−0.847290	−	0.531131i	\(-0.821767\pi\)
−0.143716	+	0.989619i	\(0.545905\pi\)
\(7\)	0.762154	−	2.74503i	0.288067	−	1.03752i	−0.667948	−	0.744208i	\(-0.732827\pi\)
							0.956015	−	0.293316i	\(-0.0947589\pi\)
\(11\)	2.16687	−	2.05257i	0.653336	−	0.618873i	−0.287481	−	0.957786i	\(-0.592818\pi\)
							0.940818	+	0.338913i	\(0.110059\pi\)
\(13\)	−3.96564	+	0.431290i	−1.09987	+	0.119618i	−0.640006	−	0.768370i	\(-0.721068\pi\)
							−0.459866	+	0.887988i	\(0.652102\pi\)
\(17\)	0.894824	+	3.22287i	0.217027	+	0.781660i	0.989241	+	0.146297i	\(0.0467356\pi\)
							−0.772214	+	0.635363i	\(0.780851\pi\)
\(19\)	3.85311	−	2.92906i	0.883963	−	0.671971i	−0.0613567	−	0.998116i	\(-0.519543\pi\)
							0.945320	+	0.326144i	\(0.105750\pi\)
\(23\)	1.84164	−	3.47369i	0.384008	−	0.724315i	−0.613872	−	0.789406i	\(-0.710389\pi\)
							0.997879	+	0.0650906i	\(0.0207336\pi\)
\(29\)	4.16956	−	6.14964i	0.774268	−	1.14196i	−0.212075	−	0.977253i	\(-0.568022\pi\)
							0.986343	−	0.164706i	\(-0.0526676\pi\)
\(31\)	1.04358	+	0.793307i	0.187432	+	0.142482i	0.694682	−	0.719317i	\(-0.255545\pi\)
							−0.507250	+	0.861799i	\(0.669338\pi\)
\(37\)	11.3919	+	2.50755i	1.87282	+	0.412239i	0.997460	−	0.0712304i	\(-0.0226926\pi\)
							0.875361	+	0.483470i	\(0.160624\pi\)
\(41\)	0.241230	+	0.455008i	0.0376738	+	0.0710603i	0.901634	−	0.432499i	\(-0.142368\pi\)
							−0.863960	+	0.503560i	\(0.832023\pi\)
\(43\)	8.66588	+	8.20876i	1.32153	+	1.25182i	0.945661	+	0.325153i	\(0.105416\pi\)
							0.375873	+	0.926671i	\(0.377343\pi\)
\(47\)	−8.03229	−	3.71613i	−1.17163	−	0.542054i	−0.265111	−	0.964218i	\(-0.585409\pi\)
							−0.906519	+	0.422164i	\(0.861271\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.a.7.2	✓	140
3.2	odd	2		531.2.i.c.361.4		140
59.17	even	29	inner	177.2.e.a.76.2	yes	140
177.17	odd	58		531.2.i.c.253.4		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.a.7.2	✓	140	1.1	even	1	trivial
177.2.e.a.76.2	yes	140	59.17	even	29	inner
531.2.i.c.253.4		140	177.17	odd	58
531.2.i.c.361.4		140	3.2	odd	2