Embedded newform 177.2.e.b.16.5

• Label: 177.2.e.b.16.5
• Level: $177$
• Weight: $2$
• Character: 177.16
• 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			16.5
Character	\(\chi\)	\(=\)	177.16
Dual form			177.2.e.b.166.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.16657 + 0.476899i) q^{2} +(0.161782 + 0.986827i) q^{3} +(2.65145 + 1.22669i) q^{4} +(-0.194787 - 0.701560i) q^{5} +(-0.120104 + 2.21519i) q^{6} +(-1.18393 - 1.12147i) q^{7} +(1.62739 + 1.23711i) q^{8} +(-0.947653 + 0.319302i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q + q^{2} + 5 q^{3} - q^{4} + 2 q^{5} - q^{6} - 2 q^{7} - 3 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{2}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.16657	+	0.476899i	1.53200	+	0.337218i	0.899353	−	0.437223i	\(-0.144038\pi\)
							0.632645	+	0.774442i	\(0.281969\pi\)
\(3\)	0.161782	+	0.986827i	0.0934049	+	0.569745i
							\(5\)	−0.194787	−	0.701560i	−0.0871115	−	0.313747i	0.907138	−	0.420832i	\(-0.138262\pi\)
−0.994250	+	0.107085i	\(0.965848\pi\)
\(7\)	−1.18393	−	1.12147i	−0.447482	−	0.423877i	0.430486	−	0.902597i	\(-0.358342\pi\)
							−0.877968	+	0.478720i	\(0.841101\pi\)
\(11\)	−0.481761	+	0.567173i	−0.145256	+	0.171009i	−0.829979	−	0.557795i	\(-0.811648\pi\)
							0.684722	+	0.728804i	\(0.259923\pi\)
\(13\)	−0.536664	−	0.180823i	−0.148844	−	0.0501513i	0.243894	−	0.969802i	\(-0.421575\pi\)
							−0.392737	+	0.919651i	\(0.628472\pi\)
\(17\)	−1.22992	+	1.16504i	−0.298300	+	0.282565i	−0.822180	−	0.569227i	\(-0.807242\pi\)
							0.523880	+	0.851792i	\(0.324484\pi\)
\(19\)	0.0691539	−	0.173563i	0.0158650	−	0.0398181i	−0.920831	−	0.389962i	\(-0.872488\pi\)
							0.936696	+	0.350144i	\(0.113867\pi\)
\(23\)	−0.102366	−	0.0111329i	−0.0213447	−	0.00232138i	0.0974404	−	0.995241i	\(-0.468934\pi\)
							−0.118785	+	0.992920i	\(0.537900\pi\)
\(29\)	−0.835101	+	0.183820i	−0.155074	+	0.0341344i	−0.291829	−	0.956471i	\(-0.594264\pi\)
							0.136755	+	0.990605i	\(0.456333\pi\)
\(31\)	2.94887	+	7.40109i	0.529632	+	1.32928i	0.913857	+	0.406036i	\(0.133089\pi\)
							−0.384225	+	0.923239i	\(0.625531\pi\)
\(37\)	4.73696	−	3.60095i	0.778752	−	0.591992i	−0.138471	−	0.990367i	\(-0.544219\pi\)
							0.917223	+	0.398375i	\(0.130426\pi\)
\(41\)	−1.30653	+	0.142094i	−0.204046	+	0.0221913i	−0.209571	−	0.977793i	\(-0.567207\pi\)
							0.00552538	+	0.999985i	\(0.498241\pi\)
\(43\)	−4.21551	−	4.96289i	−0.642860	−	0.756833i	0.340153	−	0.940370i	\(-0.389521\pi\)
							−0.983013	+	0.183537i	\(0.941245\pi\)
\(47\)	−3.13966	+	11.3080i	−0.457966	+	1.64945i	0.269275	+	0.963063i	\(0.413216\pi\)
							−0.727241	+	0.686382i	\(0.759198\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.b.16.5	✓	140
3.2	odd	2		531.2.i.b.370.1		140
59.48	even	29	inner	177.2.e.b.166.5	yes	140
177.107	odd	58		531.2.i.b.343.1		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.b.16.5	✓	140	1.1	even	1	trivial
177.2.e.b.166.5	yes	140	59.48	even	29	inner
531.2.i.b.343.1		140	177.107	odd	58
531.2.i.b.370.1		140	3.2	odd	2