Embedded newform 177.2.e.b.16.3

• Label: 177.2.e.b.16.3
• 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			16.3
Character	\(\chi\)	\(=\)	177.16
Dual form			177.2.e.b.166.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0555609 - 0.0122299i) q^{2} +(0.161782 + 0.986827i) q^{3} +(-1.81221 - 0.838419i) q^{4} +(-1.03314 - 3.72102i) q^{5} +(0.00308001 - 0.0568075i) q^{6} +(-1.19345 - 1.13050i) q^{7} +(0.181015 + 0.137604i) q^{8} +(-0.947653 + 0.319302i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q + q^{2} + 5 q^{3} - q^{4} + 2 q^{5} - q^{6} - 2 q^{7} - 3 q^{8} - 5 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0555609	−	0.0122299i	−0.0392875	−	0.00864782i	0.195283	−	0.980747i	\(-0.437437\pi\)
							−0.234571	+	0.972099i	\(0.575368\pi\)
\(3\)	0.161782	+	0.986827i	0.0934049	+	0.569745i
							\(5\)	−1.03314	−	3.72102i	−0.462033	−	1.66409i	−0.716947	−	0.697127i	\(-0.754461\pi\)
0.254915	−	0.966963i	\(-0.417953\pi\)
\(7\)	−1.19345	−	1.13050i	−0.451083	−	0.427288i	0.428130	−	0.903717i	\(-0.359172\pi\)
							−0.879213	+	0.476429i	\(0.841931\pi\)
\(11\)	0.114362	−	0.134638i	0.0344815	−	0.0405948i	−0.744651	−	0.667454i	\(-0.767384\pi\)
							0.779132	+	0.626859i	\(0.215660\pi\)
\(13\)	−3.49715	−	1.17833i	−0.969935	−	0.326809i	−0.210646	−	0.977562i	\(-0.567557\pi\)
							−0.759289	+	0.650753i	\(0.774453\pi\)
\(17\)	4.56731	−	4.32638i	1.10773	−	1.04930i	0.109264	−	0.994013i	\(-0.465151\pi\)
							0.998470	−	0.0552889i	\(-0.0176080\pi\)
\(19\)	0.115014	−	0.288662i	0.0263859	−	0.0662237i	−0.915193	−	0.403015i	\(-0.867962\pi\)
							0.941579	+	0.336791i	\(0.109342\pi\)
\(23\)	6.51437	+	0.708480i	1.35834	+	0.147728i	0.758124	−	0.652110i	\(-0.226116\pi\)
							0.600215	+	0.799838i	\(0.295082\pi\)
\(29\)	−4.01339	+	0.883414i	−0.745268	+	0.164046i	−0.571341	−	0.820713i	\(-0.693577\pi\)
							−0.173927	+	0.984759i	\(0.555646\pi\)
\(31\)	−1.57163	−	3.94450i	−0.282274	−	0.708454i	−0.999927	−	0.0120536i	\(-0.996163\pi\)
							0.717653	−	0.696400i	\(-0.245216\pi\)
\(37\)	−6.92517	+	5.26438i	−1.13849	+	0.865458i	−0.992307	−	0.123799i	\(-0.960492\pi\)
							−0.146184	+	0.989257i	\(0.546699\pi\)
\(41\)	5.87060	−	0.638467i	0.916834	−	0.0997117i	0.362479	−	0.931992i	\(-0.381930\pi\)
							0.554355	+	0.832280i	\(0.312965\pi\)
\(43\)	4.42390	+	5.20821i	0.674638	+	0.794245i	0.987845	−	0.155439i	\(-0.0496794\pi\)
							−0.313207	+	0.949685i	\(0.601403\pi\)
\(47\)	0.868365	−	3.12757i	0.126664	−	0.456203i	−0.872829	−	0.488025i	\(-0.837717\pi\)
							0.999494	+	0.0318226i	\(0.0101312\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.b.16.3	✓	140
3.2	odd	2		531.2.i.b.370.3		140
59.48	even	29	inner	177.2.e.b.166.3	yes	140
177.107	odd	58		531.2.i.b.343.3		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.b.16.3	✓	140	1.1	even	1	trivial
177.2.e.b.166.3	yes	140	59.48	even	29	inner
531.2.i.b.343.3		140	177.107	odd	58
531.2.i.b.370.3		140	3.2	odd	2