Embedded newform 177.2.e.b.130.5

• Label: 177.2.e.b.130.5
• Level: $177$
• Weight: $2$
• Character: 177.130
• 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			130.5
Character	\(\chi\)	\(=\)	177.130
Dual form			177.2.e.b.64.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.91605 - 0.645591i) q^{2} +(0.856857 - 0.515554i) q^{3} +(1.66226 - 1.26362i) q^{4} +(0.0203200 + 0.0509993i) q^{5} +(1.30894 - 1.54100i) q^{6} +(-3.48406 - 1.61190i) q^{7} +(0.0998732 - 0.147302i) q^{8} +(0.468408 - 0.883512i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q + q^{2} + 5 q^{3} - q^{4} + 2 q^{5} - q^{6} - 2 q^{7} - 3 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{26}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.91605	−	0.645591i	1.35485	−	0.456502i	0.454142	−	0.890929i	\(-0.349946\pi\)
							0.900707	+	0.434427i	\(0.143049\pi\)
\(3\)	0.856857	−	0.515554i	0.494707	−	0.297655i
							\(5\)	0.0203200	+	0.0509993i	0.00908738	+	0.0228076i	0.933450	−	0.358706i	\(-0.116782\pi\)
−0.924363	+	0.381514i	\(0.875403\pi\)
\(7\)	−3.48406	−	1.61190i	−1.31685	−	0.609241i	−0.369439	−	0.929255i	\(-0.620450\pi\)
							−0.947413	+	0.320014i	\(0.896312\pi\)
\(11\)	0.695053	+	2.50335i	0.209566	+	0.754789i	0.991351	+	0.131240i	\(0.0418960\pi\)
							−0.781784	+	0.623549i	\(0.785690\pi\)
\(13\)	2.54501	+	4.80040i	0.705860	+	1.33139i	0.933827	+	0.357724i	\(0.116447\pi\)
							−0.227968	+	0.973669i	\(0.573208\pi\)
\(17\)	0.787366	−	0.364274i	0.190964	−	0.0883495i	−0.322077	−	0.946714i	\(-0.604381\pi\)
							0.513041	+	0.858364i	\(0.328519\pi\)
\(19\)	−2.36917	−	0.521495i	−0.543526	−	0.119639i	−0.0652831	−	0.997867i	\(-0.520795\pi\)
							−0.478243	+	0.878228i	\(0.658726\pi\)
\(23\)	−1.35479	−	8.26386i	−0.282493	−	1.72313i	−0.622421	−	0.782682i	\(-0.713851\pi\)
							0.339928	−	0.940451i	\(-0.389597\pi\)
\(29\)	−6.21213	−	2.09311i	−1.15356	−	0.388681i	−0.323433	−	0.946251i	\(-0.604837\pi\)
							−0.830130	+	0.557570i	\(0.811734\pi\)
\(31\)	−0.834606	+	0.183711i	−0.149900	+	0.0329954i	−0.289286	−	0.957243i	\(-0.593418\pi\)
							0.139387	+	0.990238i	\(0.455487\pi\)
\(37\)	0.774310	+	1.14202i	0.127296	+	0.187747i	0.885996	−	0.463692i	\(-0.153476\pi\)
							−0.758700	+	0.651440i	\(0.774165\pi\)
\(41\)	0.402216	−	2.45341i	0.0628156	−	0.383158i	−0.936644	−	0.350283i	\(-0.886085\pi\)
							0.999459	−	0.0328751i	\(-0.0104664\pi\)
\(43\)	−2.51249	+	9.04919i	−0.383152	+	1.37999i	0.480984	+	0.876729i	\(0.340280\pi\)
							−0.864136	+	0.503259i	\(0.832134\pi\)
\(47\)	2.80371	−	7.03678i	0.408963	−	1.02642i	−0.569866	−	0.821738i	\(-0.693005\pi\)
							0.978829	−	0.204682i	\(-0.0656159\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.b.130.5	yes	140
3.2	odd	2		531.2.i.b.307.1		140
59.5	even	29	inner	177.2.e.b.64.5	✓	140
177.5	odd	58		531.2.i.b.64.1		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.b.64.5	✓	140	59.5	even	29	inner
177.2.e.b.130.5	yes	140	1.1	even	1	trivial
531.2.i.b.64.1		140	177.5	odd	58
531.2.i.b.307.1		140	3.2	odd	2