Embedded newform 177.2.e.b.64.5

• Label: 177.2.e.b.64.5
• Level: $177$
• Weight: $2$
• Character: 177.64
• 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			64.5
Character	\(\chi\)	\(=\)	177.64
Dual form			177.2.e.b.130.5

$q$-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 \(\left(\mathbb{Z}/177\mathbb{Z}\right)^\times\).

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{3}{29}\right)\)	\(1\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.b.64.5	✓	140	1.1	even	1	trivial
177.2.e.b.130.5	yes	140	59.12	even	29	inner
531.2.i.b.64.1		140	3.2	odd	2
531.2.i.b.307.1		140	177.71	odd	58