Embedded newform 177.2.e.b.22.4

• Label: 177.2.e.b.22.4
• Level: $177$
• Weight: $2$
• Character: 177.22
• 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			22.4
Character	\(\chi\)	\(=\)	177.22
Dual form			177.2.e.b.169.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.276207 + 1.68479i) q^{2} +(-0.267528 - 0.963550i) q^{3} +(-0.866914 + 0.292097i) q^{4} +(-2.40908 + 3.55313i) q^{5} +(1.54948 - 0.716868i) q^{6} +(0.331175 + 0.0728972i) q^{7} +(0.867838 + 1.63692i) q^{8} +(-0.856857 + 0.515554i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q + q^{2} + 5 q^{3} - q^{4} + 2 q^{5} - q^{6} - 2 q^{7} - 3 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{13}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.276207	+	1.68479i	0.195308	+	1.19132i	0.884428	+	0.466677i	\(0.154549\pi\)
							−0.689120	+	0.724647i	\(0.742003\pi\)
\(3\)	−0.267528	−	0.963550i	−0.154458	−	0.556306i
							\(5\)	−2.40908	+	3.55313i	−1.07737	+	1.58901i	−0.306666	+	0.951817i	\(0.599213\pi\)
−0.770707	+	0.637189i	\(0.780097\pi\)
\(7\)	0.331175	+	0.0728972i	0.125172	+	0.0275525i	0.277115	−	0.960837i	\(-0.410622\pi\)
							−0.151942	+	0.988389i	\(0.548553\pi\)
\(11\)	0.00152925	+	0.00116250i	0.000461085	+	0.000350508i	0.605405	−	0.795918i	\(-0.293011\pi\)
							−0.604944	+	0.796268i	\(0.706804\pi\)
\(13\)	1.61209	+	0.969960i	0.447112	+	0.269019i	0.721254	−	0.692671i	\(-0.243566\pi\)
							−0.274142	+	0.961689i	\(0.588394\pi\)
\(17\)	−1.13865	+	0.250635i	−0.276163	+	0.0607880i	−0.350892	−	0.936416i	\(-0.614122\pi\)
							0.0747297	+	0.997204i	\(0.476191\pi\)
\(19\)	3.48773	+	0.379314i	0.800141	+	0.0870205i	0.499048	−	0.866574i	\(-0.333683\pi\)
							0.301093	+	0.953595i	\(0.402649\pi\)
\(23\)	2.65197	−	3.12214i	0.552973	−	0.651010i	−0.412670	−	0.910880i	\(-0.635404\pi\)
							0.965644	+	0.259870i	\(0.0836797\pi\)
\(29\)	1.38624	−	8.45568i	0.257418	−	1.57018i	−0.469171	−	0.883107i	\(-0.655447\pi\)
							0.726589	−	0.687073i	\(-0.241105\pi\)
\(31\)	9.43823	−	1.02647i	1.69516	−	0.184359i	0.791074	−	0.611720i	\(-0.209522\pi\)
							0.904081	+	0.427361i	\(0.140557\pi\)
\(37\)	−3.16402	+	5.96797i	−0.520162	+	0.981129i	0.474725	+	0.880134i	\(0.342547\pi\)
							−0.994887	+	0.100995i	\(0.967797\pi\)
\(41\)	6.48244	+	7.63171i	1.01239	+	1.19187i	0.981325	+	0.192356i	\(0.0616128\pi\)
							0.0310612	+	0.999517i	\(0.490111\pi\)
\(43\)	−1.69305	+	1.28703i	−0.258188	+	0.196270i	−0.726259	−	0.687422i	\(-0.758743\pi\)
							0.468070	+	0.883691i	\(0.344949\pi\)
\(47\)	0.206692	+	0.304848i	0.0301491	+	0.0444666i	0.842465	−	0.538751i	\(-0.181104\pi\)
							−0.812316	+	0.583218i	\(0.801793\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.b.22.4	✓	140
3.2	odd	2		531.2.i.b.199.2		140
59.51	even	29	inner	177.2.e.b.169.4	yes	140
177.110	odd	58		531.2.i.b.523.2		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.b.22.4	✓	140	1.1	even	1	trivial
177.2.e.b.169.4	yes	140	59.51	even	29	inner
531.2.i.b.199.2		140	3.2	odd	2
531.2.i.b.523.2		140	177.110	odd	58