Embedded newform 177.2.e.a.19.1

• Label: 177.2.e.a.19.1
• Level: $177$
• Weight: $2$
• Character: 177.19
• 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			19.1
Character	\(\chi\)	\(=\)	177.19
Dual form			177.2.e.a.28.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.07727 + 2.03195i) q^{2} +(-0.725995 + 0.687699i) q^{3} +(-1.84595 - 2.72257i) q^{4} +(-3.23491 + 0.712057i) q^{5} +(-0.615278 - 2.21603i) q^{6} +(1.31099 - 0.996592i) q^{7} +(2.94796 - 0.320610i) q^{8} +(0.0541389 - 0.998533i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - q^{2} - 5 q^{3} - 9 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} - 9 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{19}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.07727	+	2.03195i	−0.761748	+	1.43681i	0.133411	+	0.991061i	\(0.457407\pi\)
							−0.895159	+	0.445748i	\(0.852938\pi\)
\(3\)	−0.725995	+	0.687699i	−0.419154	+	0.397043i
							\(5\)	−3.23491	+	0.712057i	−1.44670	+	0.318442i	−0.867781	−	0.496947i	\(-0.834454\pi\)
−0.578914	+	0.815388i	\(0.696523\pi\)
\(7\)	1.31099	−	0.996592i	0.495509	−	0.376676i	−0.327367	−	0.944897i	\(-0.606161\pi\)
							0.822876	+	0.568221i	\(0.192368\pi\)
\(11\)	−0.922770	−	2.31598i	−0.278226	−	0.698293i	−0.999981	−	0.00614541i	\(-0.998044\pi\)
							0.721756	−	0.692148i	\(-0.243335\pi\)
\(13\)	0.119964	+	2.21261i	0.0332720	+	0.613666i	0.966858	+	0.255314i	\(0.0821787\pi\)
							−0.933586	+	0.358353i	\(0.883339\pi\)
\(17\)	−2.34147	−	1.77994i	−0.567889	−	0.431698i	0.281427	−	0.959583i	\(-0.409192\pi\)
							−0.849316	+	0.527885i	\(0.822985\pi\)
\(19\)	−6.93750	+	2.33752i	−1.59157	+	0.536263i	−0.969161	−	0.246428i	\(-0.920743\pi\)
							−0.622411	+	0.782691i	\(0.713847\pi\)
\(23\)	2.37337	−	1.42801i	0.494883	−	0.297761i	−0.246148	−	0.969232i	\(-0.579165\pi\)
							0.741031	+	0.671471i	\(0.234337\pi\)
\(29\)	1.14256	+	2.15509i	0.212168	+	0.400191i	0.966639	−	0.256142i	\(-0.0824515\pi\)
							−0.754471	+	0.656333i	\(0.772107\pi\)
\(31\)	−7.57979	−	2.55393i	−1.36137	−	0.458699i	−0.458532	−	0.888678i	\(-0.651624\pi\)
							−0.902839	+	0.429979i	\(0.858521\pi\)
\(37\)	−9.97984	−	1.08537i	−1.64068	−	0.178434i	−0.759264	−	0.650783i	\(-0.774441\pi\)
							−0.881412	+	0.472349i	\(0.843406\pi\)
\(41\)	−9.34133	−	5.62049i	−1.45887	−	0.877773i	−0.458928	−	0.888474i	\(-0.651766\pi\)
							−0.999943	+	0.0107003i	\(0.996594\pi\)
\(43\)	−2.62534	+	6.58910i	−0.400360	+	1.00483i	0.581215	+	0.813750i	\(0.302578\pi\)
							−0.981575	+	0.191078i	\(0.938802\pi\)
\(47\)	6.67879	+	1.47011i	0.974201	+	0.214438i	0.673424	−	0.739256i	\(-0.264823\pi\)
							0.300777	+	0.953694i	\(0.402754\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.a.19.1	✓	140
3.2	odd	2		531.2.i.c.19.5		140
59.28	even	29	inner	177.2.e.a.28.1	yes	140
177.146	odd	58		531.2.i.c.28.5		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.a.19.1	✓	140	1.1	even	1	trivial
177.2.e.a.28.1	yes	140	59.28	even	29	inner
531.2.i.c.19.5		140	3.2	odd	2
531.2.i.c.28.5		140	177.146	odd	58