Embedded newform 177.2.e.a.16.5

• Label: 177.2.e.a.16.5
• 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			16.5
Character	\(\chi\)	\(=\)	177.16
Dual form			177.2.e.a.166.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.98298 + 0.436487i) q^{2} +(-0.161782 - 0.986827i) q^{3} +(1.92655 + 0.891316i) q^{4} +(-0.867769 - 3.12542i) q^{5} +(0.109927 - 2.02748i) q^{6} +(2.78359 + 2.63676i) q^{7} +(0.198401 + 0.150821i) q^{8} +(-0.947653 + 0.319302i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - q^{2} - 5 q^{3} - 9 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} - 9 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{2}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.98298	+	0.436487i	1.40218	+	0.308643i	0.850747	−	0.525576i	\(-0.176150\pi\)
							0.551433	+	0.834219i	\(0.314081\pi\)
\(3\)	−0.161782	−	0.986827i	−0.0934049	−	0.569745i
							\(5\)	−0.867769	−	3.12542i	−0.388078	−	1.39773i	−0.857376	−	0.514691i	\(-0.827907\pi\)
0.469298	−	0.883040i	\(-0.344507\pi\)
\(7\)	2.78359	+	2.63676i	1.05210	+	0.996600i	0.999998	−	0.00204172i	\(-0.000649902\pi\)
							0.0521001	+	0.998642i	\(0.483409\pi\)
\(11\)	−3.60006	+	4.23831i	−1.08546	+	1.27790i	−0.127236	+	0.991872i	\(0.540611\pi\)
							−0.958222	+	0.286027i	\(0.907665\pi\)
\(13\)	2.21163	+	0.745186i	0.613397	+	0.206677i	0.608815	−	0.793312i	\(-0.291645\pi\)
							0.00458157	+	0.999990i	\(0.498542\pi\)
\(17\)	2.52133	−	2.38833i	0.611511	−	0.579254i	−0.317980	−	0.948097i	\(-0.603005\pi\)
							0.929492	+	0.368843i	\(0.120246\pi\)
\(19\)	−2.59009	+	6.50063i	−0.594207	+	1.49135i	0.256944	+	0.966426i	\(0.417285\pi\)
							−0.851151	+	0.524921i	\(0.824095\pi\)
\(23\)	3.27871	+	0.356581i	0.683659	+	0.0743524i	0.443356	−	0.896346i	\(-0.353788\pi\)
							0.240303	+	0.970698i	\(0.422753\pi\)
\(29\)	−8.40855	+	1.85086i	−1.56143	+	0.343696i	−0.909867	−	0.414900i	\(-0.863816\pi\)
							−0.651561	+	0.758596i	\(0.725885\pi\)
\(31\)	−2.83848	−	7.12406i	−0.509807	−	1.27952i	−0.928480	−	0.371382i	\(-0.878884\pi\)
							0.418674	−	0.908137i	\(-0.362495\pi\)
\(37\)	5.49971	−	4.18077i	0.904146	−	0.687314i	−0.0460591	−	0.998939i	\(-0.514666\pi\)
							0.950205	+	0.311625i	\(0.100873\pi\)
\(41\)	−0.825332	+	0.0897603i	−0.128895	+	0.0140182i	−0.172339	−	0.985038i	\(-0.555133\pi\)
							0.0434440	+	0.999056i	\(0.486167\pi\)
\(43\)	−0.211125	−	0.248556i	−0.0321963	−	0.0379045i	0.745828	−	0.666138i	\(-0.232054\pi\)
							−0.778025	+	0.628234i	\(0.783778\pi\)
\(47\)	1.20289	−	4.33242i	0.175460	−	0.631948i	−0.822539	−	0.568709i	\(-0.807443\pi\)
							0.997998	−	0.0632394i	\(-0.0201432\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.a.16.5	✓	140
3.2	odd	2		531.2.i.c.370.1		140
59.48	even	29	inner	177.2.e.a.166.5	yes	140
177.107	odd	58		531.2.i.c.343.1		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.a.16.5	✓	140	1.1	even	1	trivial
177.2.e.a.166.5	yes	140	59.48	even	29	inner
531.2.i.c.343.1		140	177.107	odd	58
531.2.i.c.370.1		140	3.2	odd	2