Embedded newform 177.2.e.a.133.2

• Label: 177.2.e.a.133.2
• Level: $177$
• Weight: $2$
• Character: 177.133
• 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			133.2
Character	\(\chi\)	\(=\)	177.133
Dual form			177.2.e.a.4.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38483 + 0.150610i) q^{2} +(0.647386 + 0.762162i) q^{3} +(-0.0581619 + 0.0128024i) q^{4} +(1.49641 - 1.13754i) q^{5} +(-1.01131 - 0.957965i) q^{6} +(-1.93379 - 4.85346i) q^{7} +(2.71878 - 0.916062i) q^{8} +(-0.161782 + 0.986827i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - q^{2} - 5 q^{3} - 9 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} - 9 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{28}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.38483	+	0.150610i	−0.979225	+	0.106497i	−0.583710	−	0.811962i	\(-0.698399\pi\)
							−0.395515	+	0.918459i	\(0.629434\pi\)
\(3\)	0.647386	+	0.762162i	0.373769	+	0.440034i
							\(5\)	1.49641	−	1.13754i	0.669216	−	0.508725i	−0.214515	−	0.976721i	\(-0.568817\pi\)
0.883731	+	0.467996i	\(0.155024\pi\)
\(7\)	−1.93379	−	4.85346i	−0.730906	−	1.83443i	−0.512795	−	0.858511i	\(-0.671390\pi\)
							−0.218110	−	0.975924i	\(-0.569989\pi\)
\(11\)	4.37666	+	2.02486i	1.31961	+	0.610517i	0.948118	−	0.317918i	\(-0.102984\pi\)
							0.371494	+	0.928435i	\(0.378846\pi\)
\(13\)	0.300503	+	1.83299i	0.0833445	+	0.508379i	0.995200	+	0.0978667i	\(0.0312019\pi\)
							−0.911855	+	0.410512i	\(0.865350\pi\)
\(17\)	0.912111	−	2.28923i	0.221219	−	0.555219i	−0.775918	−	0.630834i	\(-0.782713\pi\)
							0.997137	+	0.0756154i	\(0.0240921\pi\)
\(19\)	3.76355	−	5.55083i	0.863418	−	1.27345i	−0.0971124	−	0.995273i	\(-0.530961\pi\)
							0.960531	−	0.278174i	\(-0.0897290\pi\)
\(23\)	0.0327353	+	0.603767i	0.00682578	+	0.125894i	0.999960	+	0.00891580i	\(0.00283803\pi\)
							−0.993134	+	0.116978i	\(0.962679\pi\)
\(29\)	0.0486422	+	0.00529016i	0.00903264	+	0.000982358i	0.112634	−	0.993637i	\(-0.464071\pi\)
							−0.103602	+	0.994619i	\(0.533037\pi\)
\(31\)	−0.180303	−	0.265927i	−0.0323834	−	0.0477619i	0.811153	−	0.584834i	\(-0.198841\pi\)
							−0.843536	+	0.537072i	\(0.819530\pi\)
\(37\)	−7.00462	−	2.36013i	−1.15155	−	0.388003i	−0.322169	−	0.946682i	\(-0.604412\pi\)
							−0.829384	+	0.558679i	\(0.811308\pi\)
\(41\)	−0.238995	+	4.40800i	−0.0373247	+	0.688414i	0.918669	+	0.395028i	\(0.129265\pi\)
							−0.955994	+	0.293386i	\(0.905218\pi\)
\(43\)	4.90321	−	2.26847i	0.747733	−	0.345938i	−0.00874112	−	0.999962i	\(-0.502782\pi\)
							0.756474	+	0.654024i	\(0.226920\pi\)
\(47\)	2.52934	+	1.92275i	0.368942	+	0.280462i	0.773174	−	0.634194i	\(-0.218668\pi\)
							−0.404232	+	0.914657i	\(0.632461\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.a.133.2	yes	140
3.2	odd	2		531.2.i.c.487.4		140
59.4	even	29	inner	177.2.e.a.4.2	✓	140
177.122	odd	58		531.2.i.c.181.4		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.a.4.2	✓	140	59.4	even	29	inner
177.2.e.a.133.2	yes	140	1.1	even	1	trivial
531.2.i.c.181.4		140	177.122	odd	58
531.2.i.c.487.4		140	3.2	odd	2