Embedded newform 177.2.e.a.22.2

• Label: 177.2.e.a.22.2
• 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			22.2
Character	\(\chi\)	\(=\)	177.22
Dual form			177.2.e.a.169.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.335352 - 2.04556i) q^{2} +(0.267528 + 0.963550i) q^{3} +(-2.17654 + 0.733362i) q^{4} +(0.939837 - 1.38616i) q^{5} +(1.88128 - 0.870373i) q^{6} +(4.70399 + 1.03543i) q^{7} +(0.288147 + 0.543503i) q^{8} +(-0.856857 + 0.515554i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - q^{2} - 5 q^{3} - 9 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} - 9 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.335352	−	2.04556i	−0.237130	−	1.44643i	−0.789972	−	0.613142i	\(-0.789905\pi\)
							0.552843	−	0.833286i	\(-0.313543\pi\)
\(3\)	0.267528	+	0.963550i	0.154458	+	0.556306i
							\(5\)	0.939837	−	1.38616i	0.420308	−	0.619908i	−0.557308	−	0.830306i	\(-0.688166\pi\)
0.977616	+	0.210398i	\(0.0674761\pi\)
\(7\)	4.70399	+	1.03543i	1.77794	+	0.391355i	0.977980	−	0.208697i	\(-0.0669223\pi\)
							0.799961	+	0.600052i	\(0.204853\pi\)
\(11\)	−1.41839	−	1.07823i	−0.427662	−	0.325100i	0.369104	−	0.929388i	\(-0.379665\pi\)
							−0.796766	+	0.604288i	\(0.793458\pi\)
\(13\)	−3.26031	−	1.96166i	−0.904246	−	0.544067i	−0.0141808	−	0.999899i	\(-0.504514\pi\)
							−0.890066	+	0.455832i	\(0.849342\pi\)
\(17\)	−1.03358	+	0.227509i	−0.250681	+	0.0551790i	−0.338533	−	0.940954i	\(-0.609931\pi\)
							0.0878522	+	0.996134i	\(0.472000\pi\)
\(19\)	3.52321	+	0.383172i	0.808279	+	0.0879057i	0.502922	−	0.864332i	\(-0.332258\pi\)
							0.305358	+	0.952238i	\(0.401224\pi\)
\(23\)	2.75545	−	3.24397i	0.574551	−	0.676414i	−0.395738	−	0.918364i	\(-0.629511\pi\)
							0.970289	+	0.241950i	\(0.0777870\pi\)
\(29\)	−1.14152	+	6.96298i	−0.211975	+	1.29299i	0.639579	+	0.768726i	\(0.279109\pi\)
							−0.851554	+	0.524267i	\(0.824339\pi\)
\(31\)	−7.28939	+	0.792768i	−1.30921	+	0.142385i	−0.735978	−	0.677006i	\(-0.763277\pi\)
							−0.573235	+	0.819391i	\(0.694312\pi\)
\(37\)	1.92944	−	3.63932i	0.317198	−	0.598300i	−0.672727	−	0.739891i	\(-0.734877\pi\)
							0.989925	+	0.141591i	\(0.0452219\pi\)
\(41\)	4.11714	+	4.84707i	0.642989	+	0.756985i	0.983034	−	0.183424i	\(-0.0587181\pi\)
							−0.340045	+	0.940409i	\(0.610442\pi\)
\(43\)	−5.34772	+	4.06523i	−0.815519	+	0.619942i	−0.927523	−	0.373766i	\(-0.878066\pi\)
							0.112004	+	0.993708i	\(0.464273\pi\)
\(47\)	3.53939	+	5.22021i	0.516273	+	0.761446i	0.993065	−	0.117571i	\(-0.0375106\pi\)
							−0.476791	+	0.879016i	\(0.658200\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.a.22.2	✓	140
3.2	odd	2		531.2.i.c.199.4		140
59.51	even	29	inner	177.2.e.a.169.2	yes	140
177.110	odd	58		531.2.i.c.523.4		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.a.22.2	✓	140	1.1	even	1	trivial
177.2.e.a.169.2	yes	140	59.51	even	29	inner
531.2.i.c.199.4		140	3.2	odd	2
531.2.i.c.523.4		140	177.110	odd	58