Embedded newform 177.2.e.b.22.5

• Label: 177.2.e.b.22.5
• 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			22.5
Character	\(\chi\)	\(=\)	177.22
Dual form			177.2.e.b.169.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.302557 + 1.84551i) q^{2} +(-0.267528 - 0.963550i) q^{3} +(-1.41907 + 0.478141i) q^{4} +(1.62093 - 2.39070i) q^{5} +(1.69730 - 0.785256i) q^{6} +(1.15886 + 0.255085i) q^{7} +(0.440222 + 0.830346i) q^{8} +(-0.856857 + 0.515554i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q + q^{2} + 5 q^{3} - q^{4} + 2 q^{5} - q^{6} - 2 q^{7} - 3 q^{8} - 5 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.302557	+	1.84551i	0.213940	+	1.30498i	0.847299	+	0.531117i	\(0.178227\pi\)
							−0.633359	+	0.773858i	\(0.718324\pi\)
\(3\)	−0.267528	−	0.963550i	−0.154458	−	0.556306i
							\(5\)	1.62093	−	2.39070i	0.724903	−	1.06915i	−0.269449	−	0.963015i	\(-0.586841\pi\)
0.994352	−	0.106137i	\(-0.0338482\pi\)
\(7\)	1.15886	+	0.255085i	0.438009	+	0.0964130i	0.428500	−	0.903542i	\(-0.359042\pi\)
							0.00950872	+	0.999955i	\(0.496973\pi\)
\(11\)	3.14625	+	2.39171i	0.948629	+	0.721129i	0.960348	−	0.278802i	\(-0.0899374\pi\)
							−0.0117196	+	0.999931i	\(0.503731\pi\)
\(13\)	−3.86478	−	2.32536i	−1.07190	−	0.644940i	−0.133694	−	0.991023i	\(-0.542684\pi\)
							−0.938204	+	0.346083i	\(0.887512\pi\)
\(17\)	0.853899	−	0.187957i	0.207101	−	0.0455864i	−0.110208	−	0.993909i	\(-0.535152\pi\)
							0.317309	+	0.948322i	\(0.397221\pi\)
\(19\)	−3.75273	−	0.408134i	−0.860935	−	0.0936323i	−0.333010	−	0.942923i	\(-0.608064\pi\)
							−0.527925	+	0.849291i	\(0.677030\pi\)
\(23\)	2.31962	−	2.73087i	0.483674	−	0.569425i	−0.465149	−	0.885232i	\(-0.653999\pi\)
							0.948823	+	0.315807i	\(0.102275\pi\)
\(29\)	−0.671290	+	4.09469i	−0.124655	+	0.760365i	0.848325	+	0.529476i	\(0.177612\pi\)
							−0.972980	+	0.230888i	\(0.925837\pi\)
\(31\)	−5.65169	+	0.614658i	−1.01507	+	0.110396i	−0.600504	−	0.799622i	\(-0.705033\pi\)
							−0.414570	+	0.910018i	\(0.636068\pi\)
\(37\)	−2.58442	+	4.87473i	−0.424876	+	0.801401i	−0.999859	−	0.0167639i	\(-0.994664\pi\)
							0.574984	+	0.818165i	\(0.305008\pi\)
\(41\)	−4.30020	−	5.06259i	−0.671579	−	0.790644i	0.315833	−	0.948815i	\(-0.397716\pi\)
							−0.987412	+	0.158171i	\(0.949440\pi\)
\(43\)	7.31003	−	5.55694i	1.11477	−	0.847426i	0.125293	−	0.992120i	\(-0.460013\pi\)
							0.989476	+	0.144694i	\(0.0462199\pi\)
\(47\)	5.31007	+	7.83177i	0.774554	+	1.14238i	0.986285	+	0.165053i	\(0.0527796\pi\)
							−0.211731	+	0.977328i	\(0.567910\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.b.22.5	✓	140
3.2	odd	2		531.2.i.b.199.1		140
59.51	even	29	inner	177.2.e.b.169.5	yes	140
177.110	odd	58		531.2.i.b.523.1		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.b.22.5	✓	140	1.1	even	1	trivial
177.2.e.b.169.5	yes	140	59.51	even	29	inner
531.2.i.b.199.1		140	3.2	odd	2
531.2.i.b.523.1		140	177.110	odd	58