Embedded newform 177.2.e.a.121.1

• Label: 177.2.e.a.121.1
• Level: $177$
• Weight: $2$
• Character: 177.121
• 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			121.1
Character	\(\chi\)	\(=\)	177.121
Dual form			177.2.e.a.79.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.72793 + 0.799427i) q^{2} +(-0.947653 - 0.319302i) q^{3} +(1.05189 - 1.23839i) q^{4} +(-1.39134 - 0.837139i) q^{5} +(1.89274 - 0.205848i) q^{6} +(-0.142290 + 2.62438i) q^{7} +(0.191090 - 0.688244i) q^{8} +(0.796093 + 0.605174i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - q^{2} - 5 q^{3} - 9 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} - 9 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{25}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.72793	+	0.799427i	−1.22183	+	0.565280i	−0.921434	−	0.388534i	\(-0.872982\pi\)
							−0.300398	+	0.953814i	\(0.597120\pi\)
\(3\)	−0.947653	−	0.319302i	−0.547128	−	0.184349i
							\(5\)	−1.39134	−	0.837139i	−0.622224	−	0.374380i	0.169258	−	0.985572i	\(-0.445863\pi\)
−0.791482	+	0.611192i	\(0.790690\pi\)
\(7\)	−0.142290	+	2.62438i	−0.0537804	+	0.991921i	0.839766	+	0.542949i	\(0.182692\pi\)
							−0.893546	+	0.448972i	\(0.851790\pi\)
\(11\)	0.854517	−	5.21232i	0.257647	−	1.57157i	−0.468140	−	0.883655i	\(-0.655076\pi\)
							0.725786	−	0.687920i	\(-0.241476\pi\)
\(13\)	4.55434	−	3.46212i	1.26315	−	0.960220i	0.263158	−	0.964753i	\(-0.415236\pi\)
							0.999990	+	0.00453321i	\(0.00144297\pi\)
\(17\)	0.344474	+	6.35345i	0.0835472	+	1.54094i	0.676949	+	0.736030i	\(0.263302\pi\)
							−0.593401	+	0.804907i	\(0.702215\pi\)
\(19\)	4.06852	−	3.85390i	0.933382	−	0.884146i	−0.0600938	−	0.998193i	\(-0.519140\pi\)
							0.993475	+	0.114047i	\(0.0363814\pi\)
\(23\)	6.63682	−	1.46087i	1.38387	−	0.304613i	0.540229	−	0.841518i	\(-0.318338\pi\)
							0.843643	+	0.536905i	\(0.180407\pi\)
\(29\)	−0.602619	−	0.278801i	−0.111903	−	0.0517721i	0.363139	−	0.931735i	\(-0.381705\pi\)
							−0.475042	+	0.879963i	\(0.657567\pi\)
\(31\)	−7.43368	−	7.04155i	−1.33513	−	1.26470i	−0.937867	−	0.346995i	\(-0.887202\pi\)
							−0.397261	−	0.917706i	\(-0.630039\pi\)
\(37\)	0.215223	+	0.775163i	0.0353825	+	0.127436i	0.979132	−	0.203225i	\(-0.0651422\pi\)
							−0.943750	+	0.330661i	\(0.892728\pi\)
\(41\)	1.27550	+	0.280760i	0.199200	+	0.0438473i	0.313450	−	0.949605i	\(-0.398515\pi\)
							−0.114250	+	0.993452i	\(0.536446\pi\)
\(43\)	0.229197	+	1.39804i	0.0349522	+	0.213199i	0.998528	−	0.0542358i	\(-0.0172723\pi\)
							−0.963576	+	0.267435i	\(0.913824\pi\)
\(47\)	0.00247183	−	0.00148725i	0.000360554	−	0.000216938i	−0.515374	−	0.856966i	\(-0.672347\pi\)
							0.515734	+	0.856749i	\(0.327519\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.a.121.1	yes	140
3.2	odd	2		531.2.i.c.298.5		140
59.20	even	29	inner	177.2.e.a.79.1	✓	140
177.20	odd	58		531.2.i.c.433.5		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.a.79.1	✓	140	59.20	even	29	inner
177.2.e.a.121.1	yes	140	1.1	even	1	trivial
531.2.i.c.298.5		140	3.2	odd	2
531.2.i.c.433.5		140	177.20	odd	58