Embedded newform 177.2.e.b.79.1

• Label: 177.2.e.b.79.1
• Level: $177$
• Weight: $2$
• Character: 177.79
• 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			79.1
Character	\(\chi\)	\(=\)	177.79
Dual form			177.2.e.b.121.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.13706 - 0.988707i) q^{2} +(0.947653 - 0.319302i) q^{3} +(2.29469 + 2.70152i) q^{4} +(2.84975 - 1.71464i) q^{5} +(-2.34088 - 0.254586i) q^{6} +(0.205959 + 3.79870i) q^{7} +(-0.972984 - 3.50437i) q^{8} +(0.796093 - 0.605174i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q + q^{2} + 5 q^{3} - q^{4} + 2 q^{5} - q^{6} - 2 q^{7} - 3 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{4}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.13706	−	0.988707i	−1.51113	−	0.699122i	−0.522939	−	0.852370i	\(-0.675165\pi\)
							−0.988188	+	0.153248i	\(0.951027\pi\)
\(3\)	0.947653	−	0.319302i	0.547128	−	0.184349i
							\(5\)	2.84975	−	1.71464i	1.27445	−	0.766809i	0.292554	−	0.956249i	\(-0.405495\pi\)
0.981892	+	0.189440i	\(0.0606673\pi\)
\(7\)	0.205959	+	3.79870i	0.0778453	+	1.43577i	0.732975	+	0.680256i	\(0.238131\pi\)
							−0.655129	+	0.755517i	\(0.727386\pi\)
\(11\)	0.766914	+	4.67797i	0.231233	+	1.41046i	0.805889	+	0.592066i	\(0.201688\pi\)
							−0.574656	+	0.818395i	\(0.694864\pi\)
\(13\)	1.32729	+	1.00898i	0.368123	+	0.279840i	0.772838	−	0.634603i	\(-0.218836\pi\)
							−0.404715	+	0.914443i	\(0.632629\pi\)
\(17\)	0.334992	−	6.17856i	0.0812474	−	1.49852i	−0.619435	−	0.785048i	\(-0.712638\pi\)
							0.700683	−	0.713473i	\(-0.252879\pi\)
\(19\)	−0.760634	−	0.720510i	−0.174501	−	0.165296i	0.595464	−	0.803382i	\(-0.296968\pi\)
							−0.769966	+	0.638085i	\(0.779727\pi\)
\(23\)	−6.59327	−	1.45129i	−1.37479	−	0.302614i	−0.534692	−	0.845047i	\(-0.679572\pi\)
							−0.840099	+	0.542433i	\(0.817503\pi\)
\(29\)	1.47482	−	0.682323i	0.273867	−	0.126704i	−0.278149	−	0.960538i	\(-0.589721\pi\)
							0.552016	+	0.833834i	\(0.313859\pi\)
\(31\)	3.36504	−	3.18754i	0.604379	−	0.572498i	−0.323104	−	0.946363i	\(-0.604726\pi\)
							0.927483	+	0.373865i	\(0.121968\pi\)
\(37\)	0.453724	−	1.63417i	0.0745918	−	0.268655i	−0.916845	−	0.399244i	\(-0.869273\pi\)
							0.991437	+	0.130588i	\(0.0416866\pi\)
\(41\)	−5.60800	+	1.23442i	−0.875823	+	0.192783i	−0.630060	−	0.776547i	\(-0.716970\pi\)
							−0.245763	+	0.969330i	\(0.579039\pi\)
\(43\)	−0.677758	+	4.13414i	−0.103357	+	0.630451i	0.883447	+	0.468531i	\(0.155217\pi\)
							−0.986804	+	0.161919i	\(0.948232\pi\)
\(47\)	−11.1732	−	6.72270i	−1.62978	−	0.980606i	−0.974260	−	0.225428i	\(-0.927622\pi\)
							−0.655520	−	0.755178i	\(-0.727550\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.b.79.1	✓	140	1.1	even	1	trivial
177.2.e.b.121.1	yes	140	59.3	even	29	inner
531.2.i.b.298.5		140	177.62	odd	58
531.2.i.b.433.5		140	3.2	odd	2