Embedded newform 177.2.e.b.127.5

• Label: 177.2.e.b.127.5
• Level: $177$
• Weight: $2$
• Character: 177.127
• 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			127.5
Character	\(\chi\)	\(=\)	177.127
Dual form			177.2.e.b.46.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.08904 - 1.28211i) q^{2} +(-0.796093 - 0.605174i) q^{3} +(-0.134248 - 0.818877i) q^{4} +(-1.86649 - 3.52057i) q^{5} +(-1.64287 + 0.361624i) q^{6} +(-3.71509 - 0.404041i) q^{7} +(1.68672 + 1.01487i) q^{8} +(0.267528 + 0.963550i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q + q^{2} + 5 q^{3} - q^{4} + 2 q^{5} - q^{6} - 2 q^{7} - 3 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{21}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.08904	−	1.28211i	0.770064	−	0.906590i	−0.227679	−	0.973736i	\(-0.573114\pi\)
							0.997743	+	0.0671465i	\(0.0213895\pi\)
\(3\)	−0.796093	−	0.605174i	−0.459625	−	0.349397i
							\(5\)	−1.86649	−	3.52057i	−0.834720	−	1.57445i	−0.818448	−	0.574580i	\(-0.805165\pi\)
−0.0162713	−	0.999868i	\(-0.505180\pi\)
\(7\)	−3.71509	−	0.404041i	−1.40417	−	0.152713i	−0.625540	−	0.780192i	\(-0.715121\pi\)
							−0.778634	+	0.627479i	\(0.784087\pi\)
\(11\)	5.42657	+	1.82842i	1.63617	+	0.551291i	0.979248	−	0.202666i	\(-0.0649605\pi\)
							0.656924	+	0.753956i	\(0.271857\pi\)
\(13\)	1.34895	−	4.85848i	0.374132	−	1.34750i	−0.501765	−	0.865004i	\(-0.667316\pi\)
							0.875897	−	0.482498i	\(-0.160270\pi\)
\(17\)	0.313949	−	0.0341440i	0.0761439	−	0.00828114i	−0.0699678	−	0.997549i	\(-0.522290\pi\)
							0.146112	+	0.989268i	\(0.453324\pi\)
\(19\)	−0.0262083	+	0.483384i	−0.00601261	+	0.110896i	0.993986	+	0.109503i	\(0.0349260\pi\)
							−0.999999	+	0.00139263i	\(0.999557\pi\)
\(23\)	2.19117	−	1.01375i	0.456891	−	0.211380i	−0.177924	−	0.984044i	\(-0.556938\pi\)
							0.634816	+	0.772664i	\(0.281076\pi\)
\(29\)	−0.0575069	−	0.0677024i	−0.0106788	−	0.0125720i	0.756797	−	0.653650i	\(-0.226763\pi\)
							−0.767476	+	0.641078i	\(0.778487\pi\)
\(31\)	0.229697	+	4.23651i	0.0412547	+	0.760899i	0.943843	+	0.330394i	\(0.107182\pi\)
							−0.902588	+	0.430505i	\(0.858336\pi\)
\(37\)	−2.38839	+	1.43704i	−0.392648	+	0.236249i	−0.698163	−	0.715939i	\(-0.745999\pi\)
							0.305515	+	0.952187i	\(0.401171\pi\)
\(41\)	−2.90804	−	1.34540i	−0.454160	−	0.210117i	0.179453	−	0.983766i	\(-0.442567\pi\)
							−0.633614	+	0.773650i	\(0.718429\pi\)
\(43\)	4.42772	−	1.49187i	0.675221	−	0.227508i	0.0392591	−	0.999229i	\(-0.487500\pi\)
							0.635962	+	0.771721i	\(0.280604\pi\)
\(47\)	−1.44555	+	2.72660i	−0.210855	+	0.397715i	−0.966270	−	0.257532i	\(-0.917091\pi\)
							0.755414	+	0.655247i	\(0.227436\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.b.127.5	yes	140
3.2	odd	2		531.2.i.b.127.1		140
59.46	even	29	inner	177.2.e.b.46.5	✓	140
177.164	odd	58		531.2.i.b.46.1		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.b.46.5	✓	140	59.46	even	29	inner
177.2.e.b.127.5	yes	140	1.1	even	1	trivial
531.2.i.b.46.1		140	177.164	odd	58
531.2.i.b.127.1		140	3.2	odd	2