Embedded newform 177.2.e.b.46.5

• Label: 177.2.e.b.46.5
• Level: $177$
• Weight: $2$
• Character: 177.46
• 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			46.5
Character	\(\chi\)	\(=\)	177.46
Dual form			177.2.e.b.127.5

$q$-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 \(\left(\mathbb{Z}/177\mathbb{Z}\right)^\times\).

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{8}{29}\right)\)	\(1\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.b.46.5	✓	140	1.1	even	1	trivial
177.2.e.b.127.5	yes	140	59.9	even	29	inner
531.2.i.b.46.1		140	3.2	odd	2
531.2.i.b.127.1		140	177.68	odd	58