Embedded newform 177.2.e.b.19.5

• Label: 177.2.e.b.19.5
• Level: $177$
• Weight: $2$
• Character: 177.19
• 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			19.5
Character	\(\chi\)	\(=\)	177.19
Dual form			177.2.e.b.28.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.09892 - 2.07278i) q^{2} +(0.725995 - 0.687699i) q^{3} +(-1.96641 - 2.90024i) q^{4} +(-2.10012 + 0.462272i) q^{5} +(-0.627639 - 2.26055i) q^{6} +(0.717585 - 0.545494i) q^{7} +(-3.50786 + 0.381503i) q^{8} +(0.0541389 - 0.998533i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q + q^{2} + 5 q^{3} - q^{4} + 2 q^{5} - q^{6} - 2 q^{7} - 3 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{19}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.09892	−	2.07278i	0.777052	−	1.46568i	−0.104974	−	0.994475i	\(-0.533476\pi\)
							0.882026	−	0.471200i	\(-0.156179\pi\)
\(3\)	0.725995	−	0.687699i	0.419154	−	0.397043i
							\(5\)	−2.10012	+	0.462272i	−0.939204	+	0.206734i	−0.658089	−	0.752940i	\(-0.728635\pi\)
−0.281115	+	0.959674i	\(0.590704\pi\)
\(7\)	0.717585	−	0.545494i	0.271222	−	0.206177i	−0.460718	−	0.887546i	\(-0.652408\pi\)
							0.731940	+	0.681369i	\(0.238615\pi\)
\(11\)	1.40865	+	3.53546i	0.424725	+	1.06598i	0.973230	+	0.229833i	\(0.0738179\pi\)
							−0.548505	+	0.836147i	\(0.684803\pi\)
\(13\)	−0.206281	−	3.80462i	−0.0572119	−	1.05521i	−0.876368	−	0.481642i	\(-0.840041\pi\)
							0.819156	−	0.573570i	\(-0.194442\pi\)
\(17\)	6.11677	+	4.64985i	1.48353	+	1.12775i	0.961492	+	0.274834i	\(0.0886228\pi\)
							0.522042	+	0.852920i	\(0.325170\pi\)
\(19\)	1.17308	−	0.395258i	0.269124	−	0.0906783i	−0.181504	−	0.983390i	\(-0.558097\pi\)
							0.450628	+	0.892712i	\(0.351200\pi\)
\(23\)	−3.89892	+	2.34590i	−0.812981	+	0.489155i	−0.860338	−	0.509723i	\(-0.829748\pi\)
							0.0473569	+	0.998878i	\(0.484920\pi\)
\(29\)	3.05551	+	5.76330i	0.567394	+	1.07022i	0.986727	+	0.162386i	\(0.0519190\pi\)
							−0.419334	+	0.907832i	\(0.637736\pi\)
\(31\)	−2.29118	−	0.771990i	−0.411509	−	0.138653i	0.105930	−	0.994374i	\(-0.466218\pi\)
							−0.517439	+	0.855720i	\(0.673115\pi\)
\(37\)	0.583965	+	0.0635101i	0.0960033	+	0.0104410i	0.155995	−	0.987758i	\(-0.450142\pi\)
							−0.0599913	+	0.998199i	\(0.519107\pi\)
\(41\)	−9.60910	−	5.78161i	−1.50069	−	0.902935i	−0.999288	−	0.0377202i	\(-0.987990\pi\)
							−0.501401	−	0.865215i	\(-0.667182\pi\)
\(43\)	−0.0633175	+	0.158915i	−0.00965583	+	0.0242343i	−0.933726	−	0.357990i	\(-0.883462\pi\)
							0.924070	+	0.382224i	\(0.124842\pi\)
\(47\)	−0.570219	−	0.125515i	−0.0831750	−	0.0183082i	0.173188	−	0.984889i	\(-0.444593\pi\)
							−0.256363	+	0.966581i	\(0.582524\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.b.19.5	✓	140
3.2	odd	2		531.2.i.b.19.1		140
59.28	even	29	inner	177.2.e.b.28.5	yes	140
177.146	odd	58		531.2.i.b.28.1		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.b.19.5	✓	140	1.1	even	1	trivial
177.2.e.b.28.5	yes	140	59.28	even	29	inner
531.2.i.b.19.1		140	3.2	odd	2
531.2.i.b.28.1		140	177.146	odd	58