Embedded newform 177.2.f.a.2.5

• Label: 177.2.f.a.2.5
• Level: $177$
• Weight: $2$
• Character: 177.2
• Analytic conductor: $1.413$
• Analytic rank: $0$
• Dimension: $504$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.f (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.41335211578\)
Analytic rank:	\(0\)
Dimension:	\(504\)
Relative dimension:	\(18\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			2.5
Character	\(\chi\)	\(=\)	177.2
Dual form			177.2.f.a.89.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0852373 + 1.57211i) q^{2} +(1.69601 - 0.351511i) q^{3} +(-0.475985 - 0.0517665i) q^{4} +(0.628919 - 1.86657i) q^{5} +(0.408051 + 2.69627i) q^{6} +(-0.690899 - 1.01900i) q^{7} +(-0.387471 + 2.36347i) q^{8} +(2.75288 - 1.19233i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{1}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0852373	+	1.57211i	−0.0602719	+	1.11165i	0.799399	+	0.600801i	\(0.205151\pi\)
							−0.859671	+	0.510848i	\(0.829331\pi\)
\(3\)	1.69601	−	0.351511i	0.979190	−	0.202945i
							\(5\)	0.628919	−	1.86657i	0.281261	−	0.834754i	−0.710159	−	0.704041i	\(-0.751377\pi\)
0.991420	−	0.130713i	\(-0.0417265\pi\)
\(7\)	−0.690899	−	1.01900i	−0.261135	−	0.385146i	0.674407	−	0.738359i	\(-0.264399\pi\)
							−0.935543	+	0.353214i	\(0.885089\pi\)
\(11\)	−0.567693	+	0.124959i	−0.171166	+	0.0376765i	−0.299726	−	0.954025i	\(-0.596895\pi\)
							0.128561	+	0.991702i	\(0.458964\pi\)
\(13\)	−1.39850	+	1.18790i	−0.387875	+	0.329464i	−0.820014	−	0.572343i	\(-0.806035\pi\)
							0.432140	+	0.901807i	\(0.357759\pi\)
\(17\)	−3.78800	−	2.56833i	−0.918726	−	0.622912i	0.00746277	−	0.999972i	\(-0.497625\pi\)
							−0.926189	+	0.377061i	\(0.876935\pi\)
\(19\)	−3.45636	+	6.51938i	−0.792943	+	1.49565i	0.0740099	+	0.997258i	\(0.476420\pi\)
							−0.866952	+	0.498391i	\(0.833924\pi\)
\(23\)	0.916295	−	0.867961i	0.191061	−	0.180982i	−0.586189	−	0.810174i	\(-0.699373\pi\)
							0.777250	+	0.629192i	\(0.216614\pi\)
\(29\)	−5.45088	+	0.295538i	−1.01220	+	0.0548801i	−0.552771	−	0.833333i	\(-0.686430\pi\)
							−0.459432	+	0.888213i	\(0.651947\pi\)
\(31\)	−4.06403	+	2.15461i	−0.729922	+	0.386980i	−0.791538	−	0.611120i	\(-0.790719\pi\)
							0.0616163	+	0.998100i	\(0.480375\pi\)
\(37\)	3.20340	−	0.525170i	0.526635	−	0.0863374i	0.107401	−	0.994216i	\(-0.465747\pi\)
							0.419234	+	0.907878i	\(0.362299\pi\)
\(41\)	5.26784	−	5.56119i	0.822698	−	0.868512i	−0.170451	−	0.985366i	\(-0.554522\pi\)
							0.993149	+	0.116855i	\(0.0372811\pi\)
\(43\)	1.24155	−	5.64044i	0.189335	−	0.860159i	−0.782156	−	0.623083i	\(-0.785880\pi\)
							0.971491	−	0.237076i	\(-0.0761890\pi\)
\(47\)	9.62767	−	3.24394i	1.40434	−	0.473177i	0.487742	−	0.872988i	\(-0.337821\pi\)
							0.916598	+	0.399811i	\(0.130924\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.2.5	✓	504
3.2	odd	2	inner	177.2.f.a.2.14	yes	504
59.30	odd	58	inner	177.2.f.a.89.14	yes	504
177.89	even	58	inner	177.2.f.a.89.5	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.2.5	✓	504	1.1	even	1	trivial
177.2.f.a.2.14	yes	504	3.2	odd	2	inner
177.2.f.a.89.5	yes	504	177.89	even	58	inner
177.2.f.a.89.14	yes	504	59.30	odd	58	inner