Embedded newform 177.2.f.a.11.16

• Label: 177.2.f.a.11.16
• Level: $177$
• Weight: $2$
• Character: 177.11
• 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			11.16
Character	\(\chi\)	\(=\)	177.11
Dual form			177.2.f.a.161.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.93938 - 0.426889i) q^{2} +(0.180971 - 1.72257i) q^{3} +(1.76380 - 0.816019i) q^{4} +(2.30785 + 0.640771i) q^{5} +(-0.384375 - 3.41797i) q^{6} +(-2.76285 + 2.61711i) q^{7} +(-0.0894548 + 0.0680018i) q^{8} +(-2.93450 - 0.623472i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{25}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.93938	−	0.426889i	1.37135	−	0.301856i	0.532595	−	0.846370i	\(-0.321217\pi\)
							0.838752	+	0.544514i	\(0.183286\pi\)
\(3\)	0.180971	−	1.72257i	0.104484	−	0.994527i
							\(5\)	2.30785	+	0.640771i	1.03210	+	0.286562i	0.741976	−	0.670427i	\(-0.233889\pi\)
0.290126	+	0.956988i	\(0.406303\pi\)
\(7\)	−2.76285	+	2.61711i	−1.04426	+	0.989176i	−0.999952	−	0.00984151i	\(-0.996867\pi\)
							−0.0443092	+	0.999018i	\(0.514109\pi\)
\(11\)	−2.61946	−	3.08387i	−0.789798	−	0.929822i	0.209045	−	0.977906i	\(-0.432964\pi\)
							−0.998843	+	0.0480839i	\(0.984689\pi\)
\(13\)	1.46676	+	4.35320i	0.406807	+	1.20736i	0.933894	+	0.357550i	\(0.116388\pi\)
							−0.527087	+	0.849811i	\(0.676716\pi\)
\(17\)	2.28496	−	2.41220i	0.554183	−	0.585044i	−0.387363	−	0.921927i	\(-0.626614\pi\)
							0.941547	+	0.336883i	\(0.109373\pi\)
\(19\)	−1.72291	−	4.32417i	−0.395262	−	0.992032i	−0.983103	−	0.183055i	\(-0.941401\pi\)
							0.587841	−	0.808976i	\(-0.299978\pi\)
\(23\)	6.20928	−	0.675300i	1.29472	−	0.140810i	0.565308	−	0.824880i	\(-0.308757\pi\)
							0.729416	+	0.684070i	\(0.239792\pi\)
\(29\)	0.939879	−	4.26992i	0.174531	−	0.792903i	−0.805267	−	0.592913i	\(-0.797978\pi\)
							0.979798	−	0.199991i	\(-0.0640911\pi\)
\(31\)	0.538734	+	0.214651i	0.0967594	+	0.0385525i	0.418016	−	0.908440i	\(-0.362726\pi\)
							−0.321256	+	0.946992i	\(0.604105\pi\)
\(37\)	−1.09687	+	1.44291i	−0.180324	+	0.237212i	−0.877370	−	0.479814i	\(-0.840704\pi\)
							0.697046	+	0.717026i	\(0.254497\pi\)
\(41\)	−1.08982	+	10.0207i	−0.170201	+	1.56497i	0.528109	+	0.849177i	\(0.322901\pi\)
							−0.698310	+	0.715796i	\(0.746064\pi\)
\(43\)	0.713942	+	0.606428i	0.108875	+	0.0924794i	0.700170	−	0.713976i	\(-0.253107\pi\)
							−0.591295	+	0.806455i	\(0.701383\pi\)
\(47\)	−0.678785	−	2.44476i	−0.0990109	−	0.356605i	0.897404	−	0.441211i	\(-0.145451\pi\)
							−0.996415	+	0.0846055i	\(0.973037\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.11.16	yes	504
3.2	odd	2	inner	177.2.f.a.11.3	✓	504
59.43	odd	58	inner	177.2.f.a.161.3	yes	504
177.161	even	58	inner	177.2.f.a.161.16	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.11.3	✓	504	3.2	odd	2	inner
177.2.f.a.11.16	yes	504	1.1	even	1	trivial
177.2.f.a.161.3	yes	504	59.43	odd	58	inner
177.2.f.a.161.16	yes	504	177.161	even	58	inner