Embedded newform 177.2.f.a.2.12

• Label: 177.2.f.a.2.12
• 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			2.12
Character	\(\chi\)	\(=\)	177.2
Dual form			177.2.f.a.89.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0596516 - 1.10021i) q^{2} +(-1.51993 + 0.830556i) q^{3} +(0.781375 + 0.0849796i) q^{4} +(0.182962 - 0.543013i) q^{5} +(0.823119 + 1.72178i) q^{6} +(1.24114 + 1.83055i) q^{7} +(0.496616 - 3.02923i) q^{8} +(1.62035 - 2.52477i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0596516	−	1.10021i	0.0421800	−	0.777965i	−0.898563	−	0.438845i	\(-0.855388\pi\)
							0.940743	−	0.339120i	\(-0.110129\pi\)
\(3\)	−1.51993	+	0.830556i	−0.877530	+	0.479521i
							\(5\)	0.182962	−	0.543013i	0.0818233	−	0.242843i	−0.898931	−	0.438091i	\(-0.855655\pi\)
0.980754	+	0.195248i	\(0.0625513\pi\)
\(7\)	1.24114	+	1.83055i	0.469108	+	0.691883i	0.986481	−	0.163878i	\(-0.0524002\pi\)
							−0.517373	+	0.855760i	\(0.673090\pi\)
\(11\)	5.62879	−	1.23899i	1.69714	−	0.373570i	0.742159	−	0.670223i	\(-0.233802\pi\)
							0.954985	+	0.296654i	\(0.0958707\pi\)
\(13\)	−1.65395	+	1.40488i	−0.458724	+	0.389644i	−0.846738	−	0.532010i	\(-0.821437\pi\)
							0.388014	+	0.921653i	\(0.373161\pi\)
\(17\)	−1.94709	−	1.32016i	−0.472240	−	0.320187i	0.301747	−	0.953388i	\(-0.402430\pi\)
							−0.773987	+	0.633201i	\(0.781740\pi\)
\(19\)	1.42987	−	2.69702i	0.328035	−	0.618740i	−0.663560	−	0.748123i	\(-0.730955\pi\)
							0.991595	+	0.129384i	\(0.0412999\pi\)
\(23\)	−3.51309	+	3.32778i	−0.732530	+	0.693890i	−0.960033	−	0.279888i	\(-0.909703\pi\)
							0.227502	+	0.973778i	\(0.426944\pi\)
\(29\)	−6.04680	+	0.327848i	−1.12286	+	0.0608798i	−0.606202	−	0.795311i	\(-0.707308\pi\)
							−0.516660	+	0.856190i	\(0.672825\pi\)
\(31\)	−1.61584	+	0.856662i	−0.290213	+	0.153861i	−0.607154	−	0.794584i	\(-0.707689\pi\)
							0.316941	+	0.948445i	\(0.397344\pi\)
\(37\)	−0.307783	+	0.0504585i	−0.0505992	+	0.00829532i	−0.187028	−	0.982355i	\(-0.559886\pi\)
							0.136429	+	0.990650i	\(0.456437\pi\)
\(41\)	−7.48225	+	7.89892i	−1.16853	+	1.23360i	−0.201350	+	0.979519i	\(0.564533\pi\)
							−0.967182	+	0.254084i	\(0.918226\pi\)
\(43\)	0.127755	−	0.580398i	0.0194825	−	0.0885099i	−0.965876	−	0.259005i	\(-0.916606\pi\)
							0.985359	+	0.170495i	\(0.0545366\pi\)
\(47\)	−1.56406	+	0.526993i	−0.228141	+	0.0768698i	−0.431051	−	0.902328i	\(-0.641857\pi\)
							0.202909	+	0.979198i	\(0.434960\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.2.12	yes	504
3.2	odd	2	inner	177.2.f.a.2.7	✓	504
59.30	odd	58	inner	177.2.f.a.89.7	yes	504
177.89	even	58	inner	177.2.f.a.89.12	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.2.7	✓	504	3.2	odd	2	inner
177.2.f.a.2.12	yes	504	1.1	even	1	trivial
177.2.f.a.89.7	yes	504	59.30	odd	58	inner
177.2.f.a.89.12	yes	504	177.89	even	58	inner