Embedded newform 177.2.f.a.38.13

• Label: 177.2.f.a.38.13
• Level: $177$
• Weight: $2$
• Character: 177.38
• 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			38.13
Character	\(\chi\)	\(=\)	177.38
Dual form			177.2.f.a.14.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.33854 + 0.805370i) q^{2} +(-1.36238 - 1.06955i) q^{3} +(0.206240 + 0.389009i) q^{4} +(0.277684 - 2.55326i) q^{5} +(-0.962205 - 2.52884i) q^{6} +(2.22081 + 0.748279i) q^{7} +(0.131908 - 2.43291i) q^{8} +(0.712132 + 2.91425i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{39}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.33854	+	0.805370i	0.946488	+	0.569483i	0.903032	−	0.429573i	\(-0.141336\pi\)
							0.0434553	+	0.999055i	\(0.486163\pi\)
\(3\)	−1.36238	−	1.06955i	−0.786568	−	0.617504i
							\(5\)	0.277684	−	2.55326i	0.124184	−	1.14185i	−0.751758	−	0.659439i	\(-0.770794\pi\)
0.875943	−	0.482415i	\(-0.160240\pi\)
\(7\)	2.22081	+	0.748279i	0.839388	+	0.282823i	0.705974	−	0.708238i	\(-0.250509\pi\)
							0.133414	+	0.991060i	\(0.457406\pi\)
\(11\)	−0.00960777	−	0.0141704i	−0.00289685	−	0.00427253i	0.826238	−	0.563321i	\(-0.190477\pi\)
							−0.829135	+	0.559049i	\(0.811166\pi\)
\(13\)	−0.735037	−	0.775969i	−0.203863	−	0.215215i	0.616045	−	0.787711i	\(-0.288734\pi\)
							−0.819908	+	0.572496i	\(0.805975\pi\)
\(17\)	0.403390	+	1.19722i	0.0978364	+	0.290368i	0.985384	−	0.170348i	\(-0.0544892\pi\)
							−0.887548	+	0.460716i	\(0.847593\pi\)
\(19\)	−0.426929	+	2.60415i	−0.0979441	+	0.597433i	0.891557	+	0.452908i	\(0.149613\pi\)
							−0.989501	+	0.144524i	\(0.953835\pi\)
\(23\)	−2.38236	+	8.58047i	−0.496756	+	1.78915i	0.109619	+	0.993974i	\(0.465037\pi\)
							−0.606375	+	0.795179i	\(0.707377\pi\)
\(29\)	0.278052	+	0.462125i	0.0516329	+	0.0858145i	0.881599	−	0.472000i	\(-0.156468\pi\)
							−0.829966	+	0.557814i	\(0.811640\pi\)
\(31\)	−1.33466	+	0.218807i	−0.239713	+	0.0392989i	−0.280441	−	0.959871i	\(-0.590481\pi\)
							0.0407280	+	0.999170i	\(0.487032\pi\)
\(37\)	−8.98479	+	0.487141i	−1.47709	+	0.0800855i	−0.774981	−	0.631984i	\(-0.782241\pi\)
							−0.702109	+	0.712070i	\(0.747758\pi\)
\(41\)	6.38425	−	1.77258i	0.997053	−	0.276830i	0.269588	−	0.962976i	\(-0.413112\pi\)
							0.727465	+	0.686145i	\(0.240699\pi\)
\(43\)	6.76575	+	4.58729i	1.03177	+	0.699556i	0.954730	−	0.297472i	\(-0.0961436\pi\)
							0.0770373	+	0.997028i	\(0.475454\pi\)
\(47\)	12.2856	−	1.33614i	1.79204	−	0.194896i	0.849186	−	0.528094i	\(-0.177093\pi\)
							0.942857	+	0.333198i	\(0.108128\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.38.13	yes	504
3.2	odd	2	inner	177.2.f.a.38.6	yes	504
59.14	odd	58	inner	177.2.f.a.14.6	✓	504
177.14	even	58	inner	177.2.f.a.14.13	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.14.6	✓	504	59.14	odd	58	inner
177.2.f.a.14.13	yes	504	177.14	even	58	inner
177.2.f.a.38.6	yes	504	3.2	odd	2	inner
177.2.f.a.38.13	yes	504	1.1	even	1	trivial