Embedded newform 177.2.f.a.14.6

• Label: 177.2.f.a.14.6
• Level: $177$
• Weight: $2$
• Character: 177.14
• 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			14.6
Character	\(\chi\)	\(=\)	177.14
Dual form			177.2.f.a.38.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.33854 + 0.805370i) q^{2} +(1.72461 - 0.160417i) q^{3} +(0.206240 - 0.389009i) q^{4} +(-0.277684 - 2.55326i) q^{5} +(-2.17925 + 1.60367i) q^{6} +(2.22081 - 0.748279i) q^{7} +(-0.131908 - 2.43291i) q^{8} +(2.94853 - 0.553314i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{19}{58}\right)\)	\(-1\)

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	\(-0.858664\pi\)
							−0.0434553	+	0.999055i	\(0.513837\pi\)
\(3\)	1.72461	−	0.160417i	0.995702	−	0.0926170i
							\(5\)	−0.277684	−	2.55326i	−0.124184	−	1.14185i	−0.875943	−	0.482415i	\(-0.839760\pi\)
0.751758	−	0.659439i	\(-0.229206\pi\)
\(7\)	2.22081	−	0.748279i	0.839388	−	0.282823i	0.133414	−	0.991060i	\(-0.457406\pi\)
							0.705974	+	0.708238i	\(0.250509\pi\)
\(11\)	0.00960777	−	0.0141704i	0.00289685	−	0.00427253i	−0.826238	−	0.563321i	\(-0.809523\pi\)
							0.829135	+	0.559049i	\(0.188834\pi\)
\(13\)	−0.735037	+	0.775969i	−0.203863	+	0.215215i	−0.819908	−	0.572496i	\(-0.805975\pi\)
							0.616045	+	0.787711i	\(0.288734\pi\)
\(17\)	−0.403390	+	1.19722i	−0.0978364	+	0.290368i	−0.985384	−	0.170348i	\(-0.945511\pi\)
							0.887548	+	0.460716i	\(0.152407\pi\)
\(19\)	−0.426929	−	2.60415i	−0.0979441	−	0.597433i	−0.989501	−	0.144524i	\(-0.953835\pi\)
							0.891557	−	0.452908i	\(-0.149613\pi\)
\(23\)	2.38236	+	8.58047i	0.496756	+	1.78915i	0.606375	+	0.795179i	\(0.292623\pi\)
							−0.109619	+	0.993974i	\(0.534963\pi\)
\(29\)	−0.278052	+	0.462125i	−0.0516329	+	0.0858145i	−0.881599	−	0.472000i	\(-0.843532\pi\)
							0.829966	+	0.557814i	\(0.188360\pi\)
\(31\)	−1.33466	−	0.218807i	−0.239713	−	0.0392989i	0.0407280	−	0.999170i	\(-0.487032\pi\)
							−0.280441	+	0.959871i	\(0.590481\pi\)
\(37\)	−8.98479	−	0.487141i	−1.47709	−	0.0800855i	−0.702109	−	0.712070i	\(-0.747758\pi\)
							−0.774981	+	0.631984i	\(0.782241\pi\)
\(41\)	−6.38425	−	1.77258i	−0.997053	−	0.276830i	−0.269588	−	0.962976i	\(-0.586888\pi\)
							−0.727465	+	0.686145i	\(0.759301\pi\)
\(43\)	6.76575	−	4.58729i	1.03177	−	0.699556i	0.0770373	−	0.997028i	\(-0.475454\pi\)
							0.954730	+	0.297472i	\(0.0961436\pi\)
\(47\)	−12.2856	−	1.33614i	−1.79204	−	0.194896i	−0.849186	−	0.528094i	\(-0.822907\pi\)
							−0.942857	+	0.333198i	\(0.891872\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.14.6	✓	504	1.1	even	1	trivial
177.2.f.a.14.13	yes	504	3.2	odd	2	inner
177.2.f.a.38.6	yes	504	177.38	even	58	inner
177.2.f.a.38.13	yes	504	59.38	odd	58	inner