Embedded newform 177.3.g.a.124.6

• Label: 177.3.g.a.124.6
• Level: $177$
• Weight: $3$
• Character: 177.124
• Analytic conductor: $4.823$
• Analytic rank: $0$
• Dimension: $560$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.82290067918\)
Analytic rank:	\(0\)
Dimension:	\(560\)
Relative dimension:	\(20\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			124.6
Character	\(\chi\)	\(=\)	177.124
Dual form			177.3.g.a.10.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.06048 + 0.820970i) q^{2} +(1.72190 - 0.187268i) q^{3} +(0.667604 - 0.632388i) q^{4} +(4.71743 + 5.55378i) q^{5} +(-3.39419 + 1.79949i) q^{6} +(8.57272 - 5.15803i) q^{7} +(2.86886 - 6.20094i) q^{8} +(2.92986 - 0.644911i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q + 40 q^{4} + 8 q^{7} - 60 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{51}{58}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.06048	+	0.820970i	−1.03024	+	0.410485i	−0.823142	−	0.567835i	\(-0.807781\pi\)
							−0.207098	+	0.978320i	\(0.566402\pi\)
\(3\)	1.72190	−	0.187268i	0.573966	−	0.0624225i
							\(5\)	4.71743	+	5.55378i	0.943485	+	1.11076i	0.993706	+	0.112018i	\(0.0357316\pi\)
−0.0502209	+	0.998738i	\(0.515993\pi\)
\(7\)	8.57272	−	5.15803i	1.22467	−	0.736862i	0.251706	−	0.967804i	\(-0.419008\pi\)
							0.972968	+	0.230942i	\(0.0741808\pi\)
\(11\)	−9.85322	+	0.534226i	−0.895747	+	0.0485660i	−0.496261	−	0.868173i	\(-0.665294\pi\)
							−0.399486	+	0.916739i	\(0.630811\pi\)
\(13\)	0.252568	−	1.14743i	0.0194283	−	0.0882638i	−0.965909	−	0.258883i	\(-0.916646\pi\)
							0.985337	+	0.170619i	\(0.0545767\pi\)
\(17\)	21.8646	+	13.1555i	1.28615	+	0.773852i	0.983787	−	0.179341i	\(-0.0573965\pi\)
							0.302364	+	0.953192i	\(0.402224\pi\)
\(19\)	−5.87781	−	21.1699i	−0.309358	−	1.11421i	−0.940850	−	0.338825i	\(-0.889971\pi\)
							0.631491	−	0.775383i	\(-0.282443\pi\)
\(23\)	29.3221	+	19.8809i	1.27487	+	0.864385i	0.995404	−	0.0957695i	\(-0.0305312\pi\)
							0.279469	+	0.960155i	\(0.409842\pi\)
\(29\)	−15.5744	+	39.0887i	−0.537048	+	1.34789i	0.370849	+	0.928693i	\(0.379067\pi\)
							−0.907896	+	0.419194i	\(0.862313\pi\)
\(31\)	−16.5627	−	4.59860i	−0.534280	−	0.148342i	−0.0100989	−	0.999949i	\(-0.503215\pi\)
							−0.524181	+	0.851607i	\(0.675628\pi\)
\(37\)	6.01201	+	12.9948i	0.162487	+	0.351210i	0.971668	−	0.236350i	\(-0.0759510\pi\)
							−0.809181	+	0.587559i	\(0.800089\pi\)
\(41\)	−6.55007	−	9.66063i	−0.159758	−	0.235625i	0.739383	−	0.673286i	\(-0.235118\pi\)
							−0.899140	+	0.437660i	\(0.855807\pi\)
\(43\)	−28.9840	−	1.57147i	−0.674047	−	0.0365458i	−0.286061	−	0.958211i	\(-0.592346\pi\)
							−0.387985	+	0.921666i	\(0.626829\pi\)
\(47\)	−63.5192	−	53.9537i	−1.35147	−	1.14795i	−0.975171	−	0.221454i	\(-0.928920\pi\)
							−0.376302	−	0.926497i	\(-0.622804\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.g.a.124.6	yes	560
59.10	odd	58	inner	177.3.g.a.10.6	✓	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.g.a.10.6	✓	560	59.10	odd	58	inner
177.3.g.a.124.6	yes	560	1.1	even	1	trivial