Embedded newform 177.3.g.a.10.16

• Label: 177.3.g.a.10.16
• Level: $177$
• Weight: $3$
• Character: 177.10
• 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			10.16
Character	\(\chi\)	\(=\)	177.10
Dual form			177.3.g.a.124.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.13762 + 0.851704i) q^{2} +(1.72190 + 0.187268i) q^{3} +(0.940023 + 0.890437i) q^{4} +(4.37246 - 5.14766i) q^{5} +(3.52126 + 1.86685i) q^{6} +(-0.447582 - 0.269301i) q^{7} +(-2.61371 - 5.64945i) q^{8} +(2.92986 + 0.644911i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q + 40 q^{4} + 8 q^{7} - 60 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{7}{58}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.13762	+	0.851704i	1.06881	+	0.425852i	0.837166	−	0.546949i	\(-0.184211\pi\)
							0.231642	+	0.972801i	\(0.425590\pi\)
\(3\)	1.72190	+	0.187268i	0.573966	+	0.0624225i
							\(5\)	4.37246	−	5.14766i	0.874493	−	1.02953i	−0.124809	−	0.992181i	\(-0.539832\pi\)
0.999302	−	0.0373517i	\(-0.0118922\pi\)
\(7\)	−0.447582	−	0.269301i	−0.0639403	−	0.0384716i	0.483225	−	0.875496i	\(-0.339465\pi\)
							−0.547165	+	0.837025i	\(0.684293\pi\)
\(11\)	2.30333	+	0.124883i	0.209394	+	0.0113530i	0.158537	−	0.987353i	\(-0.449322\pi\)
							0.0508563	+	0.998706i	\(0.483805\pi\)
\(13\)	3.27171	+	14.8635i	0.251670	+	1.14335i	0.917654	+	0.397381i	\(0.130081\pi\)
							−0.665984	+	0.745966i	\(0.731988\pi\)
\(17\)	−20.0176	+	12.0442i	−1.17750	+	0.708481i	−0.963331	−	0.268314i	\(-0.913533\pi\)
							−0.214173	+	0.976796i	\(0.568706\pi\)
\(19\)	−6.08850	+	21.9288i	−0.320447	+	1.15415i	0.611408	+	0.791316i	\(0.290604\pi\)
							−0.931855	+	0.362831i	\(0.881810\pi\)
\(23\)	7.90473	−	5.35954i	0.343684	−	0.233023i	−0.377117	−	0.926166i	\(-0.623085\pi\)
							0.720801	+	0.693142i	\(0.243774\pi\)
\(29\)	6.58985	+	16.5393i	0.227236	+	0.570320i	0.997746	−	0.0671086i	\(-0.0213774\pi\)
							−0.770509	+	0.637429i	\(0.779998\pi\)
\(31\)	−10.7556	+	2.98626i	−0.346953	+	0.0963311i	−0.436633	−	0.899640i	\(-0.643829\pi\)
							0.0896801	+	0.995971i	\(0.471416\pi\)
\(37\)	10.2683	−	22.1945i	0.277520	−	0.599850i	−0.717825	−	0.696224i	\(-0.754862\pi\)
							0.995345	+	0.0963733i	\(0.0307243\pi\)
\(41\)	5.39614	−	7.95871i	0.131613	−	0.194115i	−0.756172	−	0.654373i	\(-0.772933\pi\)
							0.887785	+	0.460258i	\(0.152243\pi\)
\(43\)	−39.4487	+	2.13885i	−0.917411	+	0.0497406i	−0.506794	−	0.862067i	\(-0.669170\pi\)
							−0.410617	+	0.911808i	\(0.634687\pi\)
\(47\)	4.88103	−	4.14599i	0.103852	−	0.0882125i	−0.593956	−	0.804497i	\(-0.702435\pi\)
							0.697808	+	0.716285i	\(0.254159\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.g.a.10.16	✓	560
59.6	odd	58	inner	177.3.g.a.124.16	yes	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.g.a.10.16	✓	560	1.1	even	1	trivial
177.3.g.a.124.16	yes	560	59.6	odd	58	inner