Embedded newform 177.3.g.a.124.20

• Label: 177.3.g.a.124.20
• 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			124.20
Character	\(\chi\)	\(=\)	177.124
Dual form			177.3.g.a.10.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.40493 - 1.35665i) q^{2} +(1.72190 - 0.187268i) q^{3} +(6.84907 - 6.48778i) q^{4} +(1.18743 + 1.39795i) q^{5} +(5.60888 - 2.97364i) q^{6} +(-5.85901 + 3.52525i) q^{7} +(8.36297 - 18.0763i) q^{8} +(2.92986 - 0.644911i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q + 40 q^{4} + 8 q^{7} - 60 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.40493	−	1.35665i	1.70246	−	0.678324i	0.703005	−	0.711185i	\(-0.251841\pi\)
							0.999460	+	0.0328610i	\(0.0104619\pi\)
\(3\)	1.72190	−	0.187268i	0.573966	−	0.0624225i
							\(5\)	1.18743	+	1.39795i	0.237486	+	0.279590i	0.867975	−	0.496608i	\(-0.165421\pi\)
−0.630489	+	0.776198i	\(0.717146\pi\)
\(7\)	−5.85901	+	3.52525i	−0.837001	+	0.503607i	−0.868379	−	0.495901i	\(-0.834838\pi\)
							0.0313780	+	0.999508i	\(0.490010\pi\)
\(11\)	−18.1191	+	0.982387i	−1.64719	+	0.0893079i	−0.854205	−	0.519937i	\(-0.825955\pi\)
							−0.792982	+	0.609245i	\(0.791473\pi\)
\(13\)	−2.12858	+	9.67024i	−0.163737	+	0.743864i	0.821157	+	0.570703i	\(0.193329\pi\)
							−0.984893	+	0.173161i	\(0.944602\pi\)
\(17\)	23.9564	+	14.4141i	1.40920	+	0.847886i	0.997972	−	0.0636483i	\(-0.0202736\pi\)
							0.411225	+	0.911534i	\(0.365101\pi\)
\(19\)	−3.55086	−	12.7890i	−0.186887	−	0.673107i	−0.996306	−	0.0858753i	\(-0.972631\pi\)
							0.809419	−	0.587232i	\(-0.199782\pi\)
\(23\)	−21.7336	−	14.7358i	−0.944941	−	0.640686i	−0.0116982	−	0.999932i	\(-0.503724\pi\)
							−0.933243	+	0.359245i	\(0.883034\pi\)
\(29\)	6.39647	−	16.0539i	0.220568	−	0.553584i	−0.776499	−	0.630119i	\(-0.783006\pi\)
							0.997067	+	0.0765347i	\(0.0243856\pi\)
\(31\)	−1.63201	−	0.453126i	−0.0526455	−	0.0146170i	0.241106	−	0.970499i	\(-0.422490\pi\)
							−0.293751	+	0.955882i	\(0.594904\pi\)
\(37\)	10.4600	+	22.6090i	0.282703	+	0.611053i	0.995955	−	0.0898552i	\(-0.0286404\pi\)
							−0.713252	+	0.700908i	\(0.752778\pi\)
\(41\)	29.2958	+	43.2081i	0.714533	+	1.05386i	0.995557	+	0.0941562i	\(0.0300153\pi\)
							−0.281025	+	0.959700i	\(0.590674\pi\)
\(43\)	47.9681	+	2.60076i	1.11554	+	0.0604827i	0.602674	−	0.797987i	\(-0.294102\pi\)
							0.512864	+	0.858470i	\(0.328585\pi\)
\(47\)	−50.0867	−	42.5440i	−1.06567	−	0.905192i	−0.0699488	−	0.997551i	\(-0.522284\pi\)
							−0.995726	+	0.0923584i	\(0.970559\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.g.a.124.20	yes	560
59.10	odd	58	inner	177.3.g.a.10.20	✓	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.g.a.10.20	✓	560	59.10	odd	58	inner
177.3.g.a.124.20	yes	560	1.1	even	1	trivial