Embedded newform 177.4.f.a.2.12

• Label: 177.4.f.a.2.12
• Level: $177$
• Weight: $4$
• Character: 177.2
• Analytic conductor: $10.443$
• Analytic rank: $0$
• Dimension: $1624$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.f (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.4433380710\)
Analytic rank:	\(0\)
Dimension:	\(1624\)
Relative dimension:	\(58\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			2.12
Character	\(\chi\)	\(=\)	177.2
Dual form			177.4.f.a.89.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.206723 + 3.81279i) q^{2} +(-4.04421 + 3.26257i) q^{3} +(-6.54149 - 0.711430i) q^{4} +(-3.85784 + 11.4497i) q^{5} +(-11.6035 - 16.0941i) q^{6} +(-12.2513 - 18.0693i) q^{7} +(-0.877159 + 5.35043i) q^{8} +(5.71121 - 26.3891i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 21 q^{3} - 278 q^{4} - 29 q^{6} - 42 q^{7} - 25 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.206723	+	3.81279i	−0.0730877	+	1.34802i	0.700190	+	0.713956i	\(0.253098\pi\)
							−0.773278	+	0.634067i	\(0.781384\pi\)
\(3\)	−4.04421	+	3.26257i	−0.778308	+	0.627883i
							\(5\)	−3.85784	+	11.4497i	−0.345056	+	1.02409i	0.624878	+	0.780722i	\(0.285149\pi\)
−0.969934	+	0.243367i	\(0.921748\pi\)
\(7\)	−12.2513	−	18.0693i	−0.661509	−	0.975653i	−0.999406	−	0.0344697i	\(-0.989026\pi\)
							0.337897	−	0.941183i	\(-0.390285\pi\)
\(11\)	−3.29748	+	0.725830i	−0.0903843	+	0.0198951i	−0.259932	−	0.965627i	\(-0.583700\pi\)
							0.169548	+	0.985522i	\(0.445769\pi\)
\(13\)	−1.53064	+	1.30014i	−0.0326556	+	0.0277379i	−0.663566	−	0.748118i	\(-0.730958\pi\)
							0.630910	+	0.775856i	\(0.282682\pi\)
\(17\)	1.33719	+	0.906636i	0.0190774	+	0.0129348i	0.570689	−	0.821167i	\(-0.306676\pi\)
							−0.551611	+	0.834101i	\(0.685987\pi\)
\(19\)	−28.1623	+	53.1197i	−0.340046	+	0.641395i	−0.993277	−	0.115764i	\(-0.963068\pi\)
							0.653231	+	0.757159i	\(0.273413\pi\)
\(23\)	95.7596	−	90.7083i	0.868142	−	0.822347i	−0.117170	−	0.993112i	\(-0.537382\pi\)
							0.985312	+	0.170764i	\(0.0546237\pi\)
\(29\)	−97.0117	+	5.25982i	−0.621194	+	0.0336802i	−0.362050	−	0.932159i	\(-0.617923\pi\)
							−0.259144	+	0.965839i	\(0.583440\pi\)
\(31\)	71.8215	−	38.0774i	0.416114	−	0.220610i	−0.247179	−	0.968970i	\(-0.579503\pi\)
							0.663292	+	0.748360i	\(0.269159\pi\)
\(37\)	−185.207	+	30.3631i	−0.822913	+	0.134910i	−0.558505	−	0.829501i	\(-0.688625\pi\)
							−0.264408	+	0.964411i	\(0.585177\pi\)
\(41\)	−166.875	+	176.168i	−0.635645	+	0.671043i	−0.961635	−	0.274331i	\(-0.911544\pi\)
							0.325990	+	0.945373i	\(0.394302\pi\)
\(43\)	70.3742	−	319.713i	0.249580	−	1.13386i	−0.670433	−	0.741970i	\(-0.733892\pi\)
							0.920014	−	0.391886i	\(-0.128177\pi\)
\(47\)	99.6083	−	33.5619i	0.309135	−	0.104160i	−0.160458	−	0.987043i	\(-0.551297\pi\)
							0.469593	+	0.882883i	\(0.344401\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.f.a.2.12	✓	1624
3.2	odd	2	inner	177.4.f.a.2.47	yes	1624
59.30	odd	58	inner	177.4.f.a.89.47	yes	1624
177.89	even	58	inner	177.4.f.a.89.12	yes	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.f.a.2.12	✓	1624	1.1	even	1	trivial
177.4.f.a.2.47	yes	1624	3.2	odd	2	inner
177.4.f.a.89.12	yes	1624	177.89	even	58	inner
177.4.f.a.89.47	yes	1624	59.30	odd	58	inner