Embedded newform 177.3.h.a.104.9

• Label: 177.3.h.a.104.9
• Level: $177$
• Weight: $3$
• Character: 177.104
• Analytic conductor: $4.823$
• Analytic rank: $0$
• Dimension: $1064$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.82290067918\)
Analytic rank:	\(0\)
Dimension:	\(1064\)
Relative dimension:	\(38\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			104.9
Character	\(\chi\)	\(=\)	177.104
Dual form			177.3.h.a.80.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28794 - 2.14057i) q^{2} +(-2.74380 + 1.21308i) q^{3} +(-1.04962 + 1.97979i) q^{4} +(-0.792588 - 7.28773i) q^{5} +(6.13052 + 4.31093i) q^{6} +(-2.16549 + 0.729637i) q^{7} +(-4.38828 + 0.237926i) q^{8} +(6.05689 - 6.65688i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{24}{29}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.28794	−	2.14057i	−0.643969	−	1.07028i	−0.991183	−	0.132498i	\(-0.957700\pi\)
							0.347215	−	0.937786i	\(-0.387128\pi\)
\(3\)	−2.74380	+	1.21308i	−0.914601	+	0.404359i
							\(5\)	−0.792588	−	7.28773i	−0.158518	−	1.45755i	−0.755425	−	0.655235i	\(-0.772570\pi\)
0.596907	−	0.802310i	\(-0.296396\pi\)
\(7\)	−2.16549	+	0.729637i	−0.309355	+	0.104234i	−0.469697	−	0.882828i	\(-0.655637\pi\)
							0.160342	+	0.987062i	\(0.448740\pi\)
\(11\)	4.14912	+	2.81318i	0.377193	+	0.255743i	0.735021	−	0.678044i	\(-0.237172\pi\)
							−0.357828	+	0.933788i	\(0.616483\pi\)
\(13\)	−14.8533	−	14.0698i	−1.14256	−	1.08229i	−0.995653	−	0.0931399i	\(-0.970310\pi\)
							−0.146907	−	0.989150i	\(-0.546932\pi\)
\(17\)	−5.14258	+	15.2626i	−0.302505	+	0.897802i	0.683334	+	0.730106i	\(0.260529\pi\)
							−0.985839	+	0.167696i	\(0.946367\pi\)
\(19\)	5.67234	+	34.5998i	0.298544	+	1.82104i	0.529772	+	0.848140i	\(0.322277\pi\)
							−0.231228	+	0.972900i	\(0.574274\pi\)
\(23\)	5.80238	−	1.61102i	0.252278	−	0.0700445i	−0.139087	−	0.990280i	\(-0.544417\pi\)
							0.391365	+	0.920236i	\(0.372003\pi\)
\(29\)	−9.42481	+	15.6642i	−0.324994	+	0.540143i	−0.975679	−	0.219204i	\(-0.929654\pi\)
							0.650686	+	0.759347i	\(0.274482\pi\)
\(31\)	4.77808	−	29.1450i	0.154132	−	0.940161i	−0.790604	−	0.612328i	\(-0.790233\pi\)
							0.944736	−	0.327833i	\(-0.106318\pi\)
\(37\)	0.242554	−	4.47364i	0.00655551	−	0.120909i	−0.993424	−	0.114494i	\(-0.963475\pi\)
							0.999979	−	0.00641506i	\(-0.00204199\pi\)
\(41\)	−21.6722	−	6.01724i	−0.528589	−	0.146762i	−0.00702688	−	0.999975i	\(-0.502237\pi\)
							−0.521562	+	0.853213i	\(0.674651\pi\)
\(43\)	−44.2374	−	65.2452i	−1.02878	−	1.51733i	−0.845290	−	0.534308i	\(-0.820572\pi\)
							−0.183486	−	0.983022i	\(-0.558738\pi\)
\(47\)	−5.05058	+	46.4393i	−0.107459	+	0.988071i	0.808963	+	0.587859i	\(0.200029\pi\)
							−0.916423	+	0.400212i	\(0.868937\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.104.9	yes	1064
3.2	odd	2	inner	177.3.h.a.104.30	yes	1064
59.21	even	29	inner	177.3.h.a.80.30	yes	1064
177.80	odd	58	inner	177.3.h.a.80.9	✓	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.80.9	✓	1064	177.80	odd	58	inner
177.3.h.a.80.30	yes	1064	59.21	even	29	inner
177.3.h.a.104.9	yes	1064	1.1	even	1	trivial
177.3.h.a.104.30	yes	1064	3.2	odd	2	inner