Embedded newform 177.2.f.a.56.10

• Label: 177.2.f.a.56.10
• Level: $177$
• Weight: $2$
• Character: 177.56
• Analytic conductor: $1.413$
• Analytic rank: $0$
• Dimension: $504$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.f (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.41335211578\)
Analytic rank:	\(0\)
Dimension:	\(504\)
Relative dimension:	\(18\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			56.10
Character	\(\chi\)	\(=\)	177.56
Dual form			177.2.f.a.98.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.368020 - 0.170264i) q^{2} +(-0.570529 + 1.63539i) q^{3} +(-1.18832 + 1.39900i) q^{4} +(1.50580 - 2.50266i) q^{5} +(0.0684822 + 0.698996i) q^{6} +(-0.260897 + 4.81196i) q^{7} +(-0.416091 + 1.49862i) q^{8} +(-2.34899 - 1.86607i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{21}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.368020	−	0.170264i	0.260229	−	0.120395i	−0.285437	−	0.958397i	\(-0.592139\pi\)
							0.545666	+	0.838003i	\(0.316277\pi\)
\(3\)	−0.570529	+	1.63539i	−0.329395	+	0.944192i
							\(5\)	1.50580	−	2.50266i	0.673414	−	1.11922i	−0.312169	−	0.950027i	\(-0.601055\pi\)
0.985582	−	0.169196i	\(-0.0541170\pi\)
\(7\)	−0.260897	+	4.81196i	−0.0986098	+	1.81875i	0.363133	+	0.931737i	\(0.381707\pi\)
							−0.461743	+	0.887014i	\(0.652776\pi\)
\(11\)	−0.148300	+	0.904591i	−0.0447142	+	0.272744i	−0.999715	−	0.0238576i	\(-0.992405\pi\)
							0.955001	+	0.296602i	\(0.0958535\pi\)
\(13\)	2.05813	+	2.70743i	0.570823	+	0.750905i	0.987408	−	0.158191i	\(-0.0505663\pi\)
							−0.416585	+	0.909097i	\(0.636773\pi\)
\(17\)	−2.51894	+	0.136573i	−0.610934	+	0.0331239i	−0.357010	−	0.934100i	\(-0.616204\pi\)
							−0.253923	+	0.967224i	\(0.581721\pi\)
\(19\)	1.34242	−	1.27161i	0.307973	−	0.291727i	−0.518066	−	0.855341i	\(-0.673348\pi\)
							0.826039	+	0.563613i	\(0.190589\pi\)
\(23\)	5.81897	−	1.28085i	1.21334	−	0.267076i	0.438207	−	0.898874i	\(-0.355614\pi\)
							0.775133	+	0.631798i	\(0.217683\pi\)
\(29\)	1.75430	−	3.79186i	0.325766	−	0.704131i	−0.673624	−	0.739075i	\(-0.735263\pi\)
							0.999389	+	0.0349437i	\(0.0111252\pi\)
\(31\)	2.95779	−	3.12251i	0.531235	−	0.560818i	−0.404039	−	0.914742i	\(-0.632394\pi\)
							0.935274	+	0.353923i	\(0.115153\pi\)
\(37\)	5.07310	−	1.40854i	0.834012	−	0.231562i	0.175830	−	0.984421i	\(-0.443739\pi\)
							0.658183	+	0.752858i	\(0.271325\pi\)
\(41\)	1.86984	−	8.49476i	0.292020	−	1.32666i	−0.570776	−	0.821106i	\(-0.693358\pi\)
							0.862796	−	0.505553i	\(-0.168711\pi\)
\(43\)	3.60656	−	0.591266i	0.549995	−	0.0901671i	0.119624	−	0.992819i	\(-0.461831\pi\)
							0.430371	+	0.902652i	\(0.358383\pi\)
\(47\)	−11.3202	+	6.81116i	−1.65123	+	0.993509i	−0.687661	+	0.726032i	\(0.741363\pi\)
							−0.963564	+	0.267478i	\(0.913810\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.56.10	yes	504
3.2	odd	2	inner	177.2.f.a.56.9	✓	504
59.39	odd	58	inner	177.2.f.a.98.9	yes	504
177.98	even	58	inner	177.2.f.a.98.10	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.56.9	✓	504	3.2	odd	2	inner
177.2.f.a.56.10	yes	504	1.1	even	1	trivial
177.2.f.a.98.9	yes	504	59.39	odd	58	inner
177.2.f.a.98.10	yes	504	177.98	even	58	inner