Embedded newform 177.2.f.a.101.15

• Label: 177.2.f.a.101.15
• Level: $177$
• Weight: $2$
• Character: 177.101
• 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			101.15
Character	\(\chi\)	\(=\)	177.101
Dual form			177.2.f.a.170.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.14155 - 1.68366i) q^{2} +(-1.01592 - 1.40282i) q^{3} +(-0.791296 - 1.98600i) q^{4} +(-0.506396 + 1.09456i) q^{5} +(-3.52159 + 0.109083i) q^{6} +(-1.20107 - 4.32586i) q^{7} +(-0.273835 - 0.0602756i) q^{8} +(-0.935799 + 2.85031i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{11}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.14155	−	1.68366i	0.807197	−	1.19053i	−0.171494	−	0.985185i	\(-0.554859\pi\)
							0.978690	−	0.205341i	\(-0.0658304\pi\)
\(3\)	−1.01592	−	1.40282i	−0.586544	−	0.809918i
							\(5\)	−0.506396	+	1.09456i	−0.226467	+	0.489501i	−0.987185	−	0.159582i	\(-0.948985\pi\)
0.760717	+	0.649083i	\(0.224847\pi\)
\(7\)	−1.20107	−	4.32586i	−0.453962	−	1.63502i	−0.736985	−	0.675910i	\(-0.763751\pi\)
							0.283023	−	0.959113i	\(-0.408663\pi\)
\(11\)	0.0836224	+	0.0792113i	0.0252131	+	0.0238831i	0.700202	−	0.713944i	\(-0.253093\pi\)
							−0.674989	+	0.737828i	\(0.735852\pi\)
\(13\)	0.00985323	−	0.0905989i	0.00273279	−	0.0251276i	−0.992692	−	0.120674i	\(-0.961494\pi\)
							0.995425	+	0.0955466i	\(0.0304599\pi\)
\(17\)	3.32312	+	0.922660i	0.805975	+	0.223778i	0.645994	−	0.763342i	\(-0.276443\pi\)
							0.159981	+	0.987120i	\(0.448857\pi\)
\(19\)	5.47273	+	4.16026i	1.25553	+	0.954429i	0.999924	−	0.0123521i	\(-0.00393190\pi\)
							0.255606	+	0.966781i	\(0.417725\pi\)
\(23\)	−3.24613	−	6.12285i	−0.676865	−	1.27670i	−0.949252	−	0.314515i	\(-0.898158\pi\)
							0.272387	−	0.962188i	\(-0.412187\pi\)
\(29\)	−4.61327	+	3.12787i	−0.856662	+	0.580831i	−0.908529	−	0.417823i	\(-0.862793\pi\)
							0.0518665	+	0.998654i	\(0.483483\pi\)
\(31\)	1.46140	+	1.92244i	0.262476	+	0.345281i	0.908386	−	0.418132i	\(-0.137315\pi\)
							−0.645910	+	0.763413i	\(0.723522\pi\)
\(37\)	−1.53713	−	6.98325i	−0.252703	−	1.14804i	−0.916471	−	0.400101i	\(-0.868975\pi\)
							0.663768	−	0.747938i	\(-0.268956\pi\)
\(41\)	6.63407	+	3.51716i	1.03607	+	0.549288i	0.897477	−	0.441062i	\(-0.145398\pi\)
							0.138590	+	0.990350i	\(0.455743\pi\)
\(43\)	−0.316171	−	0.333777i	−0.0482156	−	0.0509005i	0.701441	−	0.712727i	\(-0.252540\pi\)
							−0.749657	+	0.661827i	\(0.769782\pi\)
\(47\)	−3.23755	+	1.49785i	−0.472245	+	0.218484i	−0.641552	−	0.767079i	\(-0.721709\pi\)
							0.169307	+	0.985563i	\(0.445847\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.101.15	yes	504
3.2	odd	2	inner	177.2.f.a.101.4	✓	504
59.52	odd	58	inner	177.2.f.a.170.4	yes	504
177.170	even	58	inner	177.2.f.a.170.15	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.101.4	✓	504	3.2	odd	2	inner
177.2.f.a.101.15	yes	504	1.1	even	1	trivial
177.2.f.a.170.4	yes	504	59.52	odd	58	inner
177.2.f.a.170.15	yes	504	177.170	even	58	inner