Embedded newform 177.2.f.a.50.7

• Label: 177.2.f.a.50.7
• Level: $177$
• Weight: $2$
• Character: 177.50
• 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			50.7
Character	\(\chi\)	\(=\)	177.50
Dual form			177.2.f.a.131.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.740497 + 0.871780i) q^{2} +(-1.61361 + 0.629505i) q^{3} +(0.111899 + 0.682553i) q^{4} +(-1.53444 + 0.813510i) q^{5} +(0.646079 - 1.87286i) q^{6} +(0.0293475 + 0.00319173i) q^{7} +(-2.63809 - 1.58728i) q^{8} +(2.20745 - 2.03155i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{13}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.740497	+	0.871780i	−0.523610	+	0.616441i	−0.958864	−	0.283866i	\(-0.908383\pi\)
							0.435254	+	0.900308i	\(0.356659\pi\)
\(3\)	−1.61361	+	0.629505i	−0.931616	+	0.363445i
							\(5\)	−1.53444	+	0.813510i	−0.686223	+	0.363813i	−0.774756	−	0.632261i	\(-0.782127\pi\)
0.0885324	+	0.996073i	\(0.471782\pi\)
\(7\)	0.0293475	+	0.00319173i	0.0110923	+	0.00120636i	0.113663	−	0.993519i	\(-0.463741\pi\)
							−0.102571	+	0.994726i	\(0.532707\pi\)
\(11\)	0.0537278	+	0.0181030i	0.0161995	+	0.00545826i	0.327390	−	0.944889i	\(-0.393831\pi\)
							−0.311190	+	0.950348i	\(0.600728\pi\)
\(13\)	−5.37021	−	1.49103i	−1.48943	−	0.413538i	−0.574956	−	0.818184i	\(-0.694981\pi\)
							−0.914473	+	0.404646i	\(0.867395\pi\)
\(17\)	−0.278487	−	2.56064i	−0.0675429	−	0.621047i	−0.978168	−	0.207814i	\(-0.933365\pi\)
							0.910625	−	0.413233i	\(-0.135600\pi\)
\(19\)	−0.163123	+	3.00862i	−0.0374229	+	0.690225i	0.918287	+	0.395915i	\(0.129573\pi\)
							−0.955710	+	0.294310i	\(0.904910\pi\)
\(23\)	−2.31790	+	1.07238i	−0.483316	+	0.223606i	−0.646389	−	0.763008i	\(-0.723722\pi\)
							0.163073	+	0.986614i	\(0.447860\pi\)
\(29\)	−2.59690	+	2.20583i	−0.482233	+	0.409612i	−0.855253	−	0.518211i	\(-0.826598\pi\)
							0.373020	+	0.927823i	\(0.378322\pi\)
\(31\)	1.74775	−	0.0947604i	0.313906	−	0.0170195i	0.103487	−	0.994631i	\(-0.467000\pi\)
							0.210419	+	0.977611i	\(0.432517\pi\)
\(37\)	1.62628	+	2.70289i	0.267358	+	0.444353i	0.961231	−	0.275745i	\(-0.0889246\pi\)
							−0.693873	+	0.720098i	\(0.744097\pi\)
\(41\)	−4.30405	+	9.30305i	−0.672179	+	1.45289i	0.207941	+	0.978141i	\(0.433324\pi\)
							−0.880120	+	0.474751i	\(0.842538\pi\)
\(43\)	2.60680	+	7.73671i	0.397534	+	1.17984i	0.940415	+	0.340028i	\(0.110437\pi\)
							−0.542881	+	0.839809i	\(0.682667\pi\)
\(47\)	2.26735	−	4.27668i	0.330727	−	0.623817i	−0.661260	−	0.750157i	\(-0.729978\pi\)
							0.991987	+	0.126339i	\(0.0403229\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.50.7	✓	504
3.2	odd	2	inner	177.2.f.a.50.12	yes	504
59.13	odd	58	inner	177.2.f.a.131.12	yes	504
177.131	even	58	inner	177.2.f.a.131.7	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.50.7	✓	504	1.1	even	1	trivial
177.2.f.a.50.12	yes	504	3.2	odd	2	inner
177.2.f.a.131.7	yes	504	177.131	even	58	inner
177.2.f.a.131.12	yes	504	59.13	odd	58	inner