Embedded newform 177.2.f.a.8.1

• Label: 177.2.f.a.8.1
• Level: $177$
• Weight: $2$
• Character: 177.8
• 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			8.1
Character	\(\chi\)	\(=\)	177.8
Dual form			177.2.f.a.155.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.438895 + 2.67714i) q^{2} +(-0.255734 - 1.71307i) q^{3} +(-5.07914 - 1.71136i) q^{4} +(2.43047 - 1.64790i) q^{5} +(4.69836 + 0.0672212i) q^{6} +(3.44243 - 0.757736i) q^{7} +(4.26929 - 8.05274i) q^{8} +(-2.86920 + 0.876179i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{3}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.438895	+	2.67714i	−0.310345	+	1.89302i	0.124008	+	0.992281i	\(0.460425\pi\)
							−0.434354	+	0.900742i	\(0.643023\pi\)
\(3\)	−0.255734	−	1.71307i	−0.147648	−	0.989040i
							\(5\)	2.43047	−	1.64790i	1.08694	−	0.736964i	0.120207	−	0.992749i	\(-0.461644\pi\)
0.966734	+	0.255784i	\(0.0823337\pi\)
\(7\)	3.44243	−	0.757736i	1.30112	−	0.286397i	0.490226	−	0.871595i	\(-0.336914\pi\)
							0.810890	+	0.585198i	\(0.198983\pi\)
\(11\)	−1.00861	+	0.766728i	−0.304108	+	0.231177i	−0.746087	−	0.665848i	\(-0.768070\pi\)
							0.441979	+	0.897025i	\(0.354277\pi\)
\(13\)	−1.15962	−	1.92730i	−0.321621	−	0.534538i	0.653297	−	0.757101i	\(-0.273385\pi\)
							−0.974918	+	0.222563i	\(0.928558\pi\)
\(17\)	−0.288917	+	1.31256i	−0.0700727	+	0.318343i	−0.998599	−	0.0529244i	\(-0.983146\pi\)
							0.928526	+	0.371268i	\(0.121077\pi\)
\(19\)	5.94804	−	0.646888i	1.36457	−	0.148406i	0.603651	−	0.797249i	\(-0.293712\pi\)
							0.760923	+	0.648843i	\(0.224747\pi\)
\(23\)	−1.63585	−	1.92587i	−0.341098	−	0.401572i	0.564690	−	0.825303i	\(-0.308996\pi\)
							−0.905788	+	0.423732i	\(0.860720\pi\)
\(29\)	1.67522	−	0.274638i	0.311080	−	0.0509990i	−0.00421977	−	0.999991i	\(-0.501343\pi\)
							0.315300	+	0.948992i	\(0.397895\pi\)
\(31\)	−0.831501	+	7.64552i	−0.149342	+	1.37318i	0.644291	+	0.764780i	\(0.277152\pi\)
							−0.793633	+	0.608397i	\(0.791813\pi\)
\(37\)	−1.69944	+	0.900988i	−0.279387	+	0.148122i	−0.602210	−	0.798338i	\(-0.705713\pi\)
							0.322823	+	0.946459i	\(0.395368\pi\)
\(41\)	−4.74788	−	4.03288i	−0.741494	−	0.629831i	0.194883	−	0.980827i	\(-0.437567\pi\)
							−0.936377	+	0.350996i	\(0.885843\pi\)
\(43\)	−0.654505	+	0.860987i	−0.0998111	+	0.131299i	−0.843287	−	0.537463i	\(-0.819383\pi\)
							0.743476	+	0.668763i	\(0.233176\pi\)
\(47\)	−2.49212	+	3.67561i	−0.363514	+	0.536142i	−0.964832	−	0.262867i	\(-0.915332\pi\)
							0.601319	+	0.799009i	\(0.294642\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.8.1	✓	504
3.2	odd	2	inner	177.2.f.a.8.18	yes	504
59.37	odd	58	inner	177.2.f.a.155.18	yes	504
177.155	even	58	inner	177.2.f.a.155.1	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.8.1	✓	504	1.1	even	1	trivial
177.2.f.a.8.18	yes	504	3.2	odd	2	inner
177.2.f.a.155.1	yes	504	177.155	even	58	inner
177.2.f.a.155.18	yes	504	59.37	odd	58	inner