Embedded newform 177.2.f.a.77.9

• Label: 177.2.f.a.77.9
• Level: $177$
• Weight: $2$
• Character: 177.77
• 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			77.9
Character	\(\chi\)	\(=\)	177.77
Dual form			177.2.f.a.23.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0604160 - 0.0572290i) q^{2} +(1.46070 - 0.930782i) q^{3} +(-0.107903 - 1.99015i) q^{4} +(-3.23268 + 0.529971i) q^{5} +(-0.141517 - 0.0273604i) q^{6} +(-1.62584 - 3.06666i) q^{7} +(-0.215124 + 0.253264i) q^{8} +(1.26729 - 2.71919i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{43}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0604160	−	0.0572290i	−0.0427205	−	0.0404670i	0.666041	−	0.745915i	\(-0.267987\pi\)
							−0.708762	+	0.705448i	\(0.750746\pi\)
\(3\)	1.46070	−	0.930782i	0.843336	−	0.537387i
							\(5\)	−3.23268	+	0.529971i	−1.44570	+	0.237010i	−0.832986	−	0.553295i	\(-0.813370\pi\)
−0.612714	+	0.790305i	\(0.709922\pi\)
\(7\)	−1.62584	−	3.06666i	−0.614510	−	1.15909i	−0.974130	−	0.225990i	\(-0.927438\pi\)
							0.359620	−	0.933099i	\(-0.382906\pi\)
\(11\)	3.14952	+	0.342531i	0.949616	+	0.103277i	0.569799	−	0.821784i	\(-0.307021\pi\)
							0.379818	+	0.925061i	\(0.375987\pi\)
\(13\)	1.78835	+	3.86545i	0.495998	+	1.07208i	0.980360	+	0.197215i	\(0.0631896\pi\)
							−0.484362	+	0.874868i	\(0.660948\pi\)
\(17\)	1.00852	+	0.534686i	0.244603	+	0.129680i	0.586201	−	0.810166i	\(-0.300623\pi\)
							−0.341597	+	0.939846i	\(0.610968\pi\)
\(19\)	5.72353	+	3.44374i	1.31307	+	0.790047i	0.987823	−	0.155582i	\(-0.0497254\pi\)
							0.325246	+	0.945630i	\(0.394553\pi\)
\(23\)	3.32650	−	8.34888i	0.693622	−	1.74086i	0.0222369	−	0.999753i	\(-0.492921\pi\)
							0.671385	−	0.741108i	\(-0.265700\pi\)
\(29\)	3.45648	+	3.64896i	0.641852	+	0.677595i	0.963021	−	0.269427i	\(-0.0868344\pi\)
							−0.321169	+	0.947022i	\(0.604076\pi\)
\(31\)	−1.85981	−	3.09102i	−0.334031	−	0.555164i	0.643634	−	0.765333i	\(-0.277426\pi\)
							−0.977665	+	0.210170i	\(0.932598\pi\)
\(37\)	−3.97492	+	3.37633i	−0.653473	+	0.555065i	−0.911617	−	0.411041i	\(-0.865165\pi\)
							0.258143	+	0.966107i	\(0.416889\pi\)
\(41\)	−7.19479	+	2.86666i	−1.12364	+	0.447698i	−0.856596	−	0.515988i	\(-0.827425\pi\)
							−0.267041	+	0.963685i	\(0.586046\pi\)
\(43\)	−0.350980	−	3.22721i	−0.0535240	−	0.492145i	−0.989964	−	0.141318i	\(-0.954866\pi\)
							0.936440	−	0.350827i	\(-0.114100\pi\)
\(47\)	−0.457783	+	2.79236i	−0.0667746	+	0.407307i	0.932196	+	0.361953i	\(0.117890\pi\)
							−0.998971	+	0.0453541i	\(0.985558\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.77.9	yes	504
3.2	odd	2	inner	177.2.f.a.77.10	yes	504
59.23	odd	58	inner	177.2.f.a.23.10	yes	504
177.23	even	58	inner	177.2.f.a.23.9	✓	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.23.9	✓	504	177.23	even	58	inner
177.2.f.a.23.10	yes	504	59.23	odd	58	inner
177.2.f.a.77.9	yes	504	1.1	even	1	trivial
177.2.f.a.77.10	yes	504	3.2	odd	2	inner