Embedded newform 177.2.f.a.11.9

• Label: 177.2.f.a.11.9
• Level: $177$
• Weight: $2$
• Character: 177.11
• 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			11.9
Character	\(\chi\)	\(=\)	177.11
Dual form			177.2.f.a.161.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.345333 + 0.0760135i) q^{2} +(0.912831 - 1.47198i) q^{3} +(-1.70167 + 0.787278i) q^{4} +(2.86142 + 0.794469i) q^{5} +(-0.203340 + 0.577712i) q^{6} +(0.822484 - 0.779098i) q^{7} +(1.09080 - 0.829202i) q^{8} +(-1.33348 - 2.68735i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{25}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.345333	+	0.0760135i	−0.244187	+	0.0537497i	−0.335378	−	0.942084i	\(-0.608864\pi\)
							0.0911903	+	0.995833i	\(0.470933\pi\)
\(3\)	0.912831	−	1.47198i	0.527023	−	0.849851i
							\(5\)	2.86142	+	0.794469i	1.27967	+	0.355297i	0.839864	−	0.542797i	\(-0.182635\pi\)
0.439802	+	0.898095i	\(0.355049\pi\)
\(7\)	0.822484	−	0.779098i	0.310870	−	0.294471i	−0.516318	−	0.856397i	\(-0.672698\pi\)
							0.827188	+	0.561925i	\(0.189939\pi\)
\(11\)	1.99887	+	2.35325i	0.602681	+	0.709531i	0.975902	−	0.218211i	\(-0.0700222\pi\)
							−0.373220	+	0.927743i	\(0.621746\pi\)
\(13\)	−0.823215	−	2.44321i	−0.228319	−	0.677626i	−0.999175	−	0.0406164i	\(-0.987068\pi\)
							0.770856	−	0.637009i	\(-0.219829\pi\)
\(17\)	0.933460	−	0.985442i	0.226397	−	0.239005i	−0.602894	−	0.797821i	\(-0.705986\pi\)
							0.829291	+	0.558817i	\(0.188744\pi\)
\(19\)	0.313296	+	0.786313i	0.0718750	+	0.180393i	0.960525	−	0.278193i	\(-0.0897355\pi\)
							−0.888650	+	0.458586i	\(0.848356\pi\)
\(23\)	−3.91137	+	0.425387i	−0.815577	+	0.0886993i	−0.506393	−	0.862303i	\(-0.669021\pi\)
							−0.309184	+	0.951002i	\(0.600056\pi\)
\(29\)	−1.76788	+	8.03157i	−0.328288	+	1.49143i	0.466491	+	0.884526i	\(0.345518\pi\)
							−0.794779	+	0.606900i	\(0.792413\pi\)
\(31\)	0.880915	+	0.350989i	0.158217	+	0.0630394i	0.447902	−	0.894083i	\(-0.352171\pi\)
							−0.289685	+	0.957122i	\(0.593551\pi\)
\(37\)	−3.53237	+	4.64675i	−0.580718	+	0.763921i	−0.988836	−	0.149008i	\(-0.952392\pi\)
							0.408118	+	0.912929i	\(0.366185\pi\)
\(41\)	−1.15879	+	10.6549i	−0.180972	+	1.66401i	0.453645	+	0.891182i	\(0.350123\pi\)
							−0.634617	+	0.772827i	\(0.718842\pi\)
\(43\)	−9.57275	−	8.13117i	−1.45983	−	1.23999i	−0.916234	−	0.400645i	\(-0.868786\pi\)
							−0.543597	−	0.839346i	\(-0.682938\pi\)
\(47\)	−1.98760	−	7.15871i	−0.289922	−	1.04420i	−0.954799	−	0.297252i	\(-0.903930\pi\)
							0.664877	−	0.746953i	\(-0.268484\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.11.9	✓	504
3.2	odd	2	inner	177.2.f.a.11.10	yes	504
59.43	odd	58	inner	177.2.f.a.161.10	yes	504
177.161	even	58	inner	177.2.f.a.161.9	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.11.9	✓	504	1.1	even	1	trivial
177.2.f.a.11.10	yes	504	3.2	odd	2	inner
177.2.f.a.161.9	yes	504	177.161	even	58	inner
177.2.f.a.161.10	yes	504	59.43	odd	58	inner