Embedded newform 177.2.f.a.56.15

• Label: 177.2.f.a.56.15
• Level: $177$
• Weight: $2$
• Character: 177.56
• 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			56.15
Character	\(\chi\)	\(=\)	177.56
Dual form			177.2.f.a.98.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.81816 - 0.841171i) q^{2} +(1.13774 - 1.30597i) q^{3} +(1.30337 - 1.53444i) q^{4} +(-1.80043 + 2.99233i) q^{5} +(0.970053 - 3.33149i) q^{6} +(0.204672 - 3.77496i) q^{7} +(0.00711540 - 0.0256273i) q^{8} +(-0.411096 - 2.97170i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{21}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.81816	−	0.841171i	1.28563	−	0.594798i	0.346404	−	0.938086i	\(-0.387403\pi\)
							0.939230	+	0.343288i	\(0.111541\pi\)
\(3\)	1.13774	−	1.30597i	0.656874	−	0.754000i
							\(5\)	−1.80043	+	2.99233i	−0.805175	+	1.33821i	0.132671	+	0.991160i	\(0.457645\pi\)
−0.937846	+	0.347051i	\(0.887183\pi\)
\(7\)	0.204672	−	3.77496i	0.0773589	−	1.42680i	−0.659983	−	0.751280i	\(-0.729437\pi\)
							0.737342	−	0.675520i	\(-0.236081\pi\)
\(11\)	−0.661460	+	4.03473i	−0.199438	+	1.21652i	0.677358	+	0.735654i	\(0.263125\pi\)
							−0.876796	+	0.480863i	\(0.840323\pi\)
\(13\)	−0.0782869	−	0.102985i	−0.0217129	−	0.0285628i	0.785109	−	0.619358i	\(-0.212607\pi\)
							−0.806821	+	0.590795i	\(0.798814\pi\)
\(17\)	2.21810	−	0.120262i	0.537968	−	0.0291678i	0.216846	−	0.976206i	\(-0.430423\pi\)
							0.321121	+	0.947038i	\(0.395940\pi\)
\(19\)	−1.70433	+	1.61443i	−0.391001	+	0.370376i	−0.857796	−	0.513990i	\(-0.828167\pi\)
							0.466796	+	0.884365i	\(0.345408\pi\)
\(23\)	−5.79120	+	1.27474i	−1.20755	+	0.265801i	−0.772741	−	0.634722i	\(-0.781115\pi\)
							−0.434807	+	0.900523i	\(0.643184\pi\)
\(29\)	3.33172	−	7.20139i	0.618685	−	1.33727i	−0.304354	−	0.952559i	\(-0.598441\pi\)
							0.923039	−	0.384706i	\(-0.125697\pi\)
\(31\)	−2.36427	+	2.49593i	−0.424636	+	0.448282i	−0.902847	−	0.429963i	\(-0.858527\pi\)
							0.478211	+	0.878245i	\(0.341285\pi\)
\(37\)	3.94900	−	1.09643i	0.649212	−	0.180253i	0.0727156	−	0.997353i	\(-0.476833\pi\)
							0.576496	+	0.817100i	\(0.304420\pi\)
\(41\)	1.94403	−	8.83182i	0.303607	−	1.37930i	−0.539696	−	0.841860i	\(-0.681461\pi\)
							0.843303	−	0.537439i	\(-0.180608\pi\)
\(43\)	7.68877	−	1.26051i	1.17253	−	0.192226i	0.456126	−	0.889915i	\(-0.349237\pi\)
							0.716400	+	0.697689i	\(0.245788\pi\)
\(47\)	−5.56050	+	3.34564i	−0.811083	+	0.488012i	−0.859696	−	0.510805i	\(-0.829347\pi\)
							0.0486136	+	0.998818i	\(0.484520\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.56.15	yes	504
3.2	odd	2	inner	177.2.f.a.56.4	✓	504
59.39	odd	58	inner	177.2.f.a.98.4	yes	504
177.98	even	58	inner	177.2.f.a.98.15	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.56.4	✓	504	3.2	odd	2	inner
177.2.f.a.56.15	yes	504	1.1	even	1	trivial
177.2.f.a.98.4	yes	504	59.39	odd	58	inner
177.2.f.a.98.15	yes	504	177.98	even	58	inner