Embedded newform 177.2.f.a.56.4

• Label: 177.2.f.a.56.4
• 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			56.4
Character	\(\chi\)	\(=\)	177.56
Dual form			177.2.f.a.98.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.81816 + 0.841171i) q^{2} +(0.115410 + 1.72820i) q^{3} +(1.30337 - 1.53444i) q^{4} +(1.80043 - 2.99233i) q^{5} +(-1.66355 - 3.04507i) q^{6} +(0.204672 - 3.77496i) q^{7} +(-0.00711540 + 0.0256273i) q^{8} +(-2.97336 + 0.398902i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

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	−0.939230	−	0.343288i	\(-0.888459\pi\)
							−0.346404	+	0.938086i	\(0.612597\pi\)
\(3\)	0.115410	+	1.72820i	0.0666317	+	0.997778i
							\(5\)	1.80043	−	2.99233i	0.805175	−	1.33821i	−0.132671	−	0.991160i	\(-0.542355\pi\)
0.937846	−	0.347051i	\(-0.112817\pi\)
\(7\)	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	\(-0.736875\pi\)
							0.876796	−	0.480863i	\(-0.159677\pi\)
\(13\)	−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	−0.321121	−	0.947038i	\(-0.604060\pi\)
							−0.216846	+	0.976206i	\(0.569577\pi\)
\(19\)	−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	0.434807	−	0.900523i	\(-0.356816\pi\)
							0.772741	+	0.634722i	\(0.218885\pi\)
\(29\)	−3.33172	+	7.20139i	−0.618685	+	1.33727i	0.304354	+	0.952559i	\(0.401559\pi\)
							−0.923039	+	0.384706i	\(0.874303\pi\)
\(31\)	−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	\(0.318539\pi\)
							−0.843303	+	0.537439i	\(0.819392\pi\)
\(43\)	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	−0.0486136	−	0.998818i	\(-0.515480\pi\)
							0.859696	+	0.510805i	\(0.170653\pi\)

(See \(a_n\) instead)

Twists

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

    

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