Embedded newform 177.2.f.a.2.11

• Label: 177.2.f.a.2.11
• Level: $177$
• Weight: $2$
• Character: 177.2
• 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			2.11
Character	\(\chi\)	\(=\)	177.2
Dual form			177.2.f.a.89.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0266055 - 0.490710i) q^{2} +(1.36909 + 1.06093i) q^{3} +(1.74819 + 0.190127i) q^{4} +(-0.431333 + 1.28015i) q^{5} +(0.557036 - 0.643601i) q^{6} +(-1.31835 - 1.94442i) q^{7} +(0.298818 - 1.82271i) q^{8} +(0.748837 + 2.90504i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0266055	−	0.490710i	0.0188129	−	0.346984i	−0.973940	−	0.226808i	\(-0.927171\pi\)
							0.992753	−	0.120177i	\(-0.0383461\pi\)
\(3\)	1.36909	+	1.06093i	0.790447	+	0.612531i
							\(5\)	−0.431333	+	1.28015i	−0.192898	+	0.572501i	−0.999836	−	0.0181360i	\(-0.994227\pi\)
0.806938	+	0.590637i	\(0.201123\pi\)
\(7\)	−1.31835	−	1.94442i	−0.498289	−	0.734922i	0.492503	−	0.870311i	\(-0.336082\pi\)
							−0.990793	+	0.135389i	\(0.956772\pi\)
\(11\)	−1.07476	+	0.236572i	−0.324052	+	0.0713292i	−0.374017	−	0.927422i	\(-0.622020\pi\)
							0.0499654	+	0.998751i	\(0.484089\pi\)
\(13\)	−2.13603	+	1.81436i	−0.592429	+	0.503214i	−0.892707	−	0.450638i	\(-0.851197\pi\)
							0.300277	+	0.953852i	\(0.402921\pi\)
\(17\)	−4.66717	−	3.16442i	−1.13196	−	0.767485i	−0.156499	−	0.987678i	\(-0.550021\pi\)
							−0.975456	+	0.220194i	\(0.929331\pi\)
\(19\)	2.37487	−	4.47948i	0.544833	−	1.02766i	−0.446312	−	0.894878i	\(-0.647263\pi\)
							0.991145	−	0.132786i	\(-0.0423924\pi\)
\(23\)	3.97723	−	3.76743i	0.829309	−	0.785563i	−0.149839	−	0.988710i	\(-0.547876\pi\)
							0.979148	+	0.203147i	\(0.0651170\pi\)
\(29\)	−3.15255	+	0.170926i	−0.585414	+	0.0317402i	−0.344467	−	0.938798i	\(-0.611940\pi\)
							−0.240946	+	0.970538i	\(0.577458\pi\)
\(31\)	−1.86947	+	0.991129i	−0.335766	+	0.178012i	−0.627757	−	0.778409i	\(-0.716027\pi\)
							0.291991	+	0.956421i	\(0.405682\pi\)
\(37\)	−9.79294	+	1.60547i	−1.60995	+	0.263938i	−0.898557	−	0.438857i	\(-0.855383\pi\)
							−0.711393	+	0.702795i	\(0.751935\pi\)
\(41\)	2.82564	−	2.98299i	0.441291	−	0.465865i	−0.466968	−	0.884274i	\(-0.654654\pi\)
							0.908259	+	0.418409i	\(0.137412\pi\)
\(43\)	−0.996458	+	4.52695i	−0.151958	+	0.690354i	0.837607	+	0.546273i	\(0.183954\pi\)
							−0.989566	+	0.144081i	\(0.953977\pi\)
\(47\)	−7.40075	+	2.49360i	−1.07951	+	0.363729i	−0.802203	−	0.597052i	\(-0.796339\pi\)
							−0.277308	+	0.960781i	\(0.589442\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.2.11	yes	504
3.2	odd	2	inner	177.2.f.a.2.8	✓	504
59.30	odd	58	inner	177.2.f.a.89.8	yes	504
177.89	even	58	inner	177.2.f.a.89.11	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.2.8	✓	504	3.2	odd	2	inner
177.2.f.a.2.11	yes	504	1.1	even	1	trivial
177.2.f.a.89.8	yes	504	59.30	odd	58	inner
177.2.f.a.89.11	yes	504	177.89	even	58	inner