Embedded newform 177.2.e.b.163.2

• Label: 177.2.e.b.163.2
• Level: $177$
• Weight: $2$
• Character: 177.163
• Analytic conductor: $1.413$
• Analytic rank: $0$
• Dimension: $140$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.e (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.41335211578\)
Analytic rank:	\(0\)
Dimension:	\(140\)
Relative dimension:	\(5\) over \(\Q(\zeta_{29})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{29}]$

Embedding invariants

Embedding label			163.2
Character	\(\chi\)	\(=\)	177.163
Dual form			177.2.e.b.139.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.992185 + 0.596978i) q^{2} +(0.370138 + 0.928977i) q^{3} +(-0.308769 + 0.582400i) q^{4} +(-0.793835 + 0.0863348i) q^{5} +(-0.921824 - 0.700752i) q^{6} +(-4.18976 + 1.41169i) q^{7} +(-0.166703 - 3.07465i) q^{8} +(-0.725995 + 0.687699i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q + q^{2} + 5 q^{3} - q^{4} + 2 q^{5} - q^{6} - 2 q^{7} - 3 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{24}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.992185	+	0.596978i	−0.701580	+	0.422127i	−0.821160	−	0.570698i	\(-0.806673\pi\)
							0.119580	+	0.992825i	\(0.461845\pi\)
\(3\)	0.370138	+	0.928977i	0.213699	+	0.536345i
							\(5\)	−0.793835	+	0.0863348i	−0.355014	+	0.0386101i	−0.283888	−	0.958857i	\(-0.591625\pi\)
−0.0711254	+	0.997467i	\(0.522659\pi\)
\(7\)	−4.18976	+	1.41169i	−1.58358	+	0.533570i	−0.967230	−	0.253903i	\(-0.918286\pi\)
							−0.616349	+	0.787473i	\(0.711389\pi\)
\(11\)	0.744195	−	1.09760i	0.224383	−	0.330940i	−0.698789	−	0.715328i	\(-0.746277\pi\)
							0.923172	+	0.384388i	\(0.125588\pi\)
\(13\)	2.28411	+	2.16363i	0.633499	+	0.600082i	0.935543	−	0.353212i	\(-0.114911\pi\)
							−0.302045	+	0.953294i	\(0.597669\pi\)
\(17\)	−7.48680	−	2.52260i	−1.81582	−	0.611819i	−0.999410	−	0.0343397i	\(-0.989067\pi\)
							−0.816405	−	0.577480i	\(-0.804036\pi\)
\(19\)	0.837472	+	5.10835i	0.192129	+	1.17194i	0.890085	+	0.455795i	\(0.150645\pi\)
							−0.697956	+	0.716141i	\(0.745907\pi\)
\(23\)	0.662011	+	2.38435i	0.138039	+	0.497171i	0.999951	−	0.00988634i	\(-0.00314697\pi\)
							−0.861912	+	0.507057i	\(0.830733\pi\)
\(29\)	6.94296	+	4.17744i	1.28928	+	0.775732i	0.984279	−	0.176622i	\(-0.0565169\pi\)
							0.304997	+	0.952353i	\(0.401345\pi\)
\(31\)	−0.686795	+	4.18927i	−0.123352	+	0.752414i	0.850626	+	0.525771i	\(0.176223\pi\)
							−0.973978	+	0.226643i	\(0.927225\pi\)
\(37\)	0.435953	−	8.04068i	0.0716702	−	1.32188i	−0.712649	−	0.701521i	\(-0.752505\pi\)
							0.784319	−	0.620358i	\(-0.213013\pi\)
\(41\)	−1.62041	+	5.83619i	−0.253065	+	0.911459i	0.722431	+	0.691443i	\(0.243025\pi\)
							−0.975496	+	0.220016i	\(0.929389\pi\)
\(43\)	6.39177	+	9.42715i	0.974735	+	1.43763i	0.897704	+	0.440598i	\(0.145234\pi\)
							0.0770311	+	0.997029i	\(0.475456\pi\)
\(47\)	3.69898	+	0.402288i	0.539551	+	0.0586797i	0.373837	−	0.927494i	\(-0.378042\pi\)
							0.165714	+	0.986174i	\(0.447007\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.b.163.2	yes	140
3.2	odd	2		531.2.i.b.163.4		140
59.21	even	29	inner	177.2.e.b.139.2	✓	140
177.80	odd	58		531.2.i.b.316.4		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.b.139.2	✓	140	59.21	even	29	inner
177.2.e.b.163.2	yes	140	1.1	even	1	trivial
531.2.i.b.163.4		140	3.2	odd	2
531.2.i.b.316.4		140	177.80	odd	58