Embedded newform 177.2.e.a.166.2

• Label: 177.2.e.a.166.2
• Level: $177$
• Weight: $2$
• Character: 177.166
• 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			166.2
Character	\(\chi\)	\(=\)	177.166
Dual form			177.2.e.a.16.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.29523 + 0.285102i) q^{2} +(-0.161782 + 0.986827i) q^{3} +(-0.218809 + 0.101232i) q^{4} +(0.106655 - 0.384136i) q^{5} +(-0.0718011 - 1.32429i) q^{6} +(-2.67256 + 2.53159i) q^{7} +(2.36617 - 1.79871i) q^{8} +(-0.947653 - 0.319302i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - q^{2} - 5 q^{3} - 9 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} - 9 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{27}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.29523	+	0.285102i	−0.915867	+	0.201598i	−0.647807	−	0.761805i	\(-0.724314\pi\)
							−0.268060	+	0.963402i	\(0.586383\pi\)
\(3\)	−0.161782	+	0.986827i	−0.0934049	+	0.569745i
							\(5\)	0.106655	−	0.384136i	0.0476975	−	0.171791i	−0.935865	−	0.352359i	\(-0.885380\pi\)
0.983563	+	0.180568i	\(0.0577935\pi\)
\(7\)	−2.67256	+	2.53159i	−1.01013	+	0.956850i	−0.999071	−	0.0430891i	\(-0.986280\pi\)
							−0.0110627	+	0.999939i	\(0.503521\pi\)
\(11\)	−2.09462	−	2.46597i	−0.631551	−	0.743519i	0.349571	−	0.936910i	\(-0.386327\pi\)
							−0.981122	+	0.193391i	\(0.938051\pi\)
\(13\)	−5.28494	+	1.78070i	−1.46578	+	0.493878i	−0.935311	−	0.353827i	\(-0.884880\pi\)
							−0.530467	+	0.847705i	\(0.677984\pi\)
\(17\)	0.872638	+	0.826606i	0.211646	+	0.200481i	0.786177	−	0.618001i	\(-0.212057\pi\)
							−0.574532	+	0.818482i	\(0.694816\pi\)
\(19\)	−1.70286	−	4.27385i	−0.390662	−	0.980488i	−0.984418	−	0.175843i	\(-0.943735\pi\)
							0.593756	−	0.804645i	\(-0.297644\pi\)
\(23\)	−6.61815	+	0.719767i	−1.37998	+	0.150082i	−0.767826	−	0.640658i	\(-0.778662\pi\)
							−0.612152	+	0.790740i	\(0.709696\pi\)
\(29\)	−1.60158	−	0.352535i	−0.297406	−	0.0654641i	0.0637607	−	0.997965i	\(-0.479691\pi\)
							−0.361167	+	0.932501i	\(0.617622\pi\)
\(31\)	−3.12782	+	7.85025i	−0.561774	+	1.40995i	0.324101	+	0.946022i	\(0.394938\pi\)
							−0.885875	+	0.463924i	\(0.846441\pi\)
\(37\)	7.38620	+	5.61484i	1.21428	+	0.923074i	0.998574	−	0.0533836i	\(-0.0170006\pi\)
							0.215709	+	0.976458i	\(0.430794\pi\)
\(41\)	8.92679	+	0.970847i	1.39413	+	0.151621i	0.774153	−	0.632999i	\(-0.218176\pi\)
							0.619977	+	0.784620i	\(0.287142\pi\)
\(43\)	2.79150	−	3.28641i	0.425700	−	0.501173i	−0.506940	−	0.861981i	\(-0.669223\pi\)
							0.932640	+	0.360809i	\(0.117499\pi\)
\(47\)	1.31279	+	4.72825i	0.191491	+	0.689687i	0.995474	+	0.0950349i	\(0.0302963\pi\)
							−0.803983	+	0.594652i	\(0.797290\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.a.166.2	yes	140
3.2	odd	2		531.2.i.c.343.4		140
59.16	even	29	inner	177.2.e.a.16.2	✓	140
177.134	odd	58		531.2.i.c.370.4		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.a.16.2	✓	140	59.16	even	29	inner
177.2.e.a.166.2	yes	140	1.1	even	1	trivial
531.2.i.c.343.4		140	3.2	odd	2
531.2.i.c.370.4		140	177.134	odd	58