Embedded newform 177.2.e.a.16.2

• Label: 177.2.e.a.16.2
• Level: $177$
• Weight: $2$
• Character: 177.16
• 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			16.2
Character	\(\chi\)	\(=\)	177.16
Dual form			177.2.e.a.166.2

$q$-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 \(\left(\mathbb{Z}/177\mathbb{Z}\right)^\times\).

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{2}{29}\right)\)	\(1\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.a.16.2	✓	140	1.1	even	1	trivial
177.2.e.a.166.2	yes	140	59.48	even	29	inner
531.2.i.c.343.4		140	177.107	odd	58
531.2.i.c.370.4		140	3.2	odd	2