Embedded newform 177.2.e.a.22.3

• Label: 177.2.e.a.22.3
• Level: $177$
• Weight: $2$
• Character: 177.22
• 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			22.3
Character	\(\chi\)	\(=\)	177.22
Dual form			177.2.e.a.169.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0194566 + 0.118680i) q^{2} +(0.267528 + 0.963550i) q^{3} +(1.88160 - 0.633985i) q^{4} +(-1.57830 + 2.32782i) q^{5} +(-0.109149 + 0.0504978i) q^{6} +(1.03804 + 0.228489i) q^{7} +(0.224517 + 0.423484i) q^{8} +(-0.856857 + 0.515554i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - q^{2} - 5 q^{3} - 9 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} - 9 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{13}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0194566	+	0.118680i	0.0137579	+	0.0839196i	0.992813	−	0.119676i	\(-0.0381855\pi\)
							−0.979055	+	0.203595i	\(0.934737\pi\)
\(3\)	0.267528	+	0.963550i	0.154458	+	0.556306i
							\(5\)	−1.57830	+	2.32782i	−0.705838	+	1.04103i	0.290611	+	0.956841i	\(0.406141\pi\)
−0.996450	+	0.0841926i	\(0.973169\pi\)
\(7\)	1.03804	+	0.228489i	0.392341	+	0.0863608i	0.406760	−	0.913535i	\(-0.366659\pi\)
							−0.0144184	+	0.999896i	\(0.504590\pi\)
\(11\)	−0.225850	−	0.171686i	−0.0680962	−	0.0517654i	0.570572	−	0.821247i	\(-0.306721\pi\)
							−0.638669	+	0.769482i	\(0.720515\pi\)
\(13\)	0.273946	+	0.164828i	0.0759790	+	0.0457150i	0.553036	−	0.833157i	\(-0.313469\pi\)
							−0.477057	+	0.878872i	\(0.658297\pi\)
\(17\)	0.728335	−	0.160319i	0.176647	−	0.0388830i	−0.125766	−	0.992060i	\(-0.540139\pi\)
							0.302413	+	0.953177i	\(0.402208\pi\)
\(19\)	−2.09873	−	0.228251i	−0.481482	−	0.0523644i	−0.135840	−	0.990731i	\(-0.543373\pi\)
							−0.345642	+	0.938366i	\(0.612339\pi\)
\(23\)	4.27803	−	5.03649i	0.892031	−	1.05018i	−0.106407	−	0.994323i	\(-0.533935\pi\)
							0.998439	−	0.0558580i	\(-0.0177894\pi\)
\(29\)	0.142643	−	0.870082i	0.0264881	−	0.161570i	−0.970270	−	0.242024i	\(-0.922189\pi\)
							0.996758	+	0.0804539i	\(0.0256370\pi\)
\(31\)	−8.02640	+	0.872924i	−1.44158	+	0.156782i	−0.795254	−	0.606276i	\(-0.792663\pi\)
							−0.646330	+	0.763058i	\(0.723697\pi\)
\(37\)	2.76542	−	5.21613i	0.454632	−	0.857526i	−0.545184	−	0.838316i	\(-0.683540\pi\)
							0.999815	−	0.0192097i	\(-0.00611501\pi\)
\(41\)	−5.00140	−	5.88810i	−0.781087	−	0.919567i	0.217314	−	0.976102i	\(-0.430271\pi\)
							−0.998401	+	0.0565351i	\(0.981995\pi\)
\(43\)	7.12566	−	5.41679i	1.08665	−	0.826053i	0.101009	−	0.994886i	\(-0.467793\pi\)
							0.985645	+	0.168833i	\(0.0539998\pi\)
\(47\)	−4.51510	−	6.65928i	−0.658596	−	0.971356i	−0.999511	−	0.0312621i	\(-0.990047\pi\)
							0.340916	−	0.940094i	\(-0.389263\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.a.22.3	✓	140
3.2	odd	2		531.2.i.c.199.3		140
59.51	even	29	inner	177.2.e.a.169.3	yes	140
177.110	odd	58		531.2.i.c.523.3		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.a.22.3	✓	140	1.1	even	1	trivial
177.2.e.a.169.3	yes	140	59.51	even	29	inner
531.2.i.c.199.3		140	3.2	odd	2
531.2.i.c.523.3		140	177.110	odd	58