Embedded newform 177.2.e.a.166.3

• Label: 177.2.e.a.166.3
• 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			166.3
Character	\(\chi\)	\(=\)	177.166
Dual form			177.2.e.a.16.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.547213 - 0.120451i) q^{2} +(-0.161782 + 0.986827i) q^{3} +(-1.53022 + 0.707954i) q^{4} +(-0.382080 + 1.37613i) q^{5} +(0.0303347 + 0.559491i) q^{6} +(-1.74470 + 1.65266i) q^{7} +(-1.64420 + 1.24989i) q^{8} +(-0.947653 - 0.319302i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - q^{2} - 5 q^{3} - 9 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} - 9 q^{8} - 5 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.547213	−	0.120451i	0.386938	−	0.0851715i	−0.0172411	−	0.999851i	\(-0.505488\pi\)
							0.404179	+	0.914680i	\(0.367557\pi\)
\(3\)	−0.161782	+	0.986827i	−0.0934049	+	0.569745i
							\(5\)	−0.382080	+	1.37613i	−0.170871	+	0.615423i	0.827659	+	0.561231i	\(0.189672\pi\)
−0.998531	+	0.0541917i	\(0.982742\pi\)
\(7\)	−1.74470	+	1.65266i	−0.659433	+	0.624649i	−0.942403	−	0.334480i	\(-0.891439\pi\)
							0.282969	+	0.959129i	\(0.408681\pi\)
\(11\)	1.30884	+	1.54088i	0.394629	+	0.464593i	0.923213	−	0.384288i	\(-0.125553\pi\)
							−0.528585	+	0.848881i	\(0.677277\pi\)
\(13\)	4.26821	−	1.43813i	1.18379	−	0.398865i	0.342534	−	0.939505i	\(-0.388715\pi\)
							0.841254	+	0.540641i	\(0.181818\pi\)
\(17\)	−2.50206	−	2.37008i	−0.606839	−	0.574829i	0.321339	−	0.946964i	\(-0.395867\pi\)
							−0.928178	+	0.372136i	\(0.878626\pi\)
\(19\)	2.45333	+	6.15740i	0.562833	+	1.41260i	0.884853	+	0.465871i	\(0.154259\pi\)
							−0.322020	+	0.946733i	\(0.604362\pi\)
\(23\)	0.885676	−	0.0963231i	0.184676	−	0.0200847i	−0.0153135	−	0.999883i	\(-0.504875\pi\)
							0.199990	+	0.979798i	\(0.435909\pi\)
\(29\)	−1.76447	−	0.388390i	−0.327654	−	0.0721222i	0.0480971	−	0.998843i	\(-0.484684\pi\)
							−0.375751	+	0.926720i	\(0.622615\pi\)
\(31\)	1.48266	−	3.72120i	0.266294	−	0.668348i	−0.733643	−	0.679535i	\(-0.762182\pi\)
							0.999937	+	0.0111868i	\(0.00356095\pi\)
\(37\)	4.12683	+	3.13713i	0.678446	+	0.515741i	0.886707	−	0.462332i	\(-0.152987\pi\)
							−0.208261	+	0.978073i	\(0.566780\pi\)
\(41\)	−9.63133	−	1.04747i	−1.50416	−	0.163587i	−0.681373	−	0.731936i	\(-0.738617\pi\)
							−0.822788	+	0.568349i	\(0.807582\pi\)
\(43\)	2.85300	−	3.35881i	0.435079	−	0.512214i	−0.500305	−	0.865849i	\(-0.666779\pi\)
							0.935383	+	0.353635i	\(0.115055\pi\)
\(47\)	2.72206	+	9.80396i	0.397053	+	1.43006i	0.844409	+	0.535699i	\(0.179952\pi\)
							−0.447356	+	0.894356i	\(0.647634\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.a.166.3	yes	140
3.2	odd	2		531.2.i.c.343.3		140
59.16	even	29	inner	177.2.e.a.16.3	✓	140
177.134	odd	58		531.2.i.c.370.3		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.a.16.3	✓	140	59.16	even	29	inner
177.2.e.a.166.3	yes	140	1.1	even	1	trivial
531.2.i.c.343.3		140	3.2	odd	2
531.2.i.c.370.3		140	177.134	odd	58