Embedded newform 177.2.e.a.16.3

• Label: 177.2.e.a.16.3
• 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			16.3
Character	\(\chi\)	\(=\)	177.16
Dual form			177.2.e.a.166.3

$q$-expansion

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

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)

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