Embedded newform 177.2.e.b.7.5

• Label: 177.2.e.b.7.5
• Level: $177$
• Weight: $2$
• Character: 177.7
• 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			7.5
Character	\(\chi\)	\(=\)	177.7
Dual form			177.2.e.b.76.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.53455 + 2.26329i) q^{2} +(-0.0541389 + 0.998533i) q^{3} +(-2.02735 + 5.08827i) q^{4} +(1.27104 - 0.588045i) q^{5} +(-2.34304 + 1.40976i) q^{6} +(1.26600 - 4.55972i) q^{7} +(-9.28622 + 2.04405i) q^{8} +(-0.994138 - 0.108119i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q + q^{2} + 5 q^{3} - q^{4} + 2 q^{5} - q^{6} - 2 q^{7} - 3 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{9}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.53455	+	2.26329i	1.08509	+	1.60038i	0.754113	+	0.656744i	\(0.228067\pi\)
							0.330974	+	0.943640i	\(0.392623\pi\)
\(3\)	−0.0541389	+	0.998533i	−0.0312571	+	0.576504i
							\(5\)	1.27104	−	0.588045i	0.568426	−	0.262982i	−0.114556	−	0.993417i	\(-0.536545\pi\)
0.682982	+	0.730435i	\(0.260683\pi\)
\(7\)	1.26600	−	4.55972i	0.478503	−	1.72341i	−0.191900	−	0.981414i	\(-0.561465\pi\)
							0.670403	−	0.741997i	\(-0.266121\pi\)
\(11\)	2.71244	−	2.56936i	0.817832	−	0.774692i	−0.159321	−	0.987227i	\(-0.550931\pi\)
							0.977153	+	0.212535i	\(0.0681720\pi\)
\(13\)	−2.50623	+	0.272569i	−0.695102	+	0.0755969i	−0.448848	−	0.893608i	\(-0.648165\pi\)
							−0.246254	+	0.969205i	\(0.579200\pi\)
\(17\)	0.0447496	+	0.161174i	0.0108534	+	0.0390903i	0.968778	−	0.247928i	\(-0.0797497\pi\)
							−0.957925	+	0.287018i	\(0.907336\pi\)
\(19\)	−4.24286	+	3.22534i	−0.973379	+	0.739944i	−0.965602	−	0.260026i	\(-0.916269\pi\)
							−0.00777782	+	0.999970i	\(0.502476\pi\)
\(23\)	−0.992916	+	1.87284i	−0.207037	+	0.390514i	−0.965185	−	0.261569i	\(-0.915760\pi\)
							0.758147	+	0.652083i	\(0.226105\pi\)
\(29\)	1.05285	−	1.55283i	0.195509	−	0.288354i	−0.717281	−	0.696784i	\(-0.754613\pi\)
							0.912789	+	0.408431i	\(0.133924\pi\)
\(31\)	2.17666	+	1.65466i	0.390940	+	0.297185i	0.782118	−	0.623130i	\(-0.214139\pi\)
							−0.391178	+	0.920315i	\(0.627932\pi\)
\(37\)	−2.78409	−	0.612823i	−0.457701	−	0.100748i	−0.0198653	−	0.999803i	\(-0.506324\pi\)
							−0.437835	+	0.899055i	\(0.644255\pi\)
\(41\)	−3.35847	−	6.33474i	−0.524504	−	0.989320i	−0.994307	−	0.106549i	\(-0.966020\pi\)
							0.469803	−	0.882771i	\(-0.344325\pi\)
\(43\)	8.37320	+	7.93151i	1.27690	+	1.20954i	0.965835	+	0.259156i	\(0.0834444\pi\)
							0.311065	+	0.950389i	\(0.399314\pi\)
\(47\)	2.98956	+	1.38312i	0.436071	+	0.201748i	0.625628	−	0.780122i	\(-0.284843\pi\)
							−0.189556	+	0.981870i	\(0.560705\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.b.7.5	✓	140
3.2	odd	2		531.2.i.b.361.1		140
59.17	even	29	inner	177.2.e.b.76.5	yes	140
177.17	odd	58		531.2.i.b.253.1		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.b.7.5	✓	140	1.1	even	1	trivial
177.2.e.b.76.5	yes	140	59.17	even	29	inner
531.2.i.b.253.1		140	177.17	odd	58
531.2.i.b.361.1		140	3.2	odd	2