Embedded newform 177.4.e.b.133.10

• Label: 177.4.e.b.133.10
• Level: $177$
• Weight: $4$
• Character: 177.133
• Analytic conductor: $10.443$
• Analytic rank: $0$
• Dimension: $420$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.e (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.4433380710\)
Analytic rank:	\(0\)
Dimension:	\(420\)
Relative dimension:	\(15\) over \(\Q(\zeta_{29})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{29}]$

Embedding invariants

Embedding label			133.10
Character	\(\chi\)	\(=\)	177.133
Dual form			177.4.e.b.4.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.00109 - 0.217632i) q^{2} +(1.94216 + 2.28649i) q^{3} +(-3.85597 + 0.848763i) q^{4} +(-3.87099 + 2.94265i) q^{5} +(4.38404 + 4.15279i) q^{6} +(5.75041 + 14.4324i) q^{7} +(-22.7916 + 7.67938i) q^{8} +(-1.45604 + 8.88144i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q + 2 q^{2} - 45 q^{3} - 56 q^{4} - 14 q^{5} + 6 q^{6} + 6 q^{7} + 66 q^{8} - 135 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{28}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.00109	−	0.217632i	0.707492	−	0.0769444i	0.252703	−	0.967544i	\(-0.418680\pi\)
							0.454789	+	0.890599i	\(0.349715\pi\)
\(3\)	1.94216	+	2.28649i	0.373769	+	0.440034i
							\(5\)	−3.87099	+	2.94265i	−0.346232	+	0.263199i	−0.763808	−	0.645443i	\(-0.776673\pi\)
0.417576	+	0.908642i	\(0.362880\pi\)
\(7\)	5.75041	+	14.4324i	0.310493	+	0.779279i	0.998561	+	0.0536323i	\(0.0170799\pi\)
							−0.688068	+	0.725646i	\(0.741541\pi\)
\(11\)	−55.5415	−	25.6962i	−1.52240	−	0.704337i	−0.532541	−	0.846404i	\(-0.678763\pi\)
							−0.989857	+	0.142067i	\(0.954625\pi\)
\(13\)	−6.50110	−	39.6549i	−0.138698	−	0.846023i	−0.960914	−	0.276847i	\(-0.910710\pi\)
							0.822215	−	0.569176i	\(-0.192738\pi\)
\(17\)	−12.6125	+	31.6550i	−0.179940	+	0.451615i	−0.991085	−	0.133230i	\(-0.957465\pi\)
							0.811145	+	0.584845i	\(0.198845\pi\)
\(19\)	−32.7981	+	48.3736i	−0.396021	+	0.584088i	−0.972465	−	0.233049i	\(-0.925130\pi\)
							0.576444	+	0.817137i	\(0.304440\pi\)
\(23\)	7.86701	+	145.099i	0.0713211	+	1.31544i	0.786975	+	0.616985i	\(0.211646\pi\)
							−0.715654	+	0.698455i	\(0.753871\pi\)
\(29\)	57.7340	+	6.27895i	0.369687	+	0.0402059i	0.291077	−	0.956700i	\(-0.405986\pi\)
							0.0786099	+	0.996905i	\(0.474952\pi\)
\(31\)	−20.3860	−	30.0671i	−0.118111	−	0.174200i	0.764038	−	0.645171i	\(-0.223214\pi\)
							−0.882149	+	0.470971i	\(0.843904\pi\)
\(37\)	183.478	+	61.8210i	0.815233	+	0.274684i	0.695871	−	0.718167i	\(-0.255019\pi\)
							0.119362	+	0.992851i	\(0.461915\pi\)
\(41\)	−7.42872	+	137.015i	−0.0282969	+	0.521905i	0.949684	+	0.313210i	\(0.101404\pi\)
							−0.977981	+	0.208695i	\(0.933078\pi\)
\(43\)	43.8662	−	20.2947i	0.155571	−	0.0719746i	−0.340559	−	0.940223i	\(-0.610616\pi\)
							0.496129	+	0.868249i	\(0.334754\pi\)
\(47\)	−154.726	−	117.620i	−0.480195	−	0.365035i	0.336898	−	0.941541i	\(-0.390622\pi\)
							−0.817092	+	0.576507i	\(0.804415\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.e.b.133.10	yes	420
59.4	even	29	inner	177.4.e.b.4.10	✓	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.e.b.4.10	✓	420	59.4	even	29	inner
177.4.e.b.133.10	yes	420	1.1	even	1	trivial