Embedded newform 177.4.e.a.4.9

• Label: 177.4.e.a.4.9
• Level: $177$
• Weight: $4$
• Character: 177.4
• 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			4.9
Character	\(\chi\)	\(=\)	177.4
Dual form			177.4.e.a.133.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.857263 + 0.0932330i) q^{2} +(-1.94216 + 2.28649i) q^{3} +(-7.08676 - 1.55991i) q^{4} +(6.24712 + 4.74894i) q^{5} +(-1.87812 + 1.77905i) q^{6} +(0.937924 - 2.35401i) q^{7} +(-12.4672 - 4.20069i) q^{8} +(-1.45604 - 8.88144i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 2 q^{2} + 45 q^{3} - 64 q^{4} + 14 q^{5} + 6 q^{6} + 6 q^{7} + 18 q^{8} - 135 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.857263	+	0.0932330i	0.303088	+	0.0329628i	0.258400	−	0.966038i	\(-0.416805\pi\)
							0.0446887	+	0.999001i	\(0.485770\pi\)
\(3\)	−1.94216	+	2.28649i	−0.373769	+	0.440034i
							\(5\)	6.24712	+	4.74894i	0.558760	+	0.424758i	0.846064	−	0.533081i	\(-0.178966\pi\)
−0.287304	+	0.957839i	\(0.592759\pi\)
\(7\)	0.937924	−	2.35401i	0.0506431	−	0.127105i	−0.901479	−	0.432823i	\(-0.857517\pi\)
							0.952122	+	0.305719i	\(0.0988967\pi\)
\(11\)	−28.1319	+	13.0152i	−0.771100	+	0.356749i	−0.765667	−	0.643237i	\(-0.777591\pi\)
							−0.00543211	+	0.999985i	\(0.501729\pi\)
\(13\)	9.71699	−	59.2710i	0.207308	−	1.26452i	−0.654021	−	0.756477i	\(-0.726919\pi\)
							0.861329	−	0.508048i	\(-0.169633\pi\)
\(17\)	−19.2476	−	48.3078i	−0.274602	−	0.689198i	−1.00000		0.000868369i	\(-0.999724\pi\)
							0.725398	−	0.688330i	\(-0.241656\pi\)
\(19\)	−7.38592	−	10.8934i	−0.0891814	−	0.131533i	0.780481	−	0.625179i	\(-0.214974\pi\)
							−0.869663	+	0.493646i	\(0.835664\pi\)
\(23\)	10.0245	−	184.892i	0.0908809	−	1.67620i	−0.497383	−	0.867531i	\(-0.665706\pi\)
							0.588264	−	0.808669i	\(-0.299812\pi\)
\(29\)	−231.508	+	25.1780i	−1.48241	+	0.161222i	−0.813258	−	0.581903i	\(-0.802308\pi\)
							−0.669153	+	0.743125i	\(0.733343\pi\)
\(31\)	182.807	−	269.620i	1.05913	−	1.56210i	0.255656	−	0.966768i	\(-0.417709\pi\)
							0.803477	−	0.595336i	\(-0.202981\pi\)
\(37\)	−370.437	+	124.815i	−1.64593	+	0.554580i	−0.981282	−	0.192574i	\(-0.938316\pi\)
							−0.664651	+	0.747154i	\(0.731420\pi\)
\(41\)	22.2309	+	410.025i	0.0846801	+	1.56183i	0.664625	+	0.747177i	\(0.268591\pi\)
							−0.579945	+	0.814655i	\(0.696926\pi\)
\(43\)	−119.670	−	55.3653i	−0.424407	−	0.196352i	0.196047	−	0.980595i	\(-0.437190\pi\)
							−0.620454	+	0.784243i	\(0.713052\pi\)
\(47\)	298.590	−	226.982i	0.926676	−	0.704441i	−0.0287765	−	0.999586i	\(-0.509161\pi\)
							0.955452	+	0.295145i	\(0.0953680\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.e.a.4.9	✓	420
59.15	even	29	inner	177.4.e.a.133.9	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.e.a.4.9	✓	420	1.1	even	1	trivial
177.4.e.a.133.9	yes	420	59.15	even	29	inner