Embedded newform 177.4.e.b.133.11

• Label: 177.4.e.b.133.11
• 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			133.11
Character	\(\chi\)	\(=\)	177.133
Dual form			177.4.e.b.4.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.91566 - 0.317097i) q^{2} +(1.94216 + 2.28649i) q^{3} +(0.587538 - 0.129327i) q^{4} +(-2.37530 + 1.80566i) q^{5} +(6.38771 + 6.05076i) q^{6} +(4.97289 + 12.4810i) q^{7} +(-20.5625 + 6.92833i) q^{8} +(-1.45604 + 8.88144i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q + 2 q^{2} - 45 q^{3} - 56 q^{4} - 14 q^{5} + 6 q^{6} + 6 q^{7} + 66 q^{8} - 135 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.91566	−	0.317097i	1.03084	−	0.112111i	0.422967	−	0.906145i	\(-0.360989\pi\)
							0.607873	+	0.794034i	\(0.292023\pi\)
\(3\)	1.94216	+	2.28649i	0.373769	+	0.440034i
							\(5\)	−2.37530	+	1.80566i	−0.212453	+	0.161503i	−0.705989	−	0.708223i	\(-0.749497\pi\)
0.493536	+	0.869725i	\(0.335704\pi\)
\(7\)	4.97289	+	12.4810i	0.268511	+	0.673912i	0.999968	−	0.00797561i	\(-0.00253874\pi\)
							−0.731457	+	0.681887i	\(0.761159\pi\)
\(11\)	42.3122	+	19.5757i	1.15978	+	0.536573i	0.902892	−	0.429867i	\(-0.141440\pi\)
							0.256892	+	0.966440i	\(0.417302\pi\)
\(13\)	6.52008	+	39.7708i	0.139104	+	0.848494i	0.960529	+	0.278179i	\(0.0897307\pi\)
							−0.821426	+	0.570315i	\(0.806821\pi\)
\(17\)	−39.4135	+	98.9205i	−0.562305	+	1.41128i	0.323059	+	0.946379i	\(0.395289\pi\)
							−0.885363	+	0.464900i	\(0.846090\pi\)
\(19\)	62.8249	−	92.6599i	0.758581	−	1.11882i	−0.230733	−	0.973017i	\(-0.574112\pi\)
							0.989313	−	0.145806i	\(-0.0465774\pi\)
\(23\)	−7.14293	−	131.744i	−0.0647567	−	1.19437i	−0.832654	−	0.553794i	\(-0.813180\pi\)
							0.767897	−	0.640573i	\(-0.221303\pi\)
\(29\)	93.3115	+	10.1482i	0.597500	+	0.0649820i	0.401872	−	0.915696i	\(-0.368360\pi\)
							0.195628	+	0.980678i	\(0.437325\pi\)
\(31\)	−39.7748	−	58.6634i	−0.230444	−	0.339879i	0.694833	−	0.719171i	\(-0.255478\pi\)
							−0.925277	+	0.379292i	\(0.876168\pi\)
\(37\)	−211.930	−	71.4074i	−0.941649	−	0.317279i	−0.193720	−	0.981057i	\(-0.562055\pi\)
							−0.747929	+	0.663778i	\(0.768952\pi\)
\(41\)	24.2399	−	447.079i	0.0923326	−	1.70297i	−0.475407	−	0.879766i	\(-0.657699\pi\)
							0.567740	−	0.823208i	\(-0.307818\pi\)
\(43\)	269.623	−	124.741i	0.956211	−	0.442390i	0.121203	−	0.992628i	\(-0.461325\pi\)
							0.835008	+	0.550237i	\(0.185463\pi\)
\(47\)	159.010	+	120.876i	0.493490	+	0.375141i	0.822117	−	0.569318i	\(-0.192793\pi\)
							−0.328627	+	0.944460i	\(0.606586\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.e.b.133.11	yes	420
59.4	even	29	inner	177.4.e.b.4.11	✓	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.e.b.4.11	✓	420	59.4	even	29	inner
177.4.e.b.133.11	yes	420	1.1	even	1	trivial