Embedded newform 177.3.h.a.5.20

• Label: 177.3.h.a.5.20
• Level: $177$
• Weight: $3$
• Character: 177.5
• Analytic conductor: $4.823$
• Analytic rank: $0$
• Dimension: $1064$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	177.h (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.82290067918\)
Analytic rank:	\(0\)
Dimension:	\(1064\)
Relative dimension:	\(38\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			5.20
Character	\(\chi\)	\(=\)	177.5
Dual form			177.3.h.a.71.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0718854 - 0.213348i) q^{2} +(0.416138 + 2.97100i) q^{3} +(3.14402 + 2.39002i) q^{4} +(7.60675 + 3.03081i) q^{5} +(0.663771 + 0.124789i) q^{6} +(4.35627 - 2.01543i) q^{7} +(1.48128 - 1.00433i) q^{8} +(-8.65366 + 2.47269i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{3}{29}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0718854	−	0.213348i	0.0359427	−	0.106674i	−0.928180	−	0.372132i	\(-0.878627\pi\)
							0.964122	+	0.265458i	\(0.0855233\pi\)
\(3\)	0.416138	+	2.97100i	0.138713	+	0.990333i
							\(5\)	7.60675	+	3.03081i	1.52135	+	0.606161i	0.973147	−	0.230184i	\(-0.0739327\pi\)
0.548203	+	0.836345i	\(0.315312\pi\)
\(7\)	4.35627	−	2.01543i	0.622325	−	0.287918i	−0.0832773	−	0.996526i	\(-0.526539\pi\)
							0.705602	+	0.708608i	\(0.250677\pi\)
\(11\)	−20.3175	−	5.64112i	−1.84704	−	0.512829i	−0.847213	−	0.531253i	\(-0.821721\pi\)
							−0.999830	+	0.0184237i	\(0.994135\pi\)
\(13\)	10.6705	−	20.1268i	0.820811	−	1.54821i	−0.0154480	−	0.999881i	\(-0.504917\pi\)
							0.836259	−	0.548334i	\(-0.184738\pi\)
\(17\)	6.72609	−	14.5382i	0.395652	−	0.855188i	−0.602709	−	0.797961i	\(-0.705912\pi\)
							0.998361	−	0.0572271i	\(-0.0182259\pi\)
\(19\)	−20.7890	+	4.57601i	−1.09416	+	0.240843i	−0.725156	−	0.688585i	\(-0.758232\pi\)
							−0.369004	+	0.929428i	\(0.620301\pi\)
\(23\)	3.12206	+	0.511835i	0.135742	+	0.0222537i	0.229270	−	0.973363i	\(-0.426366\pi\)
							−0.0935281	+	0.995617i	\(0.529814\pi\)
\(29\)	−0.204099	−	0.605744i	−0.00703789	−	0.0208877i	0.944077	−	0.329726i	\(-0.106956\pi\)
							−0.951115	+	0.308838i	\(0.900060\pi\)
\(31\)	−39.7784	−	8.75589i	−1.28317	−	0.282448i	−0.479513	−	0.877535i	\(-0.659187\pi\)
							−0.803662	+	0.595087i	\(0.797118\pi\)
\(37\)	−15.8511	+	23.3786i	−0.428407	+	0.631854i	−0.979226	−	0.202771i	\(-0.935005\pi\)
							0.550819	+	0.834625i	\(0.314316\pi\)
\(41\)	13.9739	−	2.29091i	0.340827	−	0.0558758i	0.0110621	−	0.999939i	\(-0.496479\pi\)
							0.329765	+	0.944063i	\(0.393030\pi\)
\(43\)	−3.08193	−	11.1001i	−0.0716728	−	0.258142i	0.919030	−	0.394188i	\(-0.128974\pi\)
							−0.990702	+	0.136046i	\(0.956560\pi\)
\(47\)	11.3922	−	4.53905i	0.242386	−	0.0965755i	−0.245782	−	0.969325i	\(-0.579045\pi\)
							0.488168	+	0.872750i	\(0.337665\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.5.20	yes	1064
3.2	odd	2	inner	177.3.h.a.5.19	✓	1064
59.12	even	29	inner	177.3.h.a.71.19	yes	1064
177.71	odd	58	inner	177.3.h.a.71.20	yes	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.5.19	✓	1064	3.2	odd	2	inner
177.3.h.a.5.20	yes	1064	1.1	even	1	trivial
177.3.h.a.71.19	yes	1064	59.12	even	29	inner
177.3.h.a.71.20	yes	1064	177.71	odd	58	inner