Embedded newform 177.3.h.a.104.32

• Label: 177.3.h.a.104.32
• Level: $177$
• Weight: $3$
• Character: 177.104
• 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			104.32
Character	\(\chi\)	\(=\)	177.104
Dual form			177.3.h.a.80.32

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.30426 + 2.16769i) q^{2} +(-2.33183 + 1.88748i) q^{3} +(-1.12417 + 2.12041i) q^{4} +(0.380931 + 3.50260i) q^{5} +(-7.13278 - 2.59293i) q^{6} +(-11.3918 + 3.83833i) q^{7} +(4.04183 - 0.219142i) q^{8} +(1.87485 - 8.80255i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{24}{29}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.30426	+	2.16769i	0.652129	+	1.08385i	0.989784	+	0.142576i	\(0.0455387\pi\)
							−0.337655	+	0.941270i	\(0.609634\pi\)
\(3\)	−2.33183	+	1.88748i	−0.777276	+	0.629160i
							\(5\)	0.380931	+	3.50260i	0.0761862	+	0.700521i	0.968484	+	0.249074i	\(0.0801263\pi\)
−0.892298	+	0.451447i	\(0.850908\pi\)
\(7\)	−11.3918	+	3.83833i	−1.62739	+	0.548333i	−0.977362	−	0.211574i	\(-0.932141\pi\)
							−0.650032	+	0.759907i	\(0.725245\pi\)
\(11\)	5.10134	+	3.45879i	0.463758	+	0.314436i	0.770594	−	0.637327i	\(-0.219960\pi\)
							−0.306835	+	0.951763i	\(0.599270\pi\)
\(13\)	−12.9015	−	12.2210i	−0.992424	−	0.940074i	0.00578012	−	0.999983i	\(-0.498160\pi\)
							−0.998204	+	0.0599096i	\(0.980919\pi\)
\(17\)	−8.29088	+	24.6065i	−0.487699	+	1.44744i	0.367932	+	0.929853i	\(0.380066\pi\)
							−0.855631	+	0.517586i	\(0.826831\pi\)
\(19\)	−1.18699	−	7.24031i	−0.0624731	−	0.381069i	−0.999494	−	0.0317967i	\(-0.989877\pi\)
							0.937021	−	0.349272i	\(-0.113571\pi\)
\(23\)	−4.36314	+	1.21142i	−0.189702	+	0.0526704i	−0.361080	−	0.932535i	\(-0.617592\pi\)
							0.171378	+	0.985205i	\(0.445178\pi\)
\(29\)	−0.602694	+	1.00169i	−0.0207826	+	0.0345409i	−0.867074	−	0.498179i	\(-0.834002\pi\)
							0.846292	+	0.532720i	\(0.178830\pi\)
\(31\)	−8.00494	+	48.8280i	−0.258224	+	1.57510i	0.465526	+	0.885034i	\(0.345865\pi\)
							−0.723750	+	0.690062i	\(0.757583\pi\)
\(37\)	1.62895	−	30.0443i	0.0440258	−	0.812008i	−0.890253	−	0.455465i	\(-0.849473\pi\)
							0.934279	−	0.356542i	\(-0.116044\pi\)
\(41\)	17.9159	+	4.97432i	0.436973	+	0.121325i	0.479046	−	0.877790i	\(-0.340983\pi\)
							−0.0420735	+	0.999115i	\(0.513396\pi\)
\(43\)	42.8282	+	63.1669i	0.996004	+	1.46900i	0.879553	+	0.475800i	\(0.157841\pi\)
							0.116451	+	0.993196i	\(0.462848\pi\)
\(47\)	−4.67272	+	42.9650i	−0.0994196	+	0.914149i	0.833123	+	0.553088i	\(0.186551\pi\)
							−0.932542	+	0.361060i	\(0.882415\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.104.32	yes	1064
3.2	odd	2	inner	177.3.h.a.104.7	yes	1064
59.21	even	29	inner	177.3.h.a.80.7	✓	1064
177.80	odd	58	inner	177.3.h.a.80.32	yes	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.80.7	✓	1064	59.21	even	29	inner
177.3.h.a.80.32	yes	1064	177.80	odd	58	inner
177.3.h.a.104.7	yes	1064	3.2	odd	2	inner
177.3.h.a.104.32	yes	1064	1.1	even	1	trivial