Embedded newform 177.3.h.a.104.26

• Label: 177.3.h.a.104.26
• 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			104.26
Character	\(\chi\)	\(=\)	177.104
Dual form			177.3.h.a.80.26

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.891508 + 1.48170i) q^{2} +(2.40125 + 1.79833i) q^{3} +(0.472992 - 0.892157i) q^{4} +(-1.05004 - 9.65499i) q^{5} +(-0.523837 + 5.16116i) q^{6} +(-1.35411 + 0.456253i) q^{7} +(8.65033 - 0.469007i) q^{8} +(2.53205 + 8.63648i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.891508	+	1.48170i	0.445754	+	0.740849i	0.995528	−	0.0944643i	\(-0.0301138\pi\)
							−0.549774	+	0.835313i	\(0.685286\pi\)
\(3\)	2.40125	+	1.79833i	0.800418	+	0.599442i
							\(5\)	−1.05004	−	9.65499i	−0.210009	−	1.93100i	−0.341905	−	0.939735i	\(-0.611072\pi\)
0.131896	−	0.991264i	\(-0.457894\pi\)
\(7\)	−1.35411	+	0.456253i	−0.193444	+	0.0651790i	−0.414356	−	0.910115i	\(-0.635993\pi\)
							0.220912	+	0.975294i	\(0.429097\pi\)
\(11\)	12.7458	+	8.64189i	1.15871	+	0.785626i	0.980141	−	0.198301i	\(-0.0635424\pi\)
							0.178570	+	0.983927i	\(0.442853\pi\)
\(13\)	−9.09167	−	8.61209i	−0.699359	−	0.662468i	0.252993	−	0.967468i	\(-0.418585\pi\)
							−0.952352	+	0.305000i	\(0.901344\pi\)
\(17\)	1.14428	−	3.39610i	0.0673105	−	0.199770i	−0.908719	−	0.417408i	\(-0.862939\pi\)
							0.976030	+	0.217638i	\(0.0698351\pi\)
\(19\)	3.10314	+	18.9283i	0.163323	+	0.996228i	0.933547	+	0.358454i	\(0.116696\pi\)
							−0.770224	+	0.637774i	\(0.779856\pi\)
\(23\)	−24.8642	+	6.90351i	−1.08105	+	0.300152i	−0.761995	−	0.647583i	\(-0.775780\pi\)
							−0.319057	+	0.947736i	\(0.603366\pi\)
\(29\)	17.7971	−	29.5790i	0.613693	−	1.01997i	−0.381719	−	0.924278i	\(-0.624668\pi\)
							0.995412	−	0.0956867i	\(-0.0305047\pi\)
\(31\)	−1.76368	+	10.7580i	−0.0568930	+	0.347032i	0.943005	+	0.332778i	\(0.107986\pi\)
							−0.999898	+	0.0142546i	\(0.995462\pi\)
\(37\)	0.114216	−	2.10659i	0.00308692	−	0.0569349i	−0.996584	−	0.0825843i	\(-0.973683\pi\)
							0.999671	+	0.0256495i	\(0.00816537\pi\)
\(41\)	−44.0373	−	12.2269i	−1.07408	−	0.298217i	−0.314918	−	0.949119i	\(-0.601977\pi\)
							−0.759162	+	0.650902i	\(0.774391\pi\)
\(43\)	34.3766	+	50.7017i	0.799456	+	1.17911i	0.980671	+	0.195666i	\(0.0626868\pi\)
							−0.181215	+	0.983444i	\(0.558003\pi\)
\(47\)	1.22085	−	11.2255i	0.0259755	−	0.238841i	−0.973951	−	0.226757i	\(-0.927188\pi\)
							0.999927	−	0.0120844i	\(-0.00384667\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.104.26	yes	1064
3.2	odd	2	inner	177.3.h.a.104.13	yes	1064
59.21	even	29	inner	177.3.h.a.80.13	✓	1064
177.80	odd	58	inner	177.3.h.a.80.26	yes	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.80.13	✓	1064	59.21	even	29	inner
177.3.h.a.80.26	yes	1064	177.80	odd	58	inner
177.3.h.a.104.13	yes	1064	3.2	odd	2	inner
177.3.h.a.104.26	yes	1064	1.1	even	1	trivial