Embedded newform 177.3.h.a.5.8

• Label: 177.3.h.a.5.8
• 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			5.8
Character	\(\chi\)	\(=\)	177.5
Dual form			177.3.h.a.71.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.892372 + 2.64847i) q^{2} +(0.713097 - 2.91402i) q^{3} +(-3.03367 - 2.30614i) q^{4} +(-2.87136 - 1.14405i) q^{5} +(7.08133 + 4.48900i) q^{6} +(2.33765 - 1.08151i) q^{7} +(-0.437890 + 0.296897i) q^{8} +(-7.98299 - 4.15595i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.892372	+	2.64847i	−0.446186	+	1.32423i	0.454881	+	0.890552i	\(0.349682\pi\)
							−0.901067	+	0.433681i	\(0.857215\pi\)
\(3\)	0.713097	−	2.91402i	0.237699	−	0.971339i
							\(5\)	−2.87136	−	1.14405i	−0.574271	−	0.228811i	0.0648781	−	0.997893i	\(-0.479334\pi\)
−0.639149	+	0.769083i	\(0.720713\pi\)
\(7\)	2.33765	−	1.08151i	0.333951	−	0.154502i	−0.245746	−	0.969334i	\(-0.579033\pi\)
							0.579697	+	0.814832i	\(0.303171\pi\)
\(11\)	−5.90871	−	1.64054i	−0.537155	−	0.149140i	−0.0116521	−	0.999932i	\(-0.503709\pi\)
							−0.525503	+	0.850792i	\(0.676123\pi\)
\(13\)	11.2971	−	21.3085i	0.869006	−	1.63912i	0.104479	−	0.994527i	\(-0.466682\pi\)
							0.764527	−	0.644592i	\(-0.222973\pi\)
\(17\)	8.75474	−	18.9231i	0.514985	−	1.11312i	−0.459399	−	0.888230i	\(-0.651935\pi\)
							0.974384	−	0.224891i	\(-0.0722027\pi\)
\(19\)	1.82985	−	0.402779i	0.0963076	−	0.0211989i	−0.166555	−	0.986032i	\(-0.553264\pi\)
							0.262863	+	0.964833i	\(0.415333\pi\)
\(23\)	−21.5222	−	3.52838i	−0.935747	−	0.153408i	−0.325431	−	0.945566i	\(-0.605509\pi\)
							−0.610316	+	0.792158i	\(0.708958\pi\)
\(29\)	2.19392	+	6.51131i	0.0756523	+	0.224528i	0.978806	−	0.204789i	\(-0.0656508\pi\)
							−0.903154	+	0.429317i	\(0.858754\pi\)
\(31\)	44.5980	+	9.81675i	1.43864	+	0.316669i	0.864726	−	0.502243i	\(-0.167492\pi\)
							0.573918	+	0.818913i	\(0.305423\pi\)
\(37\)	−32.5155	+	47.9568i	−0.878798	+	1.29613i	0.0754617	+	0.997149i	\(0.475957\pi\)
							−0.954259	+	0.298981i	\(0.903353\pi\)
\(41\)	4.66989	−	0.765589i	0.113900	−	0.0186729i	−0.104563	−	0.994518i	\(-0.533344\pi\)
							0.218462	+	0.975845i	\(0.429896\pi\)
\(43\)	−12.9179	−	46.5262i	−0.300417	−	1.08201i	−0.947545	−	0.319623i	\(-0.896444\pi\)
							0.647127	−	0.762382i	\(-0.275970\pi\)
\(47\)	22.9848	−	9.15799i	0.489039	−	0.194851i	−0.112567	−	0.993644i	\(-0.535907\pi\)
							0.601606	+	0.798793i	\(0.294528\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.5.8	✓	1064
3.2	odd	2	inner	177.3.h.a.5.31	yes	1064
59.12	even	29	inner	177.3.h.a.71.31	yes	1064
177.71	odd	58	inner	177.3.h.a.71.8	yes	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.5.8	✓	1064	1.1	even	1	trivial
177.3.h.a.5.31	yes	1064	3.2	odd	2	inner
177.3.h.a.71.8	yes	1064	177.71	odd	58	inner
177.3.h.a.71.31	yes	1064	59.12	even	29	inner