Embedded newform 177.3.h.a.71.8

• Label: 177.3.h.a.71.8
• Level: $177$
• Weight: $3$
• Character: 177.71
• 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			71.8
Character	\(\chi\)	\(=\)	177.71
Dual form			177.3.h.a.5.8

$q$-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 \(\left(\mathbb{Z}/177\mathbb{Z}\right)^\times\).

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{26}{29}\right)\)	\(-1\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.5.8	✓	1064	177.5	odd	58	inner
177.3.h.a.5.31	yes	1064	59.5	even	29	inner
177.3.h.a.71.8	yes	1064	1.1	even	1	trivial
177.3.h.a.71.31	yes	1064	3.2	odd	2	inner