Embedded newform 177.3.h.a.104.37

• Label: 177.3.h.a.104.37
• 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			104.37
Character	\(\chi\)	\(=\)	177.104
Dual form			177.3.h.a.80.37

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.90323 + 3.16320i) q^{2} +(-2.63839 + 1.42790i) q^{3} +(-4.50990 + 8.50657i) q^{4} +(0.853287 + 7.84585i) q^{5} +(-9.53820 - 5.62814i) q^{6} +(12.1464 - 4.09260i) q^{7} +(-20.7465 + 1.12484i) q^{8} +(4.92223 - 7.53470i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.90323	+	3.16320i	0.951617	+	1.58160i	0.805717	+	0.592300i	\(0.201780\pi\)
							0.145900	+	0.989299i	\(0.453392\pi\)
\(3\)	−2.63839	+	1.42790i	−0.879464	+	0.475965i
							\(5\)	0.853287	+	7.84585i	0.170657	+	1.56917i	0.695851	+	0.718186i	\(0.255027\pi\)
−0.525193	+	0.850983i	\(0.676007\pi\)
\(7\)	12.1464	−	4.09260i	1.73520	−	0.584657i	0.739157	−	0.673533i	\(-0.235224\pi\)
							0.996043	+	0.0888757i	\(0.0283274\pi\)
\(11\)	−2.40152	−	1.62827i	−0.218320	−	0.148025i	0.447125	−	0.894471i	\(-0.352448\pi\)
							−0.665445	+	0.746447i	\(0.731758\pi\)
\(13\)	−1.30261	−	1.23390i	−0.100201	−	0.0949153i	0.635960	−	0.771722i	\(-0.280605\pi\)
							−0.736160	+	0.676807i	\(0.763363\pi\)
\(17\)	5.27453	−	15.6542i	0.310266	−	0.920838i	−0.673155	−	0.739502i	\(-0.735061\pi\)
							0.983421	−	0.181336i	\(-0.0580423\pi\)
\(19\)	−0.0242791	−	0.148096i	−0.00127785	−	0.00779454i	0.986180	−	0.165678i	\(-0.0529812\pi\)
							−0.987458	+	0.157883i	\(0.949533\pi\)
\(23\)	30.3429	−	8.42467i	1.31926	−	0.366290i	0.464618	−	0.885511i	\(-0.346192\pi\)
							0.854640	+	0.519221i	\(0.173778\pi\)
\(29\)	−1.00197	+	1.66529i	−0.0345508	+	0.0574239i	−0.873651	−	0.486552i	\(-0.838254\pi\)
							0.839101	+	0.543976i	\(0.183082\pi\)
\(31\)	−6.01090	+	36.6648i	−0.193900	+	1.18274i	0.693056	+	0.720883i	\(0.256264\pi\)
							−0.886956	+	0.461853i	\(0.847185\pi\)
\(37\)	−1.35330	+	24.9602i	−0.0365757	+	0.674600i	0.921551	+	0.388257i	\(0.126923\pi\)
							−0.958127	+	0.286343i	\(0.907560\pi\)
\(41\)	−64.5725	−	17.9285i	−1.57494	−	0.437280i	−0.633301	−	0.773906i	\(-0.718300\pi\)
							−0.941638	+	0.336626i	\(0.890714\pi\)
\(43\)	−19.8338	−	29.2526i	−0.461250	−	0.680293i	0.523940	−	0.851755i	\(-0.324461\pi\)
							−0.985191	+	0.171462i	\(0.945151\pi\)
\(47\)	4.02836	−	37.0401i	0.0857097	−	0.788088i	−0.869855	−	0.493307i	\(-0.835788\pi\)
							0.955565	−	0.294781i	\(-0.0952466\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.104.37	yes	1064
3.2	odd	2	inner	177.3.h.a.104.2	yes	1064
59.21	even	29	inner	177.3.h.a.80.2	✓	1064
177.80	odd	58	inner	177.3.h.a.80.37	yes	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.80.2	✓	1064	59.21	even	29	inner
177.3.h.a.80.37	yes	1064	177.80	odd	58	inner
177.3.h.a.104.2	yes	1064	3.2	odd	2	inner
177.3.h.a.104.37	yes	1064	1.1	even	1	trivial