Embedded newform 177.4.f.a.101.6

• Label: 177.4.f.a.101.6
• Level: $177$
• Weight: $4$
• Character: 177.101
• Analytic conductor: $10.443$
• Analytic rank: $0$
• Dimension: $1624$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.f (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.4433380710\)
Analytic rank:	\(0\)
Dimension:	\(1624\)
Relative dimension:	\(58\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			101.6
Character	\(\chi\)	\(=\)	177.101
Dual form			177.4.f.a.170.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.66261 + 3.92706i) q^{2} +(-2.04060 + 4.77870i) q^{3} +(-5.37119 - 13.4807i) q^{4} +(-8.08344 + 17.4721i) q^{5} +(-13.3329 - 20.7374i) q^{6} +(-2.26801 - 8.16864i) q^{7} +(30.1713 + 6.64121i) q^{8} +(-18.6719 - 19.5028i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 21 q^{3} - 278 q^{4} - 29 q^{6} - 42 q^{7} - 25 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{11}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.66261	+	3.92706i	−0.941376	+	1.38843i	−0.0199797	+	0.999800i	\(0.506360\pi\)
							−0.921396	+	0.388625i	\(0.872950\pi\)
\(3\)	−2.04060	+	4.77870i	−0.392713	+	0.919661i
							\(5\)	−8.08344	+	17.4721i	−0.723005	+	1.56275i	0.100175	+	0.994970i	\(0.468060\pi\)
−0.823180	+	0.567780i	\(0.807802\pi\)
\(7\)	−2.26801	−	8.16864i	−0.122461	−	0.441065i	0.876743	−	0.480960i	\(-0.159712\pi\)
							−0.999204	+	0.0398944i	\(0.987298\pi\)
\(11\)	1.34109	+	1.27035i	0.0367595	+	0.0348205i	0.705859	−	0.708353i	\(-0.250561\pi\)
							−0.669099	+	0.743173i	\(0.733320\pi\)
\(13\)	−3.48889	+	32.0798i	−0.0744341	+	0.684410i	0.896167	+	0.443717i	\(0.146340\pi\)
							−0.970601	+	0.240694i	\(0.922625\pi\)
\(17\)	−51.4705	−	14.2907i	−0.734320	−	0.203883i	−0.119815	−	0.992796i	\(-0.538230\pi\)
							−0.614505	+	0.788913i	\(0.710644\pi\)
\(19\)	−28.1477	−	21.3973i	−0.339870	−	0.258362i	0.421291	−	0.906926i	\(-0.361577\pi\)
							−0.761161	+	0.648563i	\(0.775370\pi\)
\(23\)	46.3993	+	87.5184i	0.420649	+	0.793428i	0.999762	−	0.0218193i	\(-0.00694585\pi\)
							−0.579113	+	0.815247i	\(0.696601\pi\)
\(29\)	−52.8663	+	35.8442i	−0.338518	+	0.229521i	−0.718587	−	0.695437i	\(-0.755211\pi\)
							0.380069	+	0.924958i	\(0.375900\pi\)
\(31\)	99.0805	+	130.338i	0.574044	+	0.755142i	0.987882	−	0.155208i	\(-0.0496049\pi\)
							−0.413837	+	0.910351i	\(0.635812\pi\)
\(37\)	71.9240	+	326.754i	0.319574	+	1.45184i	0.813108	+	0.582112i	\(0.197774\pi\)
							−0.493534	+	0.869726i	\(0.664295\pi\)
\(41\)	−229.522	−	121.685i	−0.874276	−	0.463512i	−0.0300807	−	0.999547i	\(-0.509576\pi\)
							−0.844195	+	0.536036i	\(0.819921\pi\)
\(43\)	231.628	+	244.526i	0.821463	+	0.867207i	0.993018	−	0.117967i	\(-0.0376377\pi\)
							−0.171555	+	0.985175i	\(0.554879\pi\)
\(47\)	−424.853	+	196.558i	−1.31853	+	0.610019i	−0.947843	−	0.318737i	\(-0.896741\pi\)
							−0.370691	+	0.928756i	\(0.620879\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.f.a.101.6	✓	1624
3.2	odd	2	inner	177.4.f.a.101.53	yes	1624
59.52	odd	58	inner	177.4.f.a.170.53	yes	1624
177.170	even	58	inner	177.4.f.a.170.6	yes	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.f.a.101.6	✓	1624	1.1	even	1	trivial
177.4.f.a.101.53	yes	1624	3.2	odd	2	inner
177.4.f.a.170.6	yes	1624	177.170	even	58	inner
177.4.f.a.170.53	yes	1624	59.52	odd	58	inner