Embedded newform 177.4.f.a.2.16

• Label: 177.4.f.a.2.16
• Level: $177$
• Weight: $4$
• Character: 177.2
• 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			2.16
Character	\(\chi\)	\(=\)	177.2
Dual form			177.4.f.a.89.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.176293 + 3.25153i) q^{2} +(-5.15278 - 0.670007i) q^{3} +(-2.58825 - 0.281489i) q^{4} +(-0.0532670 + 0.158091i) q^{5} +(3.08694 - 16.6363i) q^{6} +(11.3529 + 16.7443i) q^{7} +(-2.84293 + 17.3411i) q^{8} +(26.1022 + 6.90479i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 21 q^{3} - 278 q^{4} - 29 q^{6} - 42 q^{7} - 25 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.176293	+	3.25153i	−0.0623289	+	1.14959i	0.785341	+	0.619064i	\(0.212488\pi\)
							−0.847669	+	0.530525i	\(0.821995\pi\)
\(3\)	−5.15278	−	0.670007i	−0.991652	−	0.128943i
							\(5\)	−0.0532670	+	0.158091i	−0.00476434	+	0.0141401i	−0.950009	−	0.312223i	\(-0.898927\pi\)
0.945245	+	0.326363i	\(0.105823\pi\)
\(7\)	11.3529	+	16.7443i	0.612998	+	0.904105i	0.999837	−	0.0180424i	\(-0.00574337\pi\)
							−0.386839	+	0.922147i	\(0.626433\pi\)
\(11\)	54.2570	−	11.9429i	1.48719	−	0.327356i	0.604234	−	0.796807i	\(-0.293479\pi\)
							0.882958	+	0.469452i	\(0.155548\pi\)
\(13\)	−57.7437	+	49.0480i	−1.23194	+	1.04642i	−0.234709	+	0.972066i	\(0.575414\pi\)
							−0.997232	+	0.0743546i	\(0.976310\pi\)
\(17\)	−95.7613	−	64.9278i	−1.36621	−	0.926312i	−0.366216	−	0.930530i	\(-0.619347\pi\)
							−0.999991	+	0.00421794i	\(0.998657\pi\)
\(19\)	−15.9943	+	30.1684i	−0.193123	+	0.364269i	−0.961096	−	0.276213i	\(-0.910920\pi\)
							0.767973	+	0.640482i	\(0.221265\pi\)
\(23\)	6.30792	−	5.97518i	0.0571866	−	0.0541700i	−0.658573	−	0.752517i	\(-0.728839\pi\)
							0.715759	+	0.698347i	\(0.246081\pi\)
\(29\)	34.7310	−	1.88306i	0.222392	−	0.0120578i	0.0573940	−	0.998352i	\(-0.481721\pi\)
							0.164998	+	0.986294i	\(0.447238\pi\)
\(31\)	−209.518	+	111.079i	−1.21389	+	0.643563i	−0.947331	−	0.320256i	\(-0.896231\pi\)
							−0.266558	+	0.963819i	\(0.585886\pi\)
\(37\)	−206.277	+	33.8173i	−0.916532	+	0.150258i	−0.601546	−	0.798838i	\(-0.705448\pi\)
							−0.314986	+	0.949096i	\(0.602000\pi\)
\(41\)	68.2338	−	72.0336i	0.259911	−	0.274384i	−0.582960	−	0.812501i	\(-0.698105\pi\)
							0.842870	+	0.538117i	\(0.180864\pi\)
\(43\)	−50.5051	+	229.447i	−0.179115	+	0.813728i	0.798276	+	0.602291i	\(0.205745\pi\)
							−0.977392	+	0.211437i	\(0.932186\pi\)
\(47\)	−26.0856	+	8.78926i	−0.0809569	+	0.0272776i	−0.359489	−	0.933149i	\(-0.617049\pi\)
							0.278532	+	0.960427i	\(0.410152\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.f.a.2.16	✓	1624
3.2	odd	2	inner	177.4.f.a.2.43	yes	1624
59.30	odd	58	inner	177.4.f.a.89.43	yes	1624
177.89	even	58	inner	177.4.f.a.89.16	yes	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.f.a.2.16	✓	1624	1.1	even	1	trivial
177.4.f.a.2.43	yes	1624	3.2	odd	2	inner
177.4.f.a.89.16	yes	1624	177.89	even	58	inner
177.4.f.a.89.43	yes	1624	59.30	odd	58	inner