Embedded newform 177.2.f.a.155.16

• Label: 177.2.f.a.155.16
• Level: $177$
• Weight: $2$
• Character: 177.155
• Analytic conductor: $1.413$
• Analytic rank: $0$
• Dimension: $504$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.f (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.41335211578\)
Analytic rank:	\(0\)
Dimension:	\(504\)
Relative dimension:	\(18\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			155.16
Character	\(\chi\)	\(=\)	177.155
Dual form			177.2.f.a.8.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.317307 + 1.93549i) q^{2} +(-1.70744 + 0.290956i) q^{3} +(-1.75012 + 0.589685i) q^{4} +(-2.92507 - 1.98325i) q^{5} +(-1.10492 - 3.21240i) q^{6} +(-2.73852 - 0.602794i) q^{7} +(0.140747 + 0.265477i) q^{8} +(2.83069 - 0.993578i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{55}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.317307	+	1.93549i	0.224370	+	1.36860i	0.823214	+	0.567731i	\(0.192179\pi\)
							−0.598844	+	0.800866i	\(0.704373\pi\)
\(3\)	−1.70744	+	0.290956i	−0.985790	+	0.167983i
							\(5\)	−2.92507	−	1.98325i	−1.30813	−	0.886936i	−0.310166	−	0.950682i	\(-0.600385\pi\)
−0.997966	+	0.0637465i	\(0.979695\pi\)
\(7\)	−2.73852	−	0.602794i	−1.03506	−	0.227835i	−0.335226	−	0.942138i	\(-0.608813\pi\)
							−0.699837	+	0.714303i	\(0.746744\pi\)
\(11\)	−2.01160	−	1.52918i	−0.606521	−	0.461065i	0.256285	−	0.966601i	\(-0.417502\pi\)
							−0.862805	+	0.505536i	\(0.831295\pi\)
\(13\)	−2.03098	+	3.37551i	−0.563292	+	0.936199i	0.436050	+	0.899923i	\(0.356377\pi\)
							−0.999342	+	0.0362761i	\(0.988450\pi\)
\(17\)	1.04883	+	4.76487i	0.254378	+	1.15565i	0.914530	+	0.404519i	\(0.132561\pi\)
							−0.660151	+	0.751133i	\(0.729508\pi\)
\(19\)	−5.48029	−	0.596018i	−1.25727	−	0.136736i	−0.544874	−	0.838518i	\(-0.683422\pi\)
							−0.712392	+	0.701782i	\(0.752388\pi\)
\(23\)	5.39194	−	6.34788i	1.12430	−	1.32362i	0.184133	−	0.982901i	\(-0.441052\pi\)
							0.940164	−	0.340723i	\(-0.110672\pi\)
\(29\)	−4.96592	−	0.814121i	−0.922148	−	0.151179i	−0.318037	−	0.948078i	\(-0.603024\pi\)
							−0.604112	+	0.796900i	\(0.706472\pi\)
\(31\)	−0.0516452	−	0.474870i	−0.00927576	−	0.0852892i	0.988585	−	0.150664i	\(-0.0481412\pi\)
							−0.997861	+	0.0653749i	\(0.979176\pi\)
\(37\)	4.35883	+	2.31091i	0.716587	+	0.379911i	0.786457	−	0.617646i	\(-0.211913\pi\)
							−0.0698691	+	0.997556i	\(0.522258\pi\)
\(41\)	−5.04670	+	4.28671i	−0.788162	+	0.669471i	−0.948212	−	0.317638i	\(-0.897110\pi\)
							0.160050	+	0.987109i	\(0.448834\pi\)
\(43\)	−1.09884	−	1.44550i	−0.167571	−	0.220437i	0.704641	−	0.709564i	\(-0.251108\pi\)
							−0.872212	+	0.489128i	\(0.837315\pi\)
\(47\)	2.14776	+	3.16771i	0.313283	+	0.462058i	0.951434	−	0.307853i	\(-0.0996106\pi\)
							−0.638151	+	0.769912i	\(0.720300\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.155.16	yes	504
3.2	odd	2	inner	177.2.f.a.155.3	yes	504
59.8	odd	58	inner	177.2.f.a.8.3	✓	504
177.8	even	58	inner	177.2.f.a.8.16	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.8.3	✓	504	59.8	odd	58	inner
177.2.f.a.8.16	yes	504	177.8	even	58	inner
177.2.f.a.155.3	yes	504	3.2	odd	2	inner
177.2.f.a.155.16	yes	504	1.1	even	1	trivial