Embedded newform 177.3.h.a.71.16

• Label: 177.3.h.a.71.16
• 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			71.16
Character	\(\chi\)	\(=\)	177.71
Dual form			177.3.h.a.5.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.225119 - 0.668129i) q^{2} +(0.0215190 + 2.99992i) q^{3} +(2.78865 - 2.11988i) q^{4} +(-5.48264 + 2.18448i) q^{5} +(1.99949 - 0.689717i) q^{6} +(-11.1057 - 5.13804i) q^{7} +(-4.37833 - 2.96858i) q^{8} +(-8.99907 + 0.129111i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.225119	−	0.668129i	−0.112559	−	0.334065i	0.876537	−	0.481335i	\(-0.159848\pi\)
							−0.989096	+	0.147270i	\(0.952951\pi\)
\(3\)	0.0215190	+	2.99992i	0.00717300	+	0.999974i
							\(5\)	−5.48264	+	2.18448i	−1.09653	+	0.436897i	−0.847066	−	0.531487i	\(-0.821633\pi\)
−0.249463	+	0.968384i	\(0.580254\pi\)
\(7\)	−11.1057	−	5.13804i	−1.58653	−	0.734005i	−0.589263	−	0.807941i	\(-0.700582\pi\)
							−0.997263	+	0.0739364i	\(0.976444\pi\)
\(11\)	5.78043	−	1.60493i	0.525493	−	0.145903i	0.00535683	−	0.999986i	\(-0.498295\pi\)
							0.520136	+	0.854083i	\(0.325881\pi\)
\(13\)	−8.84851	−	16.6901i	−0.680654	−	1.28385i	−0.947390	−	0.320081i	\(-0.896290\pi\)
							0.266736	−	0.963770i	\(-0.414055\pi\)
\(17\)	4.58854	+	9.91797i	0.269914	+	0.583410i	0.994376	−	0.105905i	\(-0.0337738\pi\)
							−0.724462	+	0.689315i	\(0.757912\pi\)
\(19\)	14.5787	+	3.20902i	0.767302	+	0.168896i	0.581340	−	0.813661i	\(-0.302529\pi\)
							0.185962	+	0.982557i	\(0.440460\pi\)
\(23\)	−24.7858	+	4.06343i	−1.07764	+	0.176671i	−0.674359	−	0.738403i	\(-0.735580\pi\)
							−0.403285	+	0.915074i	\(0.632132\pi\)
\(29\)	−7.17466	+	21.2936i	−0.247402	+	0.734263i	0.749947	+	0.661498i	\(0.230079\pi\)
							−0.997349	+	0.0727654i	\(0.976818\pi\)
\(31\)	−30.1472	+	6.63590i	−0.972490	+	0.214061i	−0.672677	−	0.739937i	\(-0.734855\pi\)
							−0.299813	+	0.953998i	\(0.596924\pi\)
\(37\)	4.27137	+	6.29981i	0.115443	+	0.170265i	0.881020	−	0.473078i	\(-0.156857\pi\)
							−0.765578	+	0.643343i	\(0.777547\pi\)
\(41\)	−39.7713	−	6.52018i	−0.970033	−	0.159029i	−0.344123	−	0.938924i	\(-0.611824\pi\)
							−0.625910	+	0.779896i	\(0.715272\pi\)
\(43\)	−5.66638	+	20.4084i	−0.131776	+	0.474615i	−0.999758	−	0.0219804i	\(-0.993003\pi\)
							0.867982	+	0.496595i	\(0.165417\pi\)
\(47\)	56.6852	+	22.5855i	1.20607	+	0.480542i	0.884587	−	0.466374i	\(-0.154440\pi\)
							0.321481	+	0.946916i	\(0.395819\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.71.16	yes	1064
3.2	odd	2	inner	177.3.h.a.71.23	yes	1064
59.5	even	29	inner	177.3.h.a.5.23	yes	1064
177.5	odd	58	inner	177.3.h.a.5.16	✓	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.5.16	✓	1064	177.5	odd	58	inner
177.3.h.a.5.23	yes	1064	59.5	even	29	inner
177.3.h.a.71.16	yes	1064	1.1	even	1	trivial
177.3.h.a.71.23	yes	1064	3.2	odd	2	inner