Embedded newform 177.3.h.a.71.13

• Label: 177.3.h.a.71.13
• 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			71.13
Character	\(\chi\)	\(=\)	177.71
Dual form			177.3.h.a.5.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.528130 - 1.56743i) q^{2} +(-2.80203 - 1.07173i) q^{3} +(1.00644 - 0.765078i) q^{4} +(6.70916 - 2.67318i) q^{5} +(-0.200026 + 4.95802i) q^{6} +(6.64296 + 3.07336i) q^{7} +(-7.20679 - 4.88632i) q^{8} +(6.70279 + 6.00604i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.528130	−	1.56743i	−0.264065	−	0.783717i	−0.994873	−	0.101134i	\(-0.967753\pi\)
							0.730808	−	0.682583i	\(-0.239144\pi\)
\(3\)	−2.80203	−	1.07173i	−0.934011	−	0.357243i
							\(5\)	6.70916	−	2.67318i	1.34183	−	0.534635i	0.414902	−	0.909866i	\(-0.363816\pi\)
0.926931	+	0.375231i	\(0.122437\pi\)
\(7\)	6.64296	+	3.07336i	0.948995	+	0.439052i	0.832429	−	0.554131i	\(-0.186949\pi\)
							0.116566	+	0.993183i	\(0.462811\pi\)
\(11\)	15.4464	−	4.28868i	1.40422	−	0.389880i	0.518890	−	0.854841i	\(-0.326346\pi\)
							0.885331	+	0.464961i	\(0.153932\pi\)
\(13\)	0.592914	+	1.11835i	0.0456088	+	0.0860273i	0.905269	−	0.424839i	\(-0.139669\pi\)
							−0.859660	+	0.510867i	\(0.829325\pi\)
\(17\)	0.274735	+	0.593831i	0.0161609	+	0.0349312i	0.915488	−	0.402346i	\(-0.131805\pi\)
							−0.899327	+	0.437277i	\(0.855943\pi\)
\(19\)	−17.1511	−	3.77525i	−0.902691	−	0.198697i	−0.260710	−	0.965417i	\(-0.583956\pi\)
							−0.641982	+	0.766720i	\(0.721888\pi\)
\(23\)	−41.3296	+	6.77564i	−1.79694	+	0.294593i	−0.965381	−	0.260845i	\(-0.915999\pi\)
							−0.831558	+	0.555438i	\(0.812551\pi\)
\(29\)	9.51275	−	28.2328i	0.328026	−	0.973546i	−0.649067	−	0.760732i	\(-0.724840\pi\)
							0.977093	−	0.212815i	\(-0.0682630\pi\)
\(31\)	13.8187	−	3.04172i	0.445764	−	0.0981200i	0.0135845	−	0.999908i	\(-0.495676\pi\)
							0.432179	+	0.901788i	\(0.357745\pi\)
\(37\)	40.8125	+	60.1940i	1.10304	+	1.62686i	0.704482	+	0.709721i	\(0.251179\pi\)
							0.398559	+	0.917143i	\(0.369510\pi\)
\(41\)	−0.669301	−	0.109726i	−0.0163244	−	0.00267625i	0.153614	−	0.988131i	\(-0.450909\pi\)
							−0.169939	+	0.985455i	\(0.554357\pi\)
\(43\)	−19.6744	+	70.8606i	−0.457543	+	1.64792i	0.270746	+	0.962651i	\(0.412730\pi\)
							−0.728289	+	0.685270i	\(0.759684\pi\)
\(47\)	24.1972	+	9.64105i	0.514834	+	0.205129i	0.613061	−	0.790035i	\(-0.289938\pi\)
							−0.0982271	+	0.995164i	\(0.531317\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.71.13	yes	1064
3.2	odd	2	inner	177.3.h.a.71.26	yes	1064
59.5	even	29	inner	177.3.h.a.5.26	yes	1064
177.5	odd	58	inner	177.3.h.a.5.13	✓	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.5.13	✓	1064	177.5	odd	58	inner
177.3.h.a.5.26	yes	1064	59.5	even	29	inner
177.3.h.a.71.13	yes	1064	1.1	even	1	trivial
177.3.h.a.71.26	yes	1064	3.2	odd	2	inner