Embedded newform 177.2.f.a.83.9

• Label: 177.2.f.a.83.9
• Level: $177$
• Weight: $2$
• Character: 177.83
• 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			83.9
Character	\(\chi\)	\(=\)	177.83
Dual form			177.2.f.a.32.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.110481 - 0.397918i) q^{2} +(-0.426514 - 1.67872i) q^{3} +(1.56758 - 0.943183i) q^{4} +(2.16684 - 0.117483i) q^{5} +(-0.620869 + 0.355184i) q^{6} +(0.253612 + 1.54697i) q^{7} +(-1.14813 - 1.08756i) q^{8} +(-2.63617 + 1.43199i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{53}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.110481	−	0.397918i	−0.0781221	−	0.281370i	0.914163	−	0.405347i	\(-0.132849\pi\)
							−0.992285	+	0.123976i	\(0.960435\pi\)
\(3\)	−0.426514	−	1.67872i	−0.246248	−	0.969207i
							\(5\)	2.16684	−	0.117483i	0.969039	−	0.0525398i	0.437181	−	0.899373i	\(-0.355977\pi\)
0.531858	+	0.846834i	\(0.321494\pi\)
\(7\)	0.253612	+	1.54697i	0.0958565	+	0.584699i	0.990456	+	0.137830i	\(0.0440127\pi\)
							−0.894599	+	0.446869i	\(0.852539\pi\)
\(11\)	0.507538	+	0.957318i	0.153028	+	0.288642i	0.948159	−	0.317796i	\(-0.102943\pi\)
							−0.795131	+	0.606438i	\(0.792598\pi\)
\(13\)	−2.53293	−	1.00921i	−0.702507	−	0.279904i	−0.00859390	−	0.999963i	\(-0.502736\pi\)
							−0.693913	+	0.720059i	\(0.744115\pi\)
\(17\)	1.49822	+	0.245621i	0.363372	+	0.0595718i	0.340702	−	0.940171i	\(-0.389335\pi\)
							0.0226696	+	0.999743i	\(0.492783\pi\)
\(19\)	−2.26534	+	2.66696i	−0.519704	+	0.611843i	−0.957923	−	0.287025i	\(-0.907334\pi\)
							0.438219	+	0.898868i	\(0.355609\pi\)
\(23\)	−1.57110	−	1.19432i	−0.327597	−	0.249033i	0.428428	−	0.903576i	\(-0.359068\pi\)
							−0.756024	+	0.654544i	\(0.772861\pi\)
\(29\)	3.75944	+	1.04380i	0.698111	+	0.193830i	0.598413	−	0.801188i	\(-0.295798\pi\)
							0.0996987	+	0.995018i	\(0.468212\pi\)
\(31\)	−4.33588	+	3.68293i	−0.778747	+	0.661474i	−0.945900	−	0.324457i	\(-0.894818\pi\)
							0.167153	+	0.985931i	\(0.446543\pi\)
\(37\)	5.65369	+	5.96853i	0.929461	+	0.981220i	0.999867	−	0.0162980i	\(-0.00518806\pi\)
							−0.0704059	+	0.997518i	\(0.522429\pi\)
\(41\)	6.36507	+	8.37310i	0.994056	+	1.30766i	0.951199	+	0.308578i	\(0.0998529\pi\)
							0.0428571	+	0.999081i	\(0.486354\pi\)
\(43\)	−8.66795	−	4.59546i	−1.32185	−	0.700801i	−0.350078	−	0.936721i	\(-0.613845\pi\)
							−0.971773	+	0.235920i	\(0.924190\pi\)
\(47\)	0.568942	−	10.4935i	0.0829887	−	1.53064i	−0.599877	−	0.800093i	\(-0.704784\pi\)
							0.682865	−	0.730544i	\(-0.260734\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.83.9	yes	504
3.2	odd	2	inner	177.2.f.a.83.10	yes	504
59.32	odd	58	inner	177.2.f.a.32.10	yes	504
177.32	even	58	inner	177.2.f.a.32.9	✓	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.32.9	✓	504	177.32	even	58	inner
177.2.f.a.32.10	yes	504	59.32	odd	58	inner
177.2.f.a.83.9	yes	504	1.1	even	1	trivial
177.2.f.a.83.10	yes	504	3.2	odd	2	inner