Embedded newform 177.2.f.a.32.2

• Label: 177.2.f.a.32.2
• Level: $177$
• Weight: $2$
• Character: 177.32
• 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			32.2
Character	\(\chi\)	\(=\)	177.32
Dual form			177.2.f.a.83.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.660709 + 2.37966i) q^{2} +(-1.11645 + 1.32421i) q^{3} +(-3.51252 - 2.11341i) q^{4} +(-0.952806 - 0.0516596i) q^{5} +(-2.41352 - 3.53169i) q^{6} +(-0.387617 + 2.36436i) q^{7} +(3.76401 - 3.56546i) q^{8} +(-0.507069 - 2.95684i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{5}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.660709	+	2.37966i	−0.467192	+	1.68267i	0.236066	+	0.971737i	\(0.424142\pi\)
							−0.703257	+	0.710935i	\(0.748272\pi\)
\(3\)	−1.11645	+	1.32421i	−0.644584	+	0.764534i
							\(5\)	−0.952806	−	0.0516596i	−0.426108	−	0.0231029i	−0.160162	−	0.987091i	\(-0.551202\pi\)
−0.265946	+	0.963988i	\(0.585684\pi\)
\(7\)	−0.387617	+	2.36436i	−0.146505	+	0.893644i	0.806620	+	0.591070i	\(0.201294\pi\)
							−0.953126	+	0.302574i	\(0.902154\pi\)
\(11\)	1.09001	−	2.05597i	0.328650	−	0.619900i	−0.663035	−	0.748588i	\(-0.730732\pi\)
							0.991685	+	0.128689i	\(0.0410768\pi\)
\(13\)	−4.15091	+	1.65387i	−1.15126	+	0.458702i	−0.866145	−	0.499792i	\(-0.833410\pi\)
							−0.285110	+	0.958495i	\(0.592030\pi\)
\(17\)	1.79831	−	0.294817i	0.436153	−	0.0715037i	0.0602944	−	0.998181i	\(-0.480796\pi\)
							0.375859	+	0.926677i	\(0.377348\pi\)
\(19\)	3.93302	+	4.63031i	0.902296	+	1.06226i	0.997764	+	0.0668324i	\(0.0212893\pi\)
							−0.0954683	+	0.995432i	\(0.530435\pi\)
\(23\)	−4.81067	+	3.65697i	−1.00309	+	0.762532i	−0.971517	−	0.236969i	\(-0.923846\pi\)
							−0.0315765	+	0.999501i	\(0.510053\pi\)
\(29\)	−2.94190	+	0.816813i	−0.546296	+	0.151678i	−0.529702	−	0.848184i	\(-0.677696\pi\)
							−0.0165944	+	0.999862i	\(0.505282\pi\)
\(31\)	−7.25288	−	6.16066i	−1.30266	−	1.10649i	−0.987529	−	0.157437i	\(-0.949677\pi\)
							−0.315127	−	0.949049i	\(-0.602047\pi\)
\(37\)	−5.20883	+	5.49890i	−0.856327	+	0.904013i	−0.996283	−	0.0861445i	\(-0.972545\pi\)
							0.139956	+	0.990158i	\(0.455304\pi\)
\(41\)	−3.48062	+	4.57867i	−0.543581	+	0.715068i	−0.983080	−	0.183174i	\(-0.941363\pi\)
							0.439500	+	0.898243i	\(0.355156\pi\)
\(43\)	7.16569	−	3.79901i	1.09276	−	0.579343i	0.178279	−	0.983980i	\(-0.442947\pi\)
							0.914477	+	0.404637i	\(0.132602\pi\)
\(47\)	0.235486	+	4.34329i	0.0343492	+	0.633534i	0.964132	+	0.265425i	\(0.0855121\pi\)
							−0.929782	+	0.368110i	\(0.880005\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.32.2	✓	504
3.2	odd	2	inner	177.2.f.a.32.17	yes	504
59.24	odd	58	inner	177.2.f.a.83.17	yes	504
177.83	even	58	inner	177.2.f.a.83.2	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.32.2	✓	504	1.1	even	1	trivial
177.2.f.a.32.17	yes	504	3.2	odd	2	inner
177.2.f.a.83.2	yes	504	177.83	even	58	inner
177.2.f.a.83.17	yes	504	59.24	odd	58	inner