Embedded newform 177.2.e.a.4.4

• Label: 177.2.e.a.4.4
• Level: $177$
• Weight: $2$
• Character: 177.4
• Analytic conductor: $1.413$
• Analytic rank: $0$
• Dimension: $140$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.e (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.41335211578\)
Analytic rank:	\(0\)
Dimension:	\(140\)
Relative dimension:	\(5\) over \(\Q(\zeta_{29})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{29}]$

Embedding invariants

Embedding label			4.4
Character	\(\chi\)	\(=\)	177.4
Dual form			177.2.e.a.133.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.283755 + 0.0308602i) q^{2} +(0.647386 - 0.762162i) q^{3} +(-1.87368 - 0.412427i) q^{4} +(2.97558 + 2.26198i) q^{5} +(0.207220 - 0.196289i) q^{6} +(1.21093 - 3.03920i) q^{7} +(-1.05991 - 0.357126i) q^{8} +(-0.161782 - 0.986827i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - q^{2} - 5 q^{3} - 9 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} - 9 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{1}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.283755	+	0.0308602i	0.200645	+	0.0218215i	0.207890	−	0.978152i	\(-0.433341\pi\)
							−0.00724454	+	0.999974i	\(0.502306\pi\)
\(3\)	0.647386	−	0.762162i	0.373769	−	0.440034i
							\(5\)	2.97558	+	2.26198i	1.33072	+	1.01159i	0.997638	+	0.0686882i	\(0.0218814\pi\)
0.333081	+	0.942898i	\(0.391912\pi\)
\(7\)	1.21093	−	3.03920i	0.457687	−	1.14871i	−0.501304	−	0.865271i	\(-0.667146\pi\)
							0.958991	−	0.283437i	\(-0.0914746\pi\)
\(11\)	2.37040	−	1.09666i	0.714702	−	0.330656i	−0.0286197	−	0.999590i	\(-0.509111\pi\)
							0.743322	+	0.668934i	\(0.233249\pi\)
\(13\)	−0.960694	+	5.85997i	−0.266448	+	1.62526i	0.426647	+	0.904418i	\(0.359695\pi\)
							−0.693096	+	0.720846i	\(0.743754\pi\)
\(17\)	−1.39701	−	3.50622i	−0.338824	−	0.850384i	−0.995372	−	0.0960996i	\(-0.969363\pi\)
							0.656548	−	0.754284i	\(-0.272016\pi\)
\(19\)	−2.54337	−	3.75119i	−0.583489	−	0.860582i	0.415304	−	0.909682i	\(-0.363675\pi\)
							−0.998794	+	0.0491000i	\(0.984365\pi\)
\(23\)	−0.0711480	+	1.31225i	−0.0148354	+	0.273622i	0.981699	+	0.190440i	\(0.0609913\pi\)
							−0.996534	+	0.0831829i	\(0.973491\pi\)
\(29\)	−6.69107	+	0.727697i	−1.24250	+	0.135130i	−0.705658	−	0.708553i	\(-0.749348\pi\)
							−0.536842	+	0.843683i	\(0.680383\pi\)
\(31\)	−4.50223	+	6.64029i	−0.808624	+	1.19263i	0.169689	+	0.985498i	\(0.445724\pi\)
							−0.978313	+	0.207133i	\(0.933587\pi\)
\(37\)	−8.46255	+	2.85137i	−1.39124	+	0.468762i	−0.912456	−	0.409174i	\(-0.865817\pi\)
							−0.478779	+	0.877936i	\(0.658920\pi\)
\(41\)	0.164521	+	3.03441i	0.0256939	+	0.473895i	0.982860	+	0.184355i	\(0.0590196\pi\)
							−0.957166	+	0.289540i	\(0.906498\pi\)
\(43\)	−1.77119	−	0.819439i	−0.270104	−	0.124963i	0.280162	−	0.959953i	\(-0.409612\pi\)
							−0.550266	+	0.834989i	\(0.685474\pi\)
\(47\)	−3.24938	+	2.47011i	−0.473970	+	0.360303i	−0.814723	−	0.579850i	\(-0.803111\pi\)
							0.340753	+	0.940153i	\(0.389318\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.a.4.4	✓	140
3.2	odd	2		531.2.i.c.181.2		140
59.15	even	29	inner	177.2.e.a.133.4	yes	140
177.74	odd	58		531.2.i.c.487.2		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.a.4.4	✓	140	1.1	even	1	trivial
177.2.e.a.133.4	yes	140	59.15	even	29	inner
531.2.i.c.181.2		140	3.2	odd	2
531.2.i.c.487.2		140	177.74	odd	58