Embedded newform 177.2.f.a.170.7

• Label: 177.2.f.a.170.7
• Level: $177$
• Weight: $2$
• Character: 177.170
• 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			170.7
Character	\(\chi\)	\(=\)	177.170
Dual form			177.2.f.a.101.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.460764 - 0.679576i) q^{2} +(1.70902 + 0.281542i) q^{3} +(0.490756 - 1.23170i) q^{4} +(0.472167 + 1.02057i) q^{5} +(-0.596124 - 1.29113i) q^{6} +(0.500561 - 1.80286i) q^{7} +(-2.66687 + 0.587023i) q^{8} +(2.84147 + 0.962318i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{47}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.460764	−	0.679576i	−0.325810	−	0.480533i	0.629144	−	0.777289i	\(-0.283406\pi\)
							−0.954953	+	0.296756i	\(0.904095\pi\)
\(3\)	1.70902	+	0.281542i	0.986701	+	0.162548i
							\(5\)	0.472167	+	1.02057i	0.211159	+	0.456414i	0.983996	−	0.178192i	\(-0.0570247\pi\)
−0.772836	+	0.634605i	\(0.781163\pi\)
\(7\)	0.500561	−	1.80286i	0.189194	−	0.681416i	−0.806706	−	0.590953i	\(-0.798752\pi\)
							0.995900	−	0.0904624i	\(-0.0288345\pi\)
\(11\)	−1.67121	+	1.58306i	−0.503890	+	0.477310i	−0.896906	−	0.442222i	\(-0.854190\pi\)
							0.393016	+	0.919532i	\(0.371432\pi\)
\(13\)	−0.00701734	−	0.0645234i	−0.00194626	−	0.0178956i	0.993125	−	0.117062i	\(-0.0373478\pi\)
							−0.995071	+	0.0991669i	\(0.968382\pi\)
\(17\)	1.12196	−	0.311512i	0.272116	−	0.0755527i	−0.128790	−	0.991672i	\(-0.541109\pi\)
							0.400906	+	0.916119i	\(0.368695\pi\)
\(19\)	−2.84345	+	2.16154i	−0.652333	+	0.495891i	−0.878216	−	0.478264i	\(-0.841266\pi\)
							0.225883	+	0.974154i	\(0.427473\pi\)
\(23\)	0.0691925	−	0.130511i	0.0144276	−	0.0272134i	−0.876193	−	0.481960i	\(-0.839925\pi\)
							0.890621	+	0.454746i	\(0.150270\pi\)
\(29\)	−4.18080	−	2.83465i	−0.776355	−	0.526382i	0.107465	−	0.994209i	\(-0.465727\pi\)
							−0.883820	+	0.467827i	\(0.845037\pi\)
\(31\)	−1.05327	+	1.38555i	−0.189173	+	0.248853i	−0.880896	−	0.473310i	\(-0.843059\pi\)
							0.691723	+	0.722163i	\(0.256852\pi\)
\(37\)	−1.76465	+	8.01689i	−0.290107	+	1.31797i	0.575729	+	0.817641i	\(0.304718\pi\)
							−0.865836	+	0.500328i	\(0.833213\pi\)
\(41\)	−3.40200	+	1.80363i	−0.531303	+	0.281679i	−0.712385	−	0.701789i	\(-0.752385\pi\)
							0.181082	+	0.983468i	\(0.442040\pi\)
\(43\)	−7.42113	+	7.83440i	−1.13171	+	1.19473i	−0.153276	+	0.988183i	\(0.548982\pi\)
							−0.978436	+	0.206551i	\(0.933776\pi\)
\(47\)	5.61931	+	2.59977i	0.819661	+	0.379215i	0.784471	−	0.620165i	\(-0.212934\pi\)
							0.0351894	+	0.999381i	\(0.488797\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.170.7	yes	504
3.2	odd	2	inner	177.2.f.a.170.12	yes	504
59.42	odd	58	inner	177.2.f.a.101.12	yes	504
177.101	even	58	inner	177.2.f.a.101.7	✓	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.101.7	✓	504	177.101	even	58	inner
177.2.f.a.101.12	yes	504	59.42	odd	58	inner
177.2.f.a.170.7	yes	504	1.1	even	1	trivial
177.2.f.a.170.12	yes	504	3.2	odd	2	inner