Embedded newform 177.2.f.a.32.7

• Label: 177.2.f.a.32.7
• 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			32.7
Character	\(\chi\)	\(=\)	177.32
Dual form			177.2.f.a.83.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.239417 + 0.862302i) q^{2} +(1.27409 - 1.17333i) q^{3} +(1.02747 + 0.618208i) q^{4} +(-0.401260 - 0.0217557i) q^{5} +(0.706728 + 1.37956i) q^{6} +(-0.709262 + 4.32631i) q^{7} +(-2.07849 + 1.96885i) q^{8} +(0.246588 - 2.98985i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.239417	+	0.862302i	−0.169293	+	0.609740i	0.829401	+	0.558654i	\(0.188682\pi\)
							−0.998694	+	0.0510857i	\(0.983732\pi\)
\(3\)	1.27409	−	1.17333i	0.735594	−	0.677423i
							\(5\)	−0.401260	−	0.0217557i	−0.179449	−	0.00972944i	−0.0358046	−	0.999359i	\(-0.511399\pi\)
−0.143644	+	0.989629i	\(0.545882\pi\)
\(7\)	−0.709262	+	4.32631i	−0.268076	+	1.63519i	0.418560	+	0.908189i	\(0.362535\pi\)
							−0.686636	+	0.727001i	\(0.740913\pi\)
\(11\)	1.91394	−	3.61008i	0.577076	−	1.08848i	−0.407447	−	0.913229i	\(-0.633581\pi\)
							0.984524	−	0.175252i	\(-0.0560741\pi\)
\(13\)	4.37846	−	1.74454i	1.21437	−	0.483847i	0.327049	−	0.945008i	\(-0.393946\pi\)
							0.887317	+	0.461160i	\(0.152567\pi\)
\(17\)	−3.68543	+	0.604196i	−0.893848	+	0.146539i	−0.591163	−	0.806552i	\(-0.701331\pi\)
							−0.302685	+	0.953091i	\(0.597883\pi\)
\(19\)	−1.96006	−	2.30756i	−0.449669	−	0.529391i	0.489887	−	0.871786i	\(-0.337038\pi\)
							−0.939556	+	0.342394i	\(0.888762\pi\)
\(23\)	−1.95715	+	1.48779i	−0.408095	+	0.310226i	−0.789005	−	0.614386i	\(-0.789404\pi\)
							0.380911	+	0.924612i	\(0.375611\pi\)
\(29\)	6.32711	−	1.75671i	1.17492	−	0.326214i	0.375402	−	0.926862i	\(-0.377505\pi\)
							0.799513	+	0.600648i	\(0.205091\pi\)
\(31\)	−4.82897	−	4.10176i	−0.867308	−	0.736698i	0.0987258	−	0.995115i	\(-0.468523\pi\)
							−0.966034	+	0.258417i	\(0.916799\pi\)
\(37\)	−0.558828	+	0.589948i	−0.0918708	+	0.0969868i	−0.770310	−	0.637670i	\(-0.779898\pi\)
							0.678439	+	0.734657i	\(0.262657\pi\)
\(41\)	−1.47564	+	1.94117i	−0.230457	+	0.303160i	−0.896757	−	0.442523i	\(-0.854084\pi\)
							0.666301	+	0.745683i	\(0.267877\pi\)
\(43\)	5.47199	−	2.90107i	0.834471	−	0.442409i	0.00434920	−	0.999991i	\(-0.498616\pi\)
							0.830122	+	0.557582i	\(0.188271\pi\)
\(47\)	0.623338	+	11.4968i	0.0909233	+	1.67698i	0.587682	+	0.809092i	\(0.300041\pi\)
							−0.496759	+	0.867889i	\(0.665477\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.32.7	✓	504
3.2	odd	2	inner	177.2.f.a.32.12	yes	504
59.24	odd	58	inner	177.2.f.a.83.12	yes	504
177.83	even	58	inner	177.2.f.a.83.7	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.32.7	✓	504	1.1	even	1	trivial
177.2.f.a.32.12	yes	504	3.2	odd	2	inner
177.2.f.a.83.7	yes	504	177.83	even	58	inner
177.2.f.a.83.12	yes	504	59.24	odd	58	inner