Embedded newform 177.2.f.a.83.12

• Label: 177.2.f.a.83.12
• 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			83.12
Character	\(\chi\)	\(=\)	177.83
Dual form			177.2.f.a.32.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.239417 + 0.862302i) q^{2} +(-1.56158 - 0.749301i) q^{3} +(1.02747 - 0.618208i) q^{4} +(0.401260 - 0.0217557i) q^{5} +(0.272254 - 1.52595i) q^{6} +(-0.709262 - 4.32631i) q^{7} +(2.07849 + 1.96885i) q^{8} +(1.87710 + 2.34020i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.239417	+	0.862302i	0.169293	+	0.609740i	0.998694	+	0.0510857i	\(0.0162682\pi\)
							−0.829401	+	0.558654i	\(0.811318\pi\)
\(3\)	−1.56158	−	0.749301i	−0.901582	−	0.432609i
							\(5\)	0.401260	−	0.0217557i	0.179449	−	0.00972944i	0.0358046	−	0.999359i	\(-0.488601\pi\)
0.143644	+	0.989629i	\(0.454118\pi\)
\(7\)	−0.709262	−	4.32631i	−0.268076	−	1.63519i	−0.686636	−	0.727001i	\(-0.740913\pi\)
							0.418560	−	0.908189i	\(-0.362535\pi\)
\(11\)	−1.91394	−	3.61008i	−0.577076	−	1.08848i	−0.984524	−	0.175252i	\(-0.943926\pi\)
							0.407447	−	0.913229i	\(-0.366419\pi\)
\(13\)	4.37846	+	1.74454i	1.21437	+	0.483847i	0.887317	−	0.461160i	\(-0.152567\pi\)
							0.327049	+	0.945008i	\(0.393946\pi\)
\(17\)	3.68543	+	0.604196i	0.893848	+	0.146539i	0.591163	−	0.806552i	\(-0.298669\pi\)
							0.302685	+	0.953091i	\(0.402117\pi\)
\(19\)	−1.96006	+	2.30756i	−0.449669	+	0.529391i	−0.939556	−	0.342394i	\(-0.888762\pi\)
							0.489887	+	0.871786i	\(0.337038\pi\)
\(23\)	1.95715	+	1.48779i	0.408095	+	0.310226i	0.789005	−	0.614386i	\(-0.210596\pi\)
							−0.380911	+	0.924612i	\(0.624389\pi\)
\(29\)	−6.32711	−	1.75671i	−1.17492	−	0.326214i	−0.375402	−	0.926862i	\(-0.622495\pi\)
							−0.799513	+	0.600648i	\(0.794909\pi\)
\(31\)	−4.82897	+	4.10176i	−0.867308	+	0.736698i	−0.966034	−	0.258417i	\(-0.916799\pi\)
							0.0987258	+	0.995115i	\(0.468523\pi\)
\(37\)	−0.558828	−	0.589948i	−0.0918708	−	0.0969868i	0.678439	−	0.734657i	\(-0.262657\pi\)
							−0.770310	+	0.637670i	\(0.779898\pi\)
\(41\)	1.47564	+	1.94117i	0.230457	+	0.303160i	0.896757	−	0.442523i	\(-0.145916\pi\)
							−0.666301	+	0.745683i	\(0.732123\pi\)
\(43\)	5.47199	+	2.90107i	0.834471	+	0.442409i	0.830122	−	0.557582i	\(-0.188271\pi\)
							0.00434920	+	0.999991i	\(0.498616\pi\)
\(47\)	−0.623338	+	11.4968i	−0.0909233	+	1.67698i	0.496759	+	0.867889i	\(0.334523\pi\)
							−0.587682	+	0.809092i	\(0.699959\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.32.7	✓	504	59.32	odd	58	inner
177.2.f.a.32.12	yes	504	177.32	even	58	inner
177.2.f.a.83.7	yes	504	3.2	odd	2	inner
177.2.f.a.83.12	yes	504	1.1	even	1	trivial