Embedded newform 177.2.f.a.14.3

• Label: 177.2.f.a.14.3
• Level: $177$
• Weight: $2$
• Character: 177.14
• 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			14.3
Character	\(\chi\)	\(=\)	177.14
Dual form			177.2.f.a.38.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.60225 + 0.964041i) q^{2} +(-1.65741 + 0.502995i) q^{3} +(0.701009 - 1.32224i) q^{4} +(0.0291328 + 0.267871i) q^{5} +(2.17067 - 2.40373i) q^{6} +(-3.96067 + 1.33451i) q^{7} +(-0.0509641 - 0.939978i) q^{8} +(2.49399 - 1.66733i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{19}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.60225	+	0.964041i	−1.13296	+	0.681680i	−0.953293	−	0.302047i	\(-0.902330\pi\)
							−0.179668	+	0.983727i	\(0.557502\pi\)
\(3\)	−1.65741	+	0.502995i	−0.956904	+	0.290404i
							\(5\)	0.0291328	+	0.267871i	0.0130286	+	0.119796i	0.998846	−	0.0480247i	\(-0.0152926\pi\)
−0.985818	+	0.167820i	\(0.946327\pi\)
\(7\)	−3.96067	+	1.33451i	−1.49699	+	0.504396i	−0.944327	−	0.329007i	\(-0.893286\pi\)
							−0.552666	+	0.833403i	\(0.686389\pi\)
\(11\)	1.25346	−	1.84871i	0.377932	−	0.557408i	−0.590389	−	0.807119i	\(-0.701025\pi\)
							0.968320	+	0.249712i	\(0.0803358\pi\)
\(13\)	0.0885345	−	0.0934648i	0.0245551	−	0.0259225i	−0.713602	−	0.700551i	\(-0.752938\pi\)
							0.738157	+	0.674629i	\(0.235696\pi\)
\(17\)	0.877671	−	2.60483i	0.212866	−	0.631765i	−0.787023	−	0.616923i	\(-0.788379\pi\)
							0.999890	−	0.0148418i	\(-0.00472447\pi\)
\(19\)	−1.29281	−	7.88577i	−0.296590	−	1.80912i	−0.542755	−	0.839891i	\(-0.682619\pi\)
							0.246165	−	0.969228i	\(-0.420830\pi\)
\(23\)	−0.724703	−	2.61014i	−0.151111	−	0.544252i	−0.999880	−	0.0154888i	\(-0.995070\pi\)
							0.848769	−	0.528764i	\(-0.177344\pi\)
\(29\)	−3.17115	+	5.27049i	−0.588868	+	0.978706i	0.408942	+	0.912560i	\(0.365898\pi\)
							−0.997810	+	0.0661454i	\(0.978930\pi\)
\(31\)	−5.97230	−	0.979109i	−1.07266	−	0.175853i	−0.400526	−	0.916285i	\(-0.631173\pi\)
							−0.672131	+	0.740432i	\(0.734621\pi\)
\(37\)	−6.76370	−	0.366717i	−1.11195	−	0.0602879i	−0.511002	−	0.859579i	\(-0.670726\pi\)
							−0.600943	+	0.799292i	\(0.705208\pi\)
\(41\)	−6.10233	−	1.69430i	−0.953024	−	0.264606i	−0.243988	−	0.969778i	\(-0.578456\pi\)
							−0.709036	+	0.705172i	\(0.750870\pi\)
\(43\)	−2.60106	+	1.76356i	−0.396658	+	0.268941i	−0.743167	−	0.669106i	\(-0.766677\pi\)
							0.346509	+	0.938047i	\(0.387367\pi\)
\(47\)	−5.39475	−	0.586714i	−0.786905	−	0.0855811i	−0.294162	−	0.955756i	\(-0.595040\pi\)
							−0.492743	+	0.870175i	\(0.664006\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.14.3	✓	504
3.2	odd	2	inner	177.2.f.a.14.16	yes	504
59.38	odd	58	inner	177.2.f.a.38.16	yes	504
177.38	even	58	inner	177.2.f.a.38.3	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.14.3	✓	504	1.1	even	1	trivial
177.2.f.a.14.16	yes	504	3.2	odd	2	inner
177.2.f.a.38.3	yes	504	177.38	even	58	inner
177.2.f.a.38.16	yes	504	59.38	odd	58	inner