Embedded newform 177.3.h.a.104.21

• Label: 177.3.h.a.104.21
• Level: $177$
• Weight: $3$
• Character: 177.104
• Analytic conductor: $4.823$
• Analytic rank: $0$
• Dimension: $1064$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	177.h (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.82290067918\)
Analytic rank:	\(0\)
Dimension:	\(1064\)
Relative dimension:	\(38\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			104.21
Character	\(\chi\)	\(=\)	177.104
Dual form			177.3.h.a.80.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.142887 + 0.237480i) q^{2} +(-1.40022 + 2.65319i) q^{3} +(1.83765 - 3.46618i) q^{4} +(0.186389 + 1.71381i) q^{5} +(-0.830150 + 0.0465825i) q^{6} +(5.16148 - 1.73911i) q^{7} +(2.19271 - 0.118885i) q^{8} +(-5.07879 - 7.43007i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{24}{29}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.142887	+	0.237480i	0.0714434	+	0.118740i	0.890520	−	0.454945i	\(-0.150341\pi\)
							−0.819076	+	0.573685i	\(0.805513\pi\)
\(3\)	−1.40022	+	2.65319i	−0.466739	+	0.884395i
							\(5\)	0.186389	+	1.71381i	0.0372777	+	0.342763i	0.997893	+	0.0648812i	\(0.0206668\pi\)
−0.960615	+	0.277882i	\(0.910368\pi\)
\(7\)	5.16148	−	1.73911i	0.737355	−	0.248444i	0.0745305	−	0.997219i	\(-0.476254\pi\)
							0.662824	+	0.748775i	\(0.269358\pi\)
\(11\)	14.2460	+	9.65904i	1.29509	+	0.878095i	0.997072	−	0.0764650i	\(-0.0243634\pi\)
							0.298020	+	0.954560i	\(0.403674\pi\)
\(13\)	5.25705	+	4.97974i	0.404389	+	0.383057i	0.862685	−	0.505742i	\(-0.168781\pi\)
							−0.458296	+	0.888800i	\(0.651540\pi\)
\(17\)	−1.38315	+	4.10505i	−0.0813619	+	0.241474i	−0.980611	−	0.195963i	\(-0.937217\pi\)
							0.899249	+	0.437436i	\(0.144113\pi\)
\(19\)	4.02187	+	24.5323i	0.211677	+	1.29118i	0.852192	+	0.523229i	\(0.175273\pi\)
							−0.640515	+	0.767946i	\(0.721279\pi\)
\(23\)	−4.61899	+	1.28246i	−0.200826	+	0.0557590i	−0.366485	−	0.930424i	\(-0.619439\pi\)
							0.165659	+	0.986183i	\(0.447025\pi\)
\(29\)	7.81221	−	12.9840i	0.269387	−	0.447724i	−0.692404	−	0.721510i	\(-0.743448\pi\)
							0.961791	+	0.273786i	\(0.0882760\pi\)
\(31\)	−3.10523	+	18.9411i	−0.100169	+	0.611003i	0.888263	+	0.459336i	\(0.151912\pi\)
							−0.988432	+	0.151667i	\(0.951536\pi\)
\(37\)	1.03665	−	19.1198i	0.0280175	−	0.516752i	−0.950517	−	0.310671i	\(-0.899446\pi\)
							0.978535	−	0.206081i	\(-0.0660710\pi\)
\(41\)	−38.2720	−	10.6262i	−0.933462	−	0.259175i	−0.232683	−	0.972553i	\(-0.574750\pi\)
							−0.700779	+	0.713378i	\(0.747164\pi\)
\(43\)	−28.3292	−	41.7825i	−0.658819	−	0.971685i	−0.999504	−	0.0315074i	\(-0.989969\pi\)
							0.340685	−	0.940178i	\(-0.389341\pi\)
\(47\)	3.04237	−	27.9741i	0.0647312	−	0.595194i	−0.916184	−	0.400758i	\(-0.868747\pi\)
							0.980915	−	0.194436i	\(-0.0622876\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.104.21	yes	1064
3.2	odd	2	inner	177.3.h.a.104.18	yes	1064
59.21	even	29	inner	177.3.h.a.80.18	✓	1064
177.80	odd	58	inner	177.3.h.a.80.21	yes	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.80.18	✓	1064	59.21	even	29	inner
177.3.h.a.80.21	yes	1064	177.80	odd	58	inner
177.3.h.a.104.18	yes	1064	3.2	odd	2	inner
177.3.h.a.104.21	yes	1064	1.1	even	1	trivial