Embedded newform 177.4.e.a.4.11

• Label: 177.4.e.a.4.11
• Level: $177$
• Weight: $4$
• Character: 177.4
• Analytic conductor: $10.443$
• Analytic rank: $0$
• Dimension: $420$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.e (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.4433380710\)
Analytic rank:	\(0\)
Dimension:	\(420\)
Relative dimension:	\(15\) over \(\Q(\zeta_{29})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{29}]$

Embedding invariants

Embedding label			4.11
Character	\(\chi\)	\(=\)	177.4
Dual form			177.4.e.a.133.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.45810 + 0.267335i) q^{2} +(-1.94216 + 2.28649i) q^{3} +(-1.84216 - 0.405491i) q^{4} +(7.30295 + 5.55156i) q^{5} +(-5.38528 + 5.10121i) q^{6} +(-2.79914 + 7.02531i) q^{7} +(-23.1651 - 7.80525i) q^{8} +(-1.45604 - 8.88144i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 2 q^{2} + 45 q^{3} - 64 q^{4} + 14 q^{5} + 6 q^{6} + 6 q^{7} + 18 q^{8} - 135 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.45810	+	0.267335i	0.869071	+	0.0945171i	0.531779	−	0.846883i	\(-0.321524\pi\)
							0.337291	+	0.941400i	\(0.390489\pi\)
\(3\)	−1.94216	+	2.28649i	−0.373769	+	0.440034i
							\(5\)	7.30295	+	5.55156i	0.653196	+	0.496547i	0.878500	−	0.477742i	\(-0.158545\pi\)
−0.225304	+	0.974288i	\(0.572338\pi\)
\(7\)	−2.79914	+	7.02531i	−0.151139	+	0.379331i	−0.984979	−	0.172676i	\(-0.944759\pi\)
							0.833839	+	0.552007i	\(0.186138\pi\)
\(11\)	−29.6784	+	13.7307i	−0.813489	+	0.376360i	−0.782104	−	0.623147i	\(-0.785854\pi\)
							−0.0313844	+	0.999507i	\(0.509992\pi\)
\(13\)	−9.95264	+	60.7084i	−0.212336	+	1.29519i	0.638444	+	0.769668i	\(0.279578\pi\)
							−0.850780	+	0.525523i	\(0.823870\pi\)
\(17\)	12.2307	+	30.6967i	0.174493	+	0.437943i	0.990047	−	0.140737i	\(-0.0449472\pi\)
							−0.815554	+	0.578680i	\(0.803568\pi\)
\(19\)	−79.9088	−	117.857i	−0.964860	−	1.42306i	−0.905269	−	0.424840i	\(-0.860330\pi\)
							−0.0595917	−	0.998223i	\(-0.518980\pi\)
\(23\)	−9.44781	+	174.255i	−0.0856524	+	1.57977i	0.568037	+	0.823003i	\(0.307703\pi\)
							−0.653690	+	0.756763i	\(0.726780\pi\)
\(29\)	−86.9923	+	9.46098i	−0.557037	+	0.0605814i	−0.382308	−	0.924035i	\(-0.624871\pi\)
							−0.174729	+	0.984617i	\(0.555905\pi\)
\(31\)	−124.267	+	183.281i	−0.719971	+	1.06188i	0.274974	+	0.961452i	\(0.411331\pi\)
							−0.994945	+	0.100426i	\(0.967979\pi\)
\(37\)	142.006	−	47.8474i	0.630964	−	0.212596i	0.0143936	−	0.999896i	\(-0.495418\pi\)
							0.616570	+	0.787300i	\(0.288522\pi\)
\(41\)	4.32733	+	79.8129i	0.0164833	+	0.304016i	0.995137	+	0.0985003i	\(0.0314046\pi\)
							−0.978654	+	0.205516i	\(0.934113\pi\)
\(43\)	275.947	+	127.667i	0.978639	+	0.452767i	0.842954	−	0.537985i	\(-0.180814\pi\)
							0.135685	+	0.990752i	\(0.456676\pi\)
\(47\)	263.317	−	200.168i	0.817206	−	0.621224i	−0.110777	−	0.993845i	\(-0.535334\pi\)
							0.927984	+	0.372621i	\(0.121541\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.e.a.4.11	✓	420
59.15	even	29	inner	177.4.e.a.133.11	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.e.a.4.11	✓	420	1.1	even	1	trivial
177.4.e.a.133.11	yes	420	59.15	even	29	inner