Embedded newform 177.4.e.a.4.4

• Label: 177.4.e.a.4.4
• 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			4.4
Character	\(\chi\)	\(=\)	177.4
Dual form			177.4.e.a.133.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.43911 - 0.374026i) q^{2} +(-1.94216 + 2.28649i) q^{3} +(3.87463 + 0.852870i) q^{4} +(-6.75733 - 5.13679i) q^{5} +(7.53451 - 7.13706i) q^{6} +(2.82453 - 7.08903i) q^{7} +(13.2201 + 4.45438i) q^{8} +(-1.45604 - 8.88144i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 2 q^{2} + 45 q^{3} - 64 q^{4} + 14 q^{5} + 6 q^{6} + 6 q^{7} + 18 q^{8} - 135 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.43911	−	0.374026i	−1.21591	−	0.132238i	−0.522409	−	0.852695i	\(-0.674967\pi\)
							−0.693500	+	0.720457i	\(0.743932\pi\)
\(3\)	−1.94216	+	2.28649i	−0.373769	+	0.440034i
							\(5\)	−6.75733	−	5.13679i	−0.604394	−	0.459449i	0.257680	−	0.966230i	\(-0.417042\pi\)
−0.862074	+	0.506782i	\(0.830835\pi\)
\(7\)	2.82453	−	7.08903i	0.152510	−	0.382772i	−0.832793	−	0.553584i	\(-0.813260\pi\)
							0.985304	+	0.170812i	\(0.0546391\pi\)
\(11\)	39.9965	−	18.5043i	1.09631	−	0.507206i	0.213501	−	0.976943i	\(-0.431513\pi\)
							0.882807	+	0.469737i	\(0.155651\pi\)
\(13\)	−8.60583	+	52.4932i	−0.183602	+	1.11992i	0.720769	+	0.693176i	\(0.243789\pi\)
							−0.904371	+	0.426748i	\(0.859659\pi\)
\(17\)	−30.5107	−	76.5762i	−0.435291	−	1.09250i	−0.969051	−	0.246861i	\(-0.920601\pi\)
							0.533760	−	0.845636i	\(-0.320778\pi\)
\(19\)	58.8399	+	86.7824i	0.710464	+	1.04786i	0.995988	+	0.0894840i	\(0.0285218\pi\)
							−0.285525	+	0.958371i	\(0.592168\pi\)
\(23\)	4.94187	−	91.1473i	0.0448022	−	0.826328i	−0.886645	−	0.462451i	\(-0.846970\pi\)
							0.931447	−	0.363877i	\(-0.118547\pi\)
\(29\)	−286.769	+	31.1880i	−1.83626	+	0.199706i	−0.959594	−	0.281387i	\(-0.909206\pi\)
							−0.876670	+	0.481093i	\(0.840240\pi\)
\(31\)	−105.738	+	155.952i	−0.612617	+	0.903542i	−0.999830	−	0.0184483i	\(-0.994127\pi\)
							0.387213	+	0.921990i	\(0.373438\pi\)
\(37\)	−329.102	+	110.887i	−1.46227	+	0.492697i	−0.934276	−	0.356551i	\(-0.883953\pi\)
							−0.527995	+	0.849247i	\(0.677056\pi\)
\(41\)	8.35177	+	154.039i	0.0318129	+	0.586754i	0.970369	+	0.241627i	\(0.0776811\pi\)
							−0.938556	+	0.345126i	\(0.887836\pi\)
\(43\)	49.4801	+	22.8919i	0.175480	+	0.0811857i	0.505662	−	0.862731i	\(-0.331248\pi\)
							−0.330182	+	0.943917i	\(0.607110\pi\)
\(47\)	−322.673	+	245.289i	−1.00142	+	0.761258i	−0.971195	−	0.238285i	\(-0.923415\pi\)
							−0.0302230	+	0.999543i	\(0.509622\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.e.a.4.4	✓	420
59.15	even	29	inner	177.4.e.a.133.4	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.e.a.4.4	✓	420	1.1	even	1	trivial
177.4.e.a.133.4	yes	420	59.15	even	29	inner