Embedded newform 177.4.e.b.4.14

• Label: 177.4.e.b.4.14
• 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			4.14
Character	\(\chi\)	\(=\)	177.4
Dual form			177.4.e.b.133.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.06804 + 0.551183i) q^{2} +(1.94216 - 2.28649i) q^{3} +(17.5683 + 3.86707i) q^{4} +(-15.9287 - 12.1087i) q^{5} +(11.1032 - 10.5175i) q^{6} +(8.66226 - 21.7406i) q^{7} +(48.2567 + 16.2596i) q^{8} +(-1.45604 - 8.88144i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q + 2 q^{2} - 45 q^{3} - 56 q^{4} - 14 q^{5} + 6 q^{6} + 6 q^{7} + 66 q^{8} - 135 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.06804	+	0.551183i	1.79182	+	0.194872i	0.942772	−	0.333439i	\(-0.108209\pi\)
							0.849051	+	0.528311i	\(0.177174\pi\)
\(3\)	1.94216	−	2.28649i	0.373769	−	0.440034i
							\(5\)	−15.9287	−	12.1087i	−1.42470	−	1.08303i	−0.982530	−	0.186104i	\(-0.940414\pi\)
−0.442175	−	0.896929i	\(-0.645793\pi\)
\(7\)	8.66226	−	21.7406i	0.467718	−	1.17388i	−0.486209	−	0.873843i	\(-0.661620\pi\)
							0.953926	−	0.300041i	\(-0.0970002\pi\)
\(11\)	13.1894	−	6.10206i	0.361522	−	0.167258i	−0.230715	−	0.973021i	\(-0.574107\pi\)
							0.592238	+	0.805763i	\(0.298245\pi\)
\(13\)	−5.18329	+	31.6167i	−0.110583	+	0.674529i	0.872113	+	0.489305i	\(0.162749\pi\)
							−0.982696	+	0.185224i	\(0.940699\pi\)
\(17\)	41.1760	+	103.344i	0.587449	+	1.47439i	0.859007	+	0.511964i	\(0.171082\pi\)
							−0.271557	+	0.962422i	\(0.587539\pi\)
\(19\)	71.2333	+	105.061i	0.860107	+	1.26856i	0.961802	+	0.273748i	\(0.0882633\pi\)
							−0.101694	+	0.994816i	\(0.532426\pi\)
\(23\)	7.06511	−	130.308i	0.0640512	−	1.18135i	−0.773064	−	0.634328i	\(-0.781277\pi\)
							0.837115	−	0.547026i	\(-0.184240\pi\)
\(29\)	27.1282	−	2.95037i	0.173710	−	0.0188921i	−0.0208508	−	0.999783i	\(-0.506637\pi\)
							0.194560	+	0.980891i	\(0.437672\pi\)
\(31\)	−127.019	+	187.339i	−0.735911	+	1.08539i	0.256986	+	0.966415i	\(0.417271\pi\)
							−0.992897	+	0.118973i	\(0.962040\pi\)
\(37\)	129.093	−	43.4964i	0.573587	−	0.193264i	−0.0175348	−	0.999846i	\(-0.505582\pi\)
							0.591122	+	0.806582i	\(0.298685\pi\)
\(41\)	−10.0334	−	185.056i	−0.0382185	−	0.704900i	−0.953373	−	0.301795i	\(-0.902414\pi\)
							0.915154	−	0.403104i	\(-0.132069\pi\)
\(43\)	−414.796	−	191.905i	−1.47106	−	0.680587i	−0.489550	−	0.871975i	\(-0.662839\pi\)
							−0.981514	+	0.191389i	\(0.938701\pi\)
\(47\)	334.404	−	254.207i	1.03782	−	0.788934i	0.0599662	−	0.998200i	\(-0.480901\pi\)
							0.977859	+	0.209266i	\(0.0671076\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.e.b.4.14	✓	420
59.15	even	29	inner	177.4.e.b.133.14	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.e.b.4.14	✓	420	1.1	even	1	trivial
177.4.e.b.133.14	yes	420	59.15	even	29	inner