Embedded newform 177.4.f.a.101.8

• Label: 177.4.f.a.101.8
• Level: $177$
• Weight: $4$
• Character: 177.101
• Analytic conductor: $10.443$
• Analytic rank: $0$
• Dimension: $1624$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.f (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.4433380710\)
Analytic rank:	\(0\)
Dimension:	\(1624\)
Relative dimension:	\(58\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			101.8
Character	\(\chi\)	\(=\)	177.101
Dual form			177.4.f.a.170.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.62802 + 3.87605i) q^{2} +(-4.98159 - 1.47774i) q^{3} +(-5.15612 - 12.9409i) q^{4} +(1.79206 - 3.87348i) q^{5} +(18.8195 - 15.4253i) q^{6} +(-7.00710 - 25.2373i) q^{7} +(27.1219 + 5.96999i) q^{8} +(22.6326 + 14.7230i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 21 q^{3} - 278 q^{4} - 29 q^{6} - 42 q^{7} - 25 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{11}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.62802	+	3.87605i	−0.929147	+	1.37039i	−0.000270367		1.00000i	\(0.500086\pi\)
							−0.928877	+	0.370389i	\(0.879224\pi\)
\(3\)	−4.98159	−	1.47774i	−0.958708	−	0.284391i
							\(5\)	1.79206	−	3.87348i	0.160287	−	0.346454i	−0.810747	−	0.585397i	\(-0.800939\pi\)
0.971034	+	0.238943i	\(0.0768008\pi\)
\(7\)	−7.00710	−	25.2373i	−0.378348	−	1.36269i	−0.870497	−	0.492174i	\(-0.836202\pi\)
							0.492149	−	0.870511i	\(-0.336211\pi\)
\(11\)	−11.3510	−	10.7523i	−0.311133	−	0.294721i	0.516159	−	0.856493i	\(-0.327361\pi\)
							−0.827292	+	0.561772i	\(0.810120\pi\)
\(13\)	−5.74916	+	52.8627i	−0.122656	+	1.12781i	0.757405	+	0.652945i	\(0.226467\pi\)
							−0.880061	+	0.474860i	\(0.842499\pi\)
\(17\)	2.47110	+	0.686096i	0.0352546	+	0.00978840i	0.285111	−	0.958495i	\(-0.407970\pi\)
							−0.249856	+	0.968283i	\(0.580383\pi\)
\(19\)	−47.3161	−	35.9688i	−0.571319	−	0.434305i	0.279213	−	0.960229i	\(-0.409926\pi\)
							−0.850532	+	0.525924i	\(0.823720\pi\)
\(23\)	−6.59093	−	12.4318i	−0.0597523	−	0.112705i	0.851837	−	0.523807i	\(-0.175489\pi\)
							−0.911589	+	0.411102i	\(0.865144\pi\)
\(29\)	−41.0509	+	27.8332i	−0.262861	+	0.178224i	−0.685497	−	0.728075i	\(-0.740415\pi\)
							0.422636	+	0.906299i	\(0.361105\pi\)
\(31\)	−42.1795	−	55.4862i	−0.244376	−	0.321471i	0.657509	−	0.753446i	\(-0.271610\pi\)
							−0.901886	+	0.431975i	\(0.857817\pi\)
\(37\)	36.7464	+	166.941i	0.163272	+	0.741754i	0.985095	+	0.172010i	\(0.0550263\pi\)
							−0.821823	+	0.569743i	\(0.807043\pi\)
\(41\)	−303.478	−	160.894i	−1.15598	−	0.612862i	−0.223722	−	0.974653i	\(-0.571821\pi\)
							−0.932259	+	0.361791i	\(0.882166\pi\)
\(43\)	147.407	+	155.616i	0.522776	+	0.551888i	0.932897	−	0.360143i	\(-0.117272\pi\)
							−0.410121	+	0.912031i	\(0.634514\pi\)
\(47\)	−266.625	+	123.354i	−0.827475	+	0.382831i	−0.787458	−	0.616368i	\(-0.788603\pi\)
							−0.0400169	+	0.999199i	\(0.512741\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.f.a.101.8	✓	1624
3.2	odd	2	inner	177.4.f.a.101.51	yes	1624
59.52	odd	58	inner	177.4.f.a.170.51	yes	1624
177.170	even	58	inner	177.4.f.a.170.8	yes	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.f.a.101.8	✓	1624	1.1	even	1	trivial
177.4.f.a.101.51	yes	1624	3.2	odd	2	inner
177.4.f.a.170.8	yes	1624	177.170	even	58	inner
177.4.f.a.170.51	yes	1624	59.52	odd	58	inner