Embedded newform 177.4.e.a.7.5

• Label: 177.4.e.a.7.5
• Level: $177$
• Weight: $4$
• Character: 177.7
• 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			7.5
Character	\(\chi\)	\(=\)	177.7
Dual form			177.4.e.a.76.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.59837 - 2.35742i) q^{2} +(-0.162417 + 2.99560i) q^{3} +(-0.0415309 + 0.104235i) q^{4} +(14.5110 - 6.71353i) q^{5} +(7.32148 - 4.40519i) q^{6} +(0.937125 - 3.37522i) q^{7} +(-21.9407 + 4.82952i) q^{8} +(-8.94724 - 0.973071i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 2 q^{2} + 45 q^{3} - 64 q^{4} + 14 q^{5} + 6 q^{6} + 6 q^{7} + 18 q^{8} - 135 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{9}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.59837	−	2.35742i	−0.565109	−	0.833473i	0.432568	−	0.901601i	\(-0.357608\pi\)
							−0.997677	+	0.0681283i	\(0.978297\pi\)
\(3\)	−0.162417	+	2.99560i	−0.0312571	+	0.576504i
							\(5\)	14.5110	−	6.71353i	1.29791	−	0.600476i	0.355417	−	0.934708i	\(-0.384339\pi\)
0.942491	+	0.334232i	\(0.108477\pi\)
\(7\)	0.937125	−	3.37522i	0.0506000	−	0.182245i	−0.933932	−	0.357452i	\(-0.883646\pi\)
							0.984532	+	0.175207i	\(0.0560595\pi\)
\(11\)	10.9330	−	10.3563i	0.299676	−	0.283868i	−0.523055	−	0.852299i	\(-0.675208\pi\)
							0.822731	+	0.568431i	\(0.192449\pi\)
\(13\)	83.7523	−	9.10861i	1.78682	−	0.194329i	0.845990	−	0.533199i	\(-0.179010\pi\)
							0.940833	+	0.338870i	\(0.110045\pi\)
\(17\)	−3.89572	−	14.0311i	−0.0555794	−	0.200179i	0.930548	−	0.366171i	\(-0.119331\pi\)
							−0.986127	+	0.165992i	\(0.946918\pi\)
\(19\)	−83.9359	+	63.8064i	−1.01348	+	0.770431i	−0.973481	−	0.228768i	\(-0.926530\pi\)
							−0.0400038	+	0.999200i	\(0.512737\pi\)
\(23\)	61.5381	−	116.073i	0.557895	−	1.05230i	−0.430812	−	0.902442i	\(-0.641773\pi\)
							0.988707	−	0.149860i	\(-0.0478822\pi\)
\(29\)	141.278	−	208.369i	0.904644	−	1.33425i	−0.0375901	−	0.999293i	\(-0.511968\pi\)
							0.942234	−	0.334956i	\(-0.108722\pi\)
\(31\)	−220.137	−	167.344i	−1.27541	−	0.969542i	−0.999966	−	0.00822562i	\(-0.997382\pi\)
							−0.275445	−	0.961317i	\(-0.588825\pi\)
\(37\)	−114.804	−	25.2703i	−0.510099	−	0.112281i	−0.0475391	−	0.998869i	\(-0.515138\pi\)
							−0.462560	+	0.886588i	\(0.653069\pi\)
\(41\)	−100.801	−	190.131i	−0.383964	−	0.724233i	0.613912	−	0.789374i	\(-0.289595\pi\)
							−0.997876	+	0.0651415i	\(0.979250\pi\)
\(43\)	−179.351	−	169.890i	−0.636065	−	0.602512i	0.300171	−	0.953885i	\(-0.402956\pi\)
							−0.936236	+	0.351373i	\(0.885715\pi\)
\(47\)	301.937	+	139.691i	0.937064	+	0.433532i	0.828142	−	0.560519i	\(-0.189398\pi\)
							0.108922	+	0.994050i	\(0.465260\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.e.a.7.5	✓	420
59.17	even	29	inner	177.4.e.a.76.5	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.e.a.7.5	✓	420	1.1	even	1	trivial
177.4.e.a.76.5	yes	420	59.17	even	29	inner