Embedded newform 177.2.e.a.7.5

• Label: 177.2.e.a.7.5
• Level: $177$
• Weight: $2$
• Character: 177.7
• Analytic conductor: $1.413$
• Analytic rank: $0$
• Dimension: $140$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.e (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.41335211578\)
Analytic rank:	\(0\)
Dimension:	\(140\)
Relative dimension:	\(5\) 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			177.2.e.a.76.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.03249 + 1.52281i) q^{2} +(0.0541389 - 0.998533i) q^{3} +(-0.512640 + 1.28663i) q^{4} +(1.77030 - 0.819027i) q^{5} +(1.57648 - 0.948534i) q^{6} +(0.611139 - 2.20113i) q^{7} +(1.10505 - 0.243241i) q^{8} +(-0.994138 - 0.108119i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140q - q^{2} - 5q^{3} - 9q^{4} - 2q^{5} - q^{6} - 2q^{7} - 9q^{8} - 5q^{9} + O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.03249	+	1.52281i	0.730082	+	1.07679i	0.993690	+	0.112161i	\(0.0357771\pi\)
							−0.263608	+	0.964630i	\(0.584913\pi\)
\(3\)	0.0541389	−	0.998533i	0.0312571	−	0.576504i
							\(5\)	1.77030	−	0.819027i	0.791702	−	0.366280i	0.0180061	−	0.999838i	\(-0.494268\pi\)
0.773695	+	0.633558i	\(0.218406\pi\)
\(7\)	0.611139	−	2.20113i	0.230989	−	0.831947i	−0.753640	−	0.657288i	\(-0.771704\pi\)
							0.984629	−	0.174660i	\(-0.0558825\pi\)
\(11\)	−4.59227	+	4.35003i	−1.38462	+	1.31158i	−0.485266	+	0.874366i	\(0.661277\pi\)
							−0.899356	+	0.437218i	\(0.855964\pi\)
\(13\)	−4.53639	+	0.493362i	−1.25817	+	0.136834i	−0.712803	−	0.701364i	\(-0.752575\pi\)
							−0.545365	+	0.838198i	\(0.683609\pi\)
\(17\)	1.27458	+	4.59062i	0.309131	+	1.11339i	0.941026	+	0.338335i	\(0.109864\pi\)
							−0.631895	+	0.775054i	\(0.717723\pi\)
\(19\)	−2.06023	+	1.56615i	−0.472650	+	0.359299i	−0.814219	−	0.580558i	\(-0.802835\pi\)
							0.341570	+	0.939856i	\(0.389042\pi\)
\(23\)	4.04174	−	7.62352i	0.842760	−	1.58961i	0.0355371	−	0.999368i	\(-0.488686\pi\)
							0.807223	−	0.590246i	\(-0.200969\pi\)
\(29\)	3.13495	−	4.62370i	0.582145	−	0.858600i	−0.416579	−	0.909100i	\(-0.636771\pi\)
							0.998724	+	0.0504994i	\(0.0160813\pi\)
\(31\)	−1.72336	−	1.31006i	−0.309524	−	0.235294i	0.438867	−	0.898552i	\(-0.355380\pi\)
							−0.748391	+	0.663258i	\(0.769173\pi\)
\(37\)	−6.46255	−	1.42252i	−1.06244	−	0.233860i	−0.350831	−	0.936439i	\(-0.614101\pi\)
							−0.711606	+	0.702579i	\(0.752032\pi\)
\(41\)	5.00369	+	9.43796i	0.781445	+	1.47396i	0.878013	+	0.478636i	\(0.158869\pi\)
							−0.0965680	+	0.995326i	\(0.530787\pi\)
\(43\)	−3.92673	−	3.71960i	−0.598821	−	0.567234i	0.327081	−	0.944996i	\(-0.393935\pi\)
							−0.925903	+	0.377763i	\(0.876694\pi\)
\(47\)	−3.31091	−	1.53179i	−0.482945	−	0.223434i	0.163282	−	0.986579i	\(-0.447792\pi\)
							−0.646227	+	0.763145i	\(0.723654\pi\)