Embedded newform 177.2.e.a.76.5

• Label: 177.2.e.a.76.5
• Level: $177$
• Weight: $2$
• Character: 177.76
• 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			76.5
Character	\(\chi\)	\(=\)	177.76
Dual form			177.2.e.a.7.5

$q$-expansion

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

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{20}{29}\right)\)	\(1\)

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

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.a.76.5	yes	140
3.2	odd	2		531.2.i.c.253.1		140
59.7	even	29	inner	177.2.e.a.7.5	✓	140
177.125	odd	58		531.2.i.c.361.1		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.a.7.5	✓	140	59.7	even	29	inner
177.2.e.a.76.5	yes	140	1.1	even	1	trivial
531.2.i.c.253.1		140	3.2	odd	2
531.2.i.c.361.1		140	177.125	odd	58