Embedded newform 177.2.e.a.7.1

• Label: 177.2.e.a.7.1
• 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			7.1
Character	\(\chi\)	\(=\)	177.7
Dual form			177.2.e.a.76.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39179 - 2.05274i) q^{2} +(0.0541389 - 0.998533i) q^{3} +(-1.53638 + 3.85601i) q^{4} +(-1.61683 + 0.748026i) q^{5} +(-2.12508 + 1.27862i) q^{6} +(-0.975578 + 3.51371i) q^{7} +(5.20950 - 1.14670i) q^{8} +(-0.994138 - 0.108119i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - q^{2} - 5 q^{3} - 9 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} - 9 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39179	−	2.05274i	−0.984145	−	1.45151i	−0.890007	−	0.455946i	\(-0.849301\pi\)
							−0.0941374	−	0.995559i	\(-0.530009\pi\)
\(3\)	0.0541389	−	0.998533i	0.0312571	−	0.576504i
							\(5\)	−1.61683	+	0.748026i	−0.723069	+	0.334527i	−0.746670	−	0.665194i	\(-0.768349\pi\)
0.0236015	+	0.999721i	\(0.492487\pi\)
\(7\)	−0.975578	+	3.51371i	−0.368734	+	1.32806i	0.513840	+	0.857886i	\(0.328222\pi\)
							−0.882574	+	0.470173i	\(0.844191\pi\)
\(11\)	−2.09707	+	1.98645i	−0.632292	+	0.598938i	−0.935217	−	0.354076i	\(-0.884795\pi\)
							0.302925	+	0.953014i	\(0.402037\pi\)
\(13\)	−5.08799	+	0.553353i	−1.41116	+	0.153472i	−0.781744	−	0.623600i	\(-0.785670\pi\)
							−0.629412	+	0.777072i	\(0.716704\pi\)
\(17\)	−1.42624	−	5.13686i	−0.345914	−	1.24587i	−0.908063	−	0.418835i	\(-0.862439\pi\)
							0.562148	−	0.827037i	\(-0.309975\pi\)
\(19\)	3.59214	−	2.73067i	0.824093	−	0.626459i	−0.105760	−	0.994392i	\(-0.533727\pi\)
							0.929852	+	0.367933i	\(0.119934\pi\)
\(23\)	−0.197691	+	0.372884i	−0.0412214	+	0.0777518i	−0.903267	−	0.429079i	\(-0.858838\pi\)
							0.862046	+	0.506831i	\(0.169183\pi\)
\(29\)	−4.40513	+	6.49708i	−0.818012	+	1.20648i	0.157715	+	0.987485i	\(0.449587\pi\)
							−0.975727	+	0.218993i	\(0.929723\pi\)
\(31\)	−2.97159	−	2.25894i	−0.533713	−	0.405718i	0.303305	−	0.952893i	\(-0.401910\pi\)
							−0.837018	+	0.547176i	\(0.815703\pi\)
\(37\)	6.60033	+	1.45284i	1.08509	+	0.238846i	0.721291	−	0.692633i	\(-0.243549\pi\)
							0.363797	+	0.931478i	\(0.381480\pi\)
\(41\)	3.63656	+	6.85928i	0.567935	+	1.07124i	0.986609	+	0.163103i	\(0.0521503\pi\)
							−0.418674	+	0.908137i	\(0.637505\pi\)
\(43\)	−6.02922	−	5.71119i	−0.919448	−	0.870948i	0.0725288	−	0.997366i	\(-0.476893\pi\)
							−0.991977	+	0.126419i	\(0.959652\pi\)
\(47\)	−5.25263	−	2.43012i	−0.766175	−	0.354470i	−0.00243702	−	0.999997i	\(-0.500776\pi\)
							−0.763737	+	0.645527i	\(0.776638\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.a.7.1	✓	140
3.2	odd	2		531.2.i.c.361.5		140
59.17	even	29	inner	177.2.e.a.76.1	yes	140
177.17	odd	58		531.2.i.c.253.5		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.a.7.1	✓	140	1.1	even	1	trivial
177.2.e.a.76.1	yes	140	59.17	even	29	inner
531.2.i.c.253.5		140	177.17	odd	58
531.2.i.c.361.5		140	3.2	odd	2