Embedded newform 177.4.e.a.7.2

• Label: 177.4.e.a.7.2
• 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			7.2
Character	\(\chi\)	\(=\)	177.7
Dual form			177.4.e.a.76.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.77836 - 4.09778i) q^{2} +(-0.162417 + 2.99560i) q^{3} +(-6.11139 + 15.3384i) q^{4} +(11.4034 - 5.27577i) q^{5} +(12.7266 - 7.65732i) q^{6} +(-0.571992 + 2.06013i) q^{7} +(41.1522 - 9.05829i) q^{8} +(-8.94724 - 0.973071i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 2 q^{2} + 45 q^{3} - 64 q^{4} + 14 q^{5} + 6 q^{6} + 6 q^{7} + 18 q^{8} - 135 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.77836	−	4.09778i	−0.982300	−	1.44878i	−0.891556	−	0.452910i	\(-0.850386\pi\)
							−0.0907436	−	0.995874i	\(-0.528924\pi\)
\(3\)	−0.162417	+	2.99560i	−0.0312571	+	0.576504i
							\(5\)	11.4034	−	5.27577i	1.01995	−	0.471879i	0.162643	−	0.986685i	\(-0.447998\pi\)
0.857307	+	0.514806i	\(0.172136\pi\)
\(7\)	−0.571992	+	2.06013i	−0.0308846	+	0.111236i	−0.977386	−	0.211464i	\(-0.932177\pi\)
							0.946501	+	0.322701i	\(0.104591\pi\)
\(11\)	−14.9759	+	14.1859i	−0.410491	+	0.388837i	−0.864891	−	0.501960i	\(-0.832612\pi\)
							0.454400	+	0.890798i	\(0.349854\pi\)
\(13\)	−56.4345	+	6.13762i	−1.20401	+	0.130944i	−0.688046	−	0.725667i	\(-0.741531\pi\)
							−0.515963	+	0.856611i	\(0.672566\pi\)
\(17\)	19.2438	+	69.3100i	0.274548	+	0.988832i	0.964306	+	0.264792i	\(0.0853032\pi\)
							−0.689758	+	0.724040i	\(0.742283\pi\)
\(19\)	−23.2702	+	17.6896i	−0.280977	+	0.213593i	−0.736165	−	0.676802i	\(-0.763365\pi\)
							0.455188	+	0.890395i	\(0.349572\pi\)
\(23\)	−14.4763	+	27.3051i	−0.131240	+	0.247544i	−0.940419	−	0.340017i	\(-0.889567\pi\)
							0.809180	+	0.587561i	\(0.199912\pi\)
\(29\)	−66.6093	+	98.2414i	−0.426518	+	0.629068i	−0.978855	−	0.204553i	\(-0.934426\pi\)
							0.552337	+	0.833621i	\(0.313736\pi\)
\(31\)	24.1423	+	18.3525i	0.139873	+	0.106329i	0.672771	−	0.739851i	\(-0.265104\pi\)
							−0.532898	+	0.846180i	\(0.678897\pi\)
\(37\)	−126.670	−	27.8822i	−0.562823	−	0.123887i	−0.0755587	−	0.997141i	\(-0.524074\pi\)
							−0.487264	+	0.873255i	\(0.662005\pi\)
\(41\)	113.382	+	213.860i	0.431884	+	0.814619i	0.999965	−	0.00835312i	\(-0.00265891\pi\)
							−0.568081	+	0.822972i	\(0.692314\pi\)
\(43\)	153.916	+	145.797i	0.545859	+	0.517065i	0.910182	−	0.414208i	\(-0.135941\pi\)
							−0.364324	+	0.931272i	\(0.618700\pi\)
\(47\)	457.088	+	211.472i	1.41858	+	0.656304i	0.971249	−	0.238065i	\(-0.0765133\pi\)
							0.447329	+	0.894370i	\(0.352375\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.e.a.7.2	✓	420
59.17	even	29	inner	177.4.e.a.76.2	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.e.a.7.2	✓	420	1.1	even	1	trivial
177.4.e.a.76.2	yes	420	59.17	even	29	inner