Embedded newform 177.2.e.a.76.4

• Label: 177.2.e.a.76.4
• 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			76.4
Character	\(\chi\)	\(=\)	177.76
Dual form			177.2.e.a.7.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.579343 - 0.854467i) q^{2} +(0.0541389 + 0.998533i) q^{3} +(0.345801 + 0.867894i) q^{4} +(0.231383 + 0.107049i) q^{5} +(0.884579 + 0.532233i) q^{6} +(-0.661330 - 2.38190i) q^{7} +(2.95836 + 0.651184i) q^{8} +(-0.994138 + 0.108119i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - q^{2} - 5 q^{3} - 9 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} - 9 q^{8} - 5 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.579343	−	0.854467i	0.409657	−	0.604199i	−0.565759	−	0.824571i	\(-0.691417\pi\)
							0.975416	+	0.220371i	\(0.0707269\pi\)
\(3\)	0.0541389	+	0.998533i	0.0312571	+	0.576504i
							\(5\)	0.231383	+	0.107049i	0.103478	+	0.0478738i	0.470945	−	0.882163i	\(-0.343913\pi\)
−0.367467	+	0.930036i	\(0.619775\pi\)
\(7\)	−0.661330	−	2.38190i	−0.249959	−	0.900272i	−0.976922	−	0.213595i	\(-0.931483\pi\)
							0.726963	−	0.686677i	\(-0.240931\pi\)
\(11\)	4.21583	+	3.99345i	1.27112	+	1.20407i	0.967953	+	0.251132i	\(0.0808028\pi\)
							0.303168	+	0.952937i	\(0.401956\pi\)
\(13\)	0.0916376	+	0.00996619i	0.0254157	+	0.00276412i	0.120818	−	0.992675i	\(-0.461448\pi\)
							−0.0954023	+	0.995439i	\(0.530414\pi\)
\(17\)	−0.270495	+	0.974234i	−0.0656046	+	0.236286i	−0.989082	−	0.147367i	\(-0.952920\pi\)
							0.923477	+	0.383653i	\(0.125334\pi\)
\(19\)	−6.56769	−	4.99263i	−1.50673	−	1.14539i	−0.948961	−	0.315393i	\(-0.897864\pi\)
							−0.557771	−	0.829995i	\(-0.688343\pi\)
\(23\)	−1.85006	−	3.48958i	−0.385764	−	0.727628i	0.612247	−	0.790666i	\(-0.290266\pi\)
							−0.998011	+	0.0630385i	\(0.979921\pi\)
\(29\)	−4.17622	−	6.15946i	−0.775504	−	1.14378i	−0.986091	−	0.166208i	\(-0.946848\pi\)
							0.210587	−	0.977575i	\(-0.432463\pi\)
\(31\)	−5.52322	+	4.19865i	−0.992000	+	0.754099i	−0.969360	−	0.245645i	\(-0.921000\pi\)
							−0.0226405	+	0.999744i	\(0.507207\pi\)
\(37\)	7.25593	−	1.59715i	1.19287	−	0.262570i	0.426209	−	0.904625i	\(-0.359849\pi\)
							0.766659	+	0.642055i	\(0.221918\pi\)
\(41\)	−1.68983	+	3.18736i	−0.263907	+	0.497782i	−0.979684	−	0.200547i	\(-0.935728\pi\)
							0.715777	+	0.698329i	\(0.246073\pi\)
\(43\)	−1.18715	+	1.12453i	−0.181039	+	0.171490i	−0.772853	−	0.634585i	\(-0.781171\pi\)
							0.591813	+	0.806075i	\(0.298412\pi\)
\(47\)	4.56376	−	2.11142i	0.665693	−	0.307982i	−0.0577859	−	0.998329i	\(-0.518404\pi\)
							0.723479	+	0.690347i	\(0.242542\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.a.76.4	yes	140
3.2	odd	2		531.2.i.c.253.2		140
59.7	even	29	inner	177.2.e.a.7.4	✓	140
177.125	odd	58		531.2.i.c.361.2		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.a.7.4	✓	140	59.7	even	29	inner
177.2.e.a.76.4	yes	140	1.1	even	1	trivial
531.2.i.c.253.2		140	3.2	odd	2
531.2.i.c.361.2		140	177.125	odd	58