Embedded newform 177.4.e.a.7.1

• Label: 177.4.e.a.7.1
• 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			7.1
Character	\(\chi\)	\(=\)	177.7
Dual form			177.4.e.a.76.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.99760 - 4.42113i) q^{2} +(-0.162417 + 2.99560i) q^{3} +(-7.59966 + 19.0737i) q^{4} +(-18.0173 + 8.33567i) q^{5} +(13.7308 - 8.26154i) q^{6} +(-1.41944 + 5.11237i) q^{7} +(65.3749 - 14.3901i) q^{8} +(-8.94724 - 0.973071i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 2 q^{2} + 45 q^{3} - 64 q^{4} + 14 q^{5} + 6 q^{6} + 6 q^{7} + 18 q^{8} - 135 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.99760	−	4.42113i	−1.05981	−	1.56310i	−0.802385	−	0.596806i	\(-0.796436\pi\)
							−0.257426	−	0.966298i	\(-0.582874\pi\)
\(3\)	−0.162417	+	2.99560i	−0.0312571	+	0.576504i
							\(5\)	−18.0173	+	8.33567i	−1.61151	+	0.745565i	−0.998992	−	0.0448893i	\(-0.985706\pi\)
−0.612521	+	0.790455i	\(0.709844\pi\)
\(7\)	−1.41944	+	5.11237i	−0.0766427	+	0.276042i	−0.991935	−	0.126748i	\(-0.959546\pi\)
							0.915292	+	0.402791i	\(0.131960\pi\)
\(11\)	−45.4443	+	43.0471i	−1.24563	+	1.17993i	−0.269441	+	0.963017i	\(0.586839\pi\)
							−0.976192	+	0.216909i	\(0.930403\pi\)
\(13\)	60.9006	−	6.62333i	1.29929	−	0.141306i	0.567805	−	0.823163i	\(-0.307793\pi\)
							0.731486	+	0.681857i	\(0.238827\pi\)
\(17\)	−27.8867	−	100.439i	−0.397854	−	1.43294i	−0.843209	−	0.537586i	\(-0.819336\pi\)
							0.445355	−	0.895354i	\(-0.353077\pi\)
\(19\)	−92.9384	+	70.6499i	−1.12219	+	0.853063i	−0.990402	−	0.138215i	\(-0.955863\pi\)
							−0.131783	+	0.991279i	\(0.542070\pi\)
\(23\)	24.4179	−	46.0571i	0.221369	−	0.417546i	−0.747807	−	0.663916i	\(-0.768893\pi\)
							0.969176	+	0.246370i	\(0.0792379\pi\)
\(29\)	−49.9476	+	73.6671i	−0.319829	+	0.471712i	−0.953287	−	0.302064i	\(-0.902324\pi\)
							0.633459	+	0.773776i	\(0.281635\pi\)
\(31\)	41.2375	+	31.3479i	0.238918	+	0.181621i	0.717781	−	0.696269i	\(-0.245158\pi\)
							−0.478863	+	0.877890i	\(0.658951\pi\)
\(37\)	52.0584	+	11.4589i	0.231307	+	0.0509144i	0.329111	−	0.944291i	\(-0.393251\pi\)
							−0.0978044	+	0.995206i	\(0.531182\pi\)
\(41\)	−4.82417	−	9.09935i	−0.0183758	−	0.0346605i	0.874154	−	0.485649i	\(-0.161417\pi\)
							−0.892530	+	0.450988i	\(0.851072\pi\)
\(43\)	−189.422	−	179.430i	−0.671780	−	0.636344i	0.273780	−	0.961792i	\(-0.411726\pi\)
							−0.945560	+	0.325448i	\(0.894485\pi\)
\(47\)	33.2655	+	15.3902i	0.103240	+	0.0477638i	0.470829	−	0.882224i	\(-0.343955\pi\)
							−0.367590	+	0.929988i	\(0.619817\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.e.a.7.1	✓	420
59.17	even	29	inner	177.4.e.a.76.1	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.e.a.7.1	✓	420	1.1	even	1	trivial
177.4.e.a.76.1	yes	420	59.17	even	29	inner