Embedded newform 177.4.e.b.4.12

• Label: 177.4.e.b.4.12
• Level: $177$
• Weight: $4$
• Character: 177.4
• 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			4.12
Character	\(\chi\)	\(=\)	177.4
Dual form			177.4.e.b.133.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.13622 + 0.341085i) q^{2} +(1.94216 - 2.28649i) q^{3} +(1.90660 + 0.419674i) q^{4} +(-14.9247 - 11.3455i) q^{5} +(6.87093 - 6.50849i) q^{6} +(-10.7338 + 26.9398i) q^{7} +(-18.0802 - 6.09194i) q^{8} +(-1.45604 - 8.88144i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q + 2 q^{2} - 45 q^{3} - 56 q^{4} - 14 q^{5} + 6 q^{6} + 6 q^{7} + 66 q^{8} - 135 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.13622	+	0.341085i	1.10882	+	0.120592i	0.644156	−	0.764894i	\(-0.277209\pi\)
							0.464667	+	0.885486i	\(0.346174\pi\)
\(3\)	1.94216	−	2.28649i	0.373769	−	0.440034i
							\(5\)	−14.9247	−	11.3455i	−1.33491	−	1.01477i	−0.997296	−	0.0734847i	\(-0.976588\pi\)
−0.337611	−	0.941286i	\(-0.609619\pi\)
\(7\)	−10.7338	+	26.9398i	−0.579571	+	1.45461i	0.288162	+	0.957582i	\(0.406956\pi\)
							−0.867733	+	0.497031i	\(0.834423\pi\)
\(11\)	−8.96676	+	4.14846i	−0.245780	+	0.113710i	−0.538912	−	0.842362i	\(-0.681164\pi\)
							0.293132	+	0.956072i	\(0.405302\pi\)
\(13\)	12.5770	−	76.7162i	0.268325	−	1.63671i	−0.417309	−	0.908765i	\(-0.637027\pi\)
							0.685634	−	0.727946i	\(-0.259525\pi\)
\(17\)	21.7046	+	54.4745i	0.309656	+	0.777177i	0.998627	+	0.0523882i	\(0.0166833\pi\)
							−0.688971	+	0.724789i	\(0.741937\pi\)
\(19\)	−65.6467	−	96.8217i	−0.792652	−	1.16907i	−0.982317	−	0.187226i	\(-0.940050\pi\)
							0.189665	−	0.981849i	\(-0.439260\pi\)
\(23\)	−5.77306	+	106.478i	−0.0523376	+	0.965311i	0.847967	+	0.530050i	\(0.177827\pi\)
							−0.900304	+	0.435261i	\(0.856656\pi\)
\(29\)	−184.443	+	20.0594i	−1.18104	+	0.128446i	−0.677499	−	0.735524i	\(-0.736936\pi\)
							−0.503544	+	0.863970i	\(0.667971\pi\)
\(31\)	170.418	−	251.347i	0.987353	−	1.45624i	0.100087	−	0.994979i	\(-0.468088\pi\)
							0.887266	−	0.461258i	\(-0.152602\pi\)
\(37\)	204.283	−	68.8310i	0.907674	−	0.305831i	0.173533	−	0.984828i	\(-0.444482\pi\)
							0.734141	+	0.678997i	\(0.237585\pi\)
\(41\)	1.98400	+	36.5927i	0.00755727	+	0.139386i	0.999877	+	0.0156860i	\(0.00499320\pi\)
							−0.992320	+	0.123700i	\(0.960524\pi\)
\(43\)	−81.8963	−	37.8892i	−0.290443	−	0.134373i	0.269257	−	0.963068i	\(-0.413222\pi\)
							−0.559701	+	0.828695i	\(0.689084\pi\)
\(47\)	119.280	−	90.6745i	0.370188	−	0.281409i	−0.403496	−	0.914981i	\(-0.632205\pi\)
							0.773684	+	0.633572i	\(0.218412\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.e.b.4.12	✓	420
59.15	even	29	inner	177.4.e.b.133.12	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.e.b.4.12	✓	420	1.1	even	1	trivial
177.4.e.b.133.12	yes	420	59.15	even	29	inner