Embedded newform 177.4.e.a.4.1

• Label: 177.4.e.a.4.1
• 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			4.1
Character	\(\chi\)	\(=\)	177.4
Dual form			177.4.e.a.133.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.26239 - 0.572319i) q^{2} +(-1.94216 + 2.28649i) q^{3} +(19.5522 + 4.30376i) q^{4} +(-16.6552 - 12.6610i) q^{5} +(11.5290 - 10.9208i) q^{6} +(-1.92301 + 4.82639i) q^{7} +(-60.2975 - 20.3166i) q^{8} +(-1.45604 - 8.88144i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 2 q^{2} + 45 q^{3} - 64 q^{4} + 14 q^{5} + 6 q^{6} + 6 q^{7} + 18 q^{8} - 135 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.26239	−	0.572319i	−1.86053	−	0.202345i	−0.892138	−	0.451762i	\(-0.850795\pi\)
							−0.968396	+	0.249417i	\(0.919761\pi\)
\(3\)	−1.94216	+	2.28649i	−0.373769	+	0.440034i
							\(5\)	−16.6552	−	12.6610i	−1.48969	−	1.13243i	−0.958470	−	0.285195i	\(-0.907942\pi\)
−0.531217	−	0.847236i	\(-0.678265\pi\)
\(7\)	−1.92301	+	4.82639i	−0.103833	+	0.260601i	−0.971710	−	0.236179i	\(-0.924105\pi\)
							0.867877	+	0.496779i	\(0.165484\pi\)
\(11\)	11.4273	−	5.28684i	0.313224	−	0.144913i	−0.256978	−	0.966417i	\(-0.582727\pi\)
							0.570202	+	0.821504i	\(0.306865\pi\)
\(13\)	10.4569	−	63.7845i	0.223095	−	1.36082i	−0.603205	−	0.797586i	\(-0.706110\pi\)
							0.826299	−	0.563231i	\(-0.190442\pi\)
\(17\)	−10.9701	−	27.5329i	−0.156508	−	0.392806i	0.829724	−	0.558174i	\(-0.188498\pi\)
							−0.986232	+	0.165368i	\(0.947119\pi\)
\(19\)	−82.0305	−	120.986i	−0.990479	−	1.46085i	−0.884535	−	0.466475i	\(-0.845524\pi\)
							−0.105944	−	0.994372i	\(-0.533786\pi\)
\(23\)	−1.40856	+	25.9793i	−0.0127697	+	0.235524i	0.985180	+	0.171523i	\(0.0548688\pi\)
							−0.997950	+	0.0640009i	\(0.979614\pi\)
\(29\)	170.874	−	18.5837i	1.09415	−	0.118996i	0.456802	−	0.889569i	\(-0.348995\pi\)
							0.637353	+	0.770572i	\(0.280029\pi\)
\(31\)	−53.7278	+	79.2425i	−0.311284	+	0.459109i	−0.950861	−	0.309618i	\(-0.899799\pi\)
							0.639578	+	0.768727i	\(0.279109\pi\)
\(37\)	−252.909	+	85.2149i	−1.12373	+	0.378628i	−0.818990	−	0.573808i	\(-0.805466\pi\)
							−0.304738	+	0.952436i	\(0.598569\pi\)
\(41\)	14.4188	+	265.939i	0.0549229	+	1.01299i	0.888002	+	0.459840i	\(0.152093\pi\)
							−0.833079	+	0.553154i	\(0.813424\pi\)
\(43\)	−329.650	−	152.512i	−1.16910	−	0.540881i	−0.263349	−	0.964701i	\(-0.584827\pi\)
							−0.905747	+	0.423820i	\(0.860689\pi\)
\(47\)	−290.828	+	221.082i	−0.902589	+	0.686130i	−0.949834	−	0.312753i	\(-0.898749\pi\)
							0.0472458	+	0.998883i	\(0.484956\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.e.a.4.1	✓	420
59.15	even	29	inner	177.4.e.a.133.1	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.e.a.4.1	✓	420	1.1	even	1	trivial
177.4.e.a.133.1	yes	420	59.15	even	29	inner