Embedded newform 177.4.e.b.4.4

• Label: 177.4.e.b.4.4
• 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			4.4
Character	\(\chi\)	\(=\)	177.4
Dual form			177.4.e.b.133.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.05964 - 0.332755i) q^{2} +(1.94216 - 2.28649i) q^{3} +(1.43768 + 0.316458i) q^{4} +(12.7861 + 9.71975i) q^{5} +(-6.70314 + 6.34955i) q^{6} +(6.84669 - 17.1839i) q^{7} +(19.0391 + 6.41502i) q^{8} +(-1.45604 - 8.88144i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q + 2 q^{2} - 45 q^{3} - 56 q^{4} - 14 q^{5} + 6 q^{6} + 6 q^{7} + 66 q^{8} - 135 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.05964	−	0.332755i	−1.08174	−	0.117647i	−0.450156	−	0.892950i	\(-0.648631\pi\)
							−0.631589	+	0.775303i	\(0.717597\pi\)
\(3\)	1.94216	−	2.28649i	0.373769	−	0.440034i
							\(5\)	12.7861	+	9.71975i	1.14362	+	0.869361i	0.992869	−	0.119210i	\(-0.0380363\pi\)
0.150755	+	0.988571i	\(0.451829\pi\)
\(7\)	6.84669	−	17.1839i	0.369687	−	0.927843i	−0.620001	−	0.784601i	\(-0.712868\pi\)
							0.989688	−	0.143242i	\(-0.0457528\pi\)
\(11\)	−10.9276	+	5.05567i	−0.299528	+	0.138576i	−0.563897	−	0.825845i	\(-0.690698\pi\)
							0.264369	+	0.964422i	\(0.414836\pi\)
\(13\)	−9.50641	+	57.9866i	−0.202816	+	1.23712i	0.667493	+	0.744616i	\(0.267367\pi\)
							−0.870309	+	0.492506i	\(0.836081\pi\)
\(17\)	12.5073	+	31.3909i	0.178439	+	0.447848i	0.990805	−	0.135300i	\(-0.0431999\pi\)
							−0.812366	+	0.583148i	\(0.801821\pi\)
\(19\)	66.0033	+	97.3475i	0.796957	+	1.17542i	0.981285	+	0.192559i	\(0.0616787\pi\)
							−0.184328	+	0.982865i	\(0.559011\pi\)
\(23\)	9.75166	−	179.859i	0.0884070	−	1.63057i	−0.532319	−	0.846544i	\(-0.678679\pi\)
							0.620726	−	0.784027i	\(-0.286838\pi\)
\(29\)	280.238	−	30.4777i	1.79445	−	0.195158i	0.850660	−	0.525716i	\(-0.176203\pi\)
							0.943786	+	0.330558i	\(0.107237\pi\)
\(31\)	14.7460	−	21.7487i	0.0854340	−	0.126006i	−0.782571	−	0.622562i	\(-0.786092\pi\)
							0.868005	+	0.496556i	\(0.165402\pi\)
\(37\)	116.275	−	39.1777i	0.516636	−	0.174075i	−0.0488919	−	0.998804i	\(-0.515569\pi\)
							0.565528	+	0.824729i	\(0.308672\pi\)
\(41\)	12.7945	+	235.981i	0.0487357	+	0.898878i	0.916037	+	0.401094i	\(0.131370\pi\)
							−0.867301	+	0.497784i	\(0.834147\pi\)
\(43\)	146.521	+	67.7880i	0.519635	+	0.240409i	0.662133	−	0.749386i	\(-0.269651\pi\)
							−0.142498	+	0.989795i	\(0.545513\pi\)
\(47\)	437.043	−	332.231i	1.35637	−	1.03108i	0.361241	−	0.932472i	\(-0.382353\pi\)
							0.995126	−	0.0986114i	\(-0.0314401\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.e.b.4.4	✓	420
59.15	even	29	inner	177.4.e.b.133.4	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.e.b.4.4	✓	420	1.1	even	1	trivial
177.4.e.b.133.4	yes	420	59.15	even	29	inner