Embedded newform 177.4.e.b.133.4

• Label: 177.4.e.b.133.4
• Level: $177$
• Weight: $4$
• Character: 177.133
• 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			133.4
Character	\(\chi\)	\(=\)	177.133
Dual form			177.4.e.b.4.4

$q$-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 \(\left(\mathbb{Z}/177\mathbb{Z}\right)^\times\).

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{28}{29}\right)\)	\(1\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.e.b.4.4	✓	420	59.4	even	29	inner
177.4.e.b.133.4	yes	420	1.1	even	1	trivial