Embedded newform 177.4.e.b.133.7

• Label: 177.4.e.b.133.7
• 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			133.7
Character	\(\chi\)	\(=\)	177.133
Dual form			177.4.e.b.4.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.488701 + 0.0531494i) q^{2} +(1.94216 + 2.28649i) q^{3} +(-7.57696 + 1.66782i) q^{4} +(-7.57411 + 5.75769i) q^{5} +(-1.07066 - 1.01418i) q^{6} +(-3.71388 - 9.32114i) q^{7} +(7.34103 - 2.47348i) q^{8} +(-1.45604 + 8.88144i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q + 2 q^{2} - 45 q^{3} - 56 q^{4} - 14 q^{5} + 6 q^{6} + 6 q^{7} + 66 q^{8} - 135 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.488701	+	0.0531494i	−0.172782	+	0.0187912i	−0.194101	−	0.980982i	\(-0.562179\pi\)
							0.0213190	+	0.999773i	\(0.493213\pi\)
\(3\)	1.94216	+	2.28649i	0.373769	+	0.440034i
							\(5\)	−7.57411	+	5.75769i	−0.677449	+	0.514983i	−0.886387	−	0.462946i	\(-0.846792\pi\)
0.208938	+	0.977929i	\(0.432999\pi\)
\(7\)	−3.71388	−	9.32114i	−0.200531	−	0.503294i	0.793978	−	0.607946i	\(-0.208006\pi\)
							−0.994509	+	0.104652i	\(0.966627\pi\)
\(11\)	4.97846	+	2.30328i	0.136460	+	0.0631333i	0.486931	−	0.873440i	\(-0.338116\pi\)
							−0.350471	+	0.936574i	\(0.613978\pi\)
\(13\)	−6.56104	−	40.0206i	−0.139977	−	0.853823i	−0.959692	−	0.281052i	\(-0.909317\pi\)
							0.819715	−	0.572771i	\(-0.194132\pi\)
\(17\)	20.8605	−	52.3560i	0.297613	−	0.746953i	−0.701789	−	0.712385i	\(-0.747615\pi\)
							0.999402	−	0.0345680i	\(-0.0110055\pi\)
\(19\)	22.1537	−	32.6743i	0.267495	−	0.394526i	−0.670091	−	0.742279i	\(-0.733745\pi\)
							0.937586	+	0.347753i	\(0.113055\pi\)
\(23\)	−5.88001	−	108.450i	−0.0533073	−	0.983195i	−0.895792	−	0.444474i	\(-0.853391\pi\)
							0.842484	−	0.538721i	\(-0.181092\pi\)
\(29\)	84.0618	+	9.14228i	0.538272	+	0.0585406i	0.373218	−	0.927744i	\(-0.378254\pi\)
							0.165055	+	0.986284i	\(0.447220\pi\)
\(31\)	−83.0912	−	122.550i	−0.481407	−	0.710023i	0.506983	−	0.861956i	\(-0.330761\pi\)
							−0.988391	+	0.151933i	\(0.951450\pi\)
\(37\)	−176.052	−	59.3188i	−0.782236	−	0.263566i	−0.100280	−	0.994959i	\(-0.531974\pi\)
							−0.681956	+	0.731393i	\(0.738871\pi\)
\(41\)	19.0132	−	350.678i	0.0724234	−	1.33577i	−0.706080	−	0.708132i	\(-0.749538\pi\)
							0.778504	−	0.627640i	\(-0.215979\pi\)
\(43\)	−224.356	+	103.798i	−0.795673	+	0.368118i	−0.775235	−	0.631673i	\(-0.782368\pi\)
							−0.0204388	+	0.999791i	\(0.506506\pi\)
\(47\)	67.3632	+	51.2082i	0.209062	+	0.158925i	0.704466	−	0.709738i	\(-0.251187\pi\)
							−0.495403	+	0.868663i	\(0.664980\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.e.b.133.7	yes	420
59.4	even	29	inner	177.4.e.b.4.7	✓	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.e.b.4.7	✓	420	59.4	even	29	inner
177.4.e.b.133.7	yes	420	1.1	even	1	trivial