Embedded newform 177.2.e.a.130.1

• Label: 177.2.e.a.130.1
• Level: $177$
• Weight: $2$
• Character: 177.130
• Analytic conductor: $1.413$
• Analytic rank: $0$
• Dimension: $140$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.e (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.41335211578\)
Analytic rank:	\(0\)
Dimension:	\(140\)
Relative dimension:	\(5\) over \(\Q(\zeta_{29})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{29}]$

Embedding invariants

Embedding label			130.1
Character	\(\chi\)	\(=\)	177.130
Dual form			177.2.e.a.64.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.61370 + 0.880659i) q^{2} +(-0.856857 + 0.515554i) q^{3} +(4.46370 - 3.39321i) q^{4} +(0.422705 + 1.06091i) q^{5} +(1.78554 - 2.10210i) q^{6} +(-3.24336 - 1.50054i) q^{7} +(-5.58292 + 8.23419i) q^{8} +(0.468408 - 0.883512i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - q^{2} - 5 q^{3} - 9 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} - 9 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{26}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.61370	+	0.880659i	−1.84817	+	0.622720i	−0.853313	+	0.521398i	\(0.825411\pi\)
							−0.994854	+	0.101322i	\(0.967693\pi\)
\(3\)	−0.856857	+	0.515554i	−0.494707	+	0.297655i
							\(5\)	0.422705	+	1.06091i	0.189039	+	0.474453i	0.992696	−	0.120640i	\(-0.0384946\pi\)
−0.803657	+	0.595093i	\(0.797115\pi\)
\(7\)	−3.24336	−	1.50054i	−1.22587	−	0.567150i	−0.303273	−	0.952904i	\(-0.598079\pi\)
							−0.922602	+	0.385754i	\(0.873941\pi\)
\(11\)	−1.24288	−	4.47645i	−0.374743	−	1.34970i	−0.875125	−	0.483897i	\(-0.839221\pi\)
							0.500382	−	0.865805i	\(-0.333193\pi\)
\(13\)	0.233736	+	0.440872i	0.0648266	+	0.122276i	0.913807	−	0.406148i	\(-0.133128\pi\)
							−0.848981	+	0.528424i	\(0.822783\pi\)
\(17\)	3.12785	−	1.44710i	0.758614	−	0.350972i	−0.00215268	−	0.999998i	\(-0.500685\pi\)
							0.760767	+	0.649025i	\(0.224823\pi\)
\(19\)	3.33146	+	0.733309i	0.764289	+	0.168233i	0.579974	−	0.814635i	\(-0.303063\pi\)
							0.184314	+	0.982867i	\(0.440994\pi\)
\(23\)	−1.36389	−	8.31938i	−0.284391	−	1.73471i	−0.612856	−	0.790195i	\(-0.709979\pi\)
							0.328465	−	0.944516i	\(-0.393469\pi\)
\(29\)	−4.38942	−	1.47897i	−0.815095	−	0.274638i	−0.119282	−	0.992860i	\(-0.538059\pi\)
							−0.695813	+	0.718223i	\(0.744956\pi\)
\(31\)	−1.07570	+	0.236779i	−0.193201	+	0.0425268i	−0.310517	−	0.950568i	\(-0.600502\pi\)
							0.117316	+	0.993095i	\(0.462571\pi\)
\(37\)	−1.37727	−	2.03133i	−0.226422	−	0.333948i	0.697461	−	0.716623i	\(-0.254313\pi\)
							−0.923883	+	0.382675i	\(0.875003\pi\)
\(41\)	−0.568080	+	3.46513i	−0.0887192	+	0.541163i	0.904646	+	0.426163i	\(0.140135\pi\)
							−0.993366	+	0.115000i	\(0.963313\pi\)
\(43\)	0.990022	−	3.56574i	0.150977	−	0.543770i	−0.848907	−	0.528542i	\(-0.822739\pi\)
							0.999884	−	0.0152278i	\(-0.00484735\pi\)
\(47\)	0.881582	−	2.21260i	0.128592	−	0.322742i	−0.850558	−	0.525882i	\(-0.823735\pi\)
							0.979150	+	0.203140i	\(0.0651147\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.a.130.1	yes	140
3.2	odd	2		531.2.i.c.307.5		140
59.5	even	29	inner	177.2.e.a.64.1	✓	140
177.5	odd	58		531.2.i.c.64.5		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.a.64.1	✓	140	59.5	even	29	inner
177.2.e.a.130.1	yes	140	1.1	even	1	trivial
531.2.i.c.64.5		140	177.5	odd	58
531.2.i.c.307.5		140	3.2	odd	2