Embedded newform 177.3.h.a.5.17

• Label: 177.3.h.a.5.17
• Level: $177$
• Weight: $3$
• Character: 177.5
• Analytic conductor: $4.823$
• Analytic rank: $0$
• Dimension: $1064$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	177.h (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.82290067918\)
Analytic rank:	\(0\)
Dimension:	\(1064\)
Relative dimension:	\(38\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			5.17
Character	\(\chi\)	\(=\)	177.5
Dual form			177.3.h.a.71.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.221328 + 0.656877i) q^{2} +(-2.72451 - 1.25582i) q^{3} +(2.80187 + 2.12993i) q^{4} +(-2.12339 - 0.846038i) q^{5} +(1.42792 - 1.51172i) q^{6} +(-1.48778 + 0.688319i) q^{7} +(-4.31412 + 2.92505i) q^{8} +(5.84586 + 6.84295i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{3}{29}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.221328	+	0.656877i	−0.110664	+	0.328438i	−0.988648	−	0.150252i	\(-0.951992\pi\)
							0.877984	+	0.478690i	\(0.158888\pi\)
\(3\)	−2.72451	−	1.25582i	−0.908168	−	0.418605i
							\(5\)	−2.12339	−	0.846038i	−0.424679	−	0.169208i	0.148000	−	0.988987i	\(-0.452716\pi\)
−0.572679	+	0.819780i	\(0.694096\pi\)
\(7\)	−1.48778	+	0.688319i	−0.212540	+	0.0983312i	−0.523270	−	0.852167i	\(-0.675288\pi\)
							0.310731	+	0.950498i	\(0.399426\pi\)
\(11\)	−14.8806	−	4.13157i	−1.35278	−	0.375597i	−0.485856	−	0.874039i	\(-0.661492\pi\)
							−0.866923	+	0.498442i	\(0.833906\pi\)
\(13\)	2.12693	−	4.01182i	0.163610	−	0.308602i	−0.788129	−	0.615511i	\(-0.788950\pi\)
							0.951739	+	0.306909i	\(0.0992947\pi\)
\(17\)	−12.2260	+	26.4262i	−0.719179	+	1.55448i	0.109016	+	0.994040i	\(0.465230\pi\)
							−0.828195	+	0.560440i	\(0.810632\pi\)
\(19\)	−19.7529	+	4.34795i	−1.03963	+	0.228839i	−0.701804	−	0.712370i	\(-0.747622\pi\)
							−0.337824	+	0.941209i	\(0.609691\pi\)
\(23\)	−35.2531	−	5.77945i	−1.53274	−	0.251281i	−0.664451	−	0.747332i	\(-0.731334\pi\)
							−0.868293	+	0.496052i	\(0.834783\pi\)
\(29\)	0.767237	+	2.27708i	0.0264565	+	0.0785200i	0.960068	−	0.279767i	\(-0.0902574\pi\)
							−0.933611	+	0.358287i	\(0.883361\pi\)
\(31\)	44.6306	+	9.82394i	1.43970	+	0.316901i	0.865127	−	0.501553i	\(-0.167238\pi\)
							0.574571	+	0.818455i	\(0.305169\pi\)
\(37\)	−17.7258	+	26.1436i	−0.479076	+	0.706584i	−0.988039	−	0.154204i	\(-0.950719\pi\)
							0.508963	+	0.860788i	\(0.330029\pi\)
\(41\)	74.9043	−	12.2799i	1.82694	−	0.299511i	0.852257	−	0.523123i	\(-0.175233\pi\)
							0.974678	+	0.223612i	\(0.0717849\pi\)
\(43\)	3.40711	+	12.2713i	0.0792351	+	0.285379i	0.992544	−	0.121889i	\(-0.0388953\pi\)
							−0.913309	+	0.407268i	\(0.866481\pi\)
\(47\)	30.7997	−	12.2717i	0.655313	−	0.261101i	−0.0186941	−	0.999825i	\(-0.505951\pi\)
							0.674007	+	0.738725i	\(0.264572\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.5.17	✓	1064
3.2	odd	2	inner	177.3.h.a.5.22	yes	1064
59.12	even	29	inner	177.3.h.a.71.22	yes	1064
177.71	odd	58	inner	177.3.h.a.71.17	yes	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.5.17	✓	1064	1.1	even	1	trivial
177.3.h.a.5.22	yes	1064	3.2	odd	2	inner
177.3.h.a.71.17	yes	1064	177.71	odd	58	inner
177.3.h.a.71.22	yes	1064	59.12	even	29	inner