Embedded newform 177.3.h.a.5.12

• Label: 177.3.h.a.5.12
• 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			5.12
Character	\(\chi\)	\(=\)	177.5
Dual form			177.3.h.a.71.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.551931 + 1.63807i) q^{2} +(-1.84525 + 2.36538i) q^{3} +(0.805720 + 0.612492i) q^{4} +(-3.83005 - 1.52603i) q^{5} +(-2.85622 - 4.32819i) q^{6} +(-0.768529 + 0.355559i) q^{7} +(-7.17084 + 4.86195i) q^{8} +(-2.19007 - 8.72947i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.551931	+	1.63807i	−0.275965	+	0.819036i	0.716618	+	0.697466i	\(0.245689\pi\)
							−0.992583	+	0.121570i	\(0.961207\pi\)
\(3\)	−1.84525	+	2.36538i	−0.615085	+	0.788461i
							\(5\)	−3.83005	−	1.52603i	−0.766009	−	0.305206i	−0.0457887	−	0.998951i	\(-0.514580\pi\)
−0.720221	+	0.693745i	\(0.755959\pi\)
\(7\)	−0.768529	+	0.355559i	−0.109790	+	0.0507942i	−0.474015	−	0.880517i	\(-0.657196\pi\)
							0.364225	+	0.931311i	\(0.381334\pi\)
\(11\)	−5.86735	−	1.62906i	−0.533395	−	0.148096i	−0.00962113	−	0.999954i	\(-0.503063\pi\)
							−0.523774	+	0.851857i	\(0.675476\pi\)
\(13\)	−0.796435	+	1.50224i	−0.0612643	+	0.115557i	−0.912253	−	0.409627i	\(-0.865659\pi\)
							0.850989	+	0.525184i	\(0.176004\pi\)
\(17\)	3.84674	−	8.31459i	0.226279	−	0.489093i	−0.760869	−	0.648906i	\(-0.775227\pi\)
							0.987147	+	0.159812i	\(0.0510890\pi\)
\(19\)	−11.0978	+	2.44280i	−0.584093	+	0.128569i	−0.497180	−	0.867647i	\(-0.665631\pi\)
							−0.0869127	+	0.996216i	\(0.527700\pi\)
\(23\)	18.5671	+	3.04392i	0.807264	+	0.132344i	0.551262	−	0.834332i	\(-0.314146\pi\)
							0.256002	+	0.966676i	\(0.417595\pi\)
\(29\)	−5.32592	−	15.8068i	−0.183652	−	0.545061i	0.815794	−	0.578343i	\(-0.196300\pi\)
							−0.999446	+	0.0332825i	\(0.989404\pi\)
\(31\)	−18.2646	−	4.02034i	−0.589180	−	0.129688i	−0.0896325	−	0.995975i	\(-0.528569\pi\)
							−0.499547	+	0.866287i	\(0.666500\pi\)
\(37\)	−20.1906	+	29.7789i	−0.545692	+	0.804836i	−0.996123	−	0.0879755i	\(-0.971960\pi\)
							0.450430	+	0.892812i	\(0.351271\pi\)
\(41\)	−63.2254	+	10.3653i	−1.54208	+	0.252811i	−0.872011	−	0.489486i	\(-0.837184\pi\)
							−0.670071	+	0.742297i	\(0.733736\pi\)
\(43\)	9.03107	+	32.5270i	0.210025	+	0.756441i	0.991228	+	0.132164i	\(0.0421926\pi\)
							−0.781203	+	0.624277i	\(0.785394\pi\)
\(47\)	−45.4641	+	18.1146i	−0.967322	+	0.385416i	−0.799677	−	0.600431i	\(-0.794996\pi\)
							−0.167645	+	0.985847i	\(0.553616\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.5.12	✓	1064
3.2	odd	2	inner	177.3.h.a.5.27	yes	1064
59.12	even	29	inner	177.3.h.a.71.27	yes	1064
177.71	odd	58	inner	177.3.h.a.71.12	yes	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.5.12	✓	1064	1.1	even	1	trivial
177.3.h.a.5.27	yes	1064	3.2	odd	2	inner
177.3.h.a.71.12	yes	1064	177.71	odd	58	inner
177.3.h.a.71.27	yes	1064	59.12	even	29	inner