Embedded newform 177.3.h.a.71.12

• Label: 177.3.h.a.71.12
• Level: $177$
• Weight: $3$
• Character: 177.71
• 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			71.12
Character	\(\chi\)	\(=\)	177.71
Dual form			177.3.h.a.5.12

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

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.5.12	✓	1064	177.5	odd	58	inner
177.3.h.a.5.27	yes	1064	59.5	even	29	inner
177.3.h.a.71.12	yes	1064	1.1	even	1	trivial
177.3.h.a.71.27	yes	1064	3.2	odd	2	inner