Embedded newform 177.3.h.a.71.6

• Label: 177.3.h.a.71.6
• 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			71.6
Character	\(\chi\)	\(=\)	177.71
Dual form			177.3.h.a.5.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.964344 - 2.86207i) q^{2} +(-0.263179 - 2.98843i) q^{3} +(-4.07712 + 3.09934i) q^{4} +(5.65262 - 2.25221i) q^{5} +(-8.29931 + 3.63511i) q^{6} +(-4.32408 - 2.00053i) q^{7} +(2.80324 + 1.90065i) q^{8} +(-8.86147 + 1.57299i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.964344	−	2.86207i	−0.482172	−	1.43104i	−0.862440	−	0.506159i	\(-0.831065\pi\)
							0.380268	−	0.924876i	\(-0.375832\pi\)
\(3\)	−0.263179	−	2.98843i	−0.0877263	−	0.996145i
							\(5\)	5.65262	−	2.25221i	1.13052	−	0.450442i	0.271531	−	0.962430i	\(-0.412470\pi\)
0.858992	+	0.511988i	\(0.171091\pi\)
\(7\)	−4.32408	−	2.00053i	−0.617725	−	0.285790i	0.0859637	−	0.996298i	\(-0.472603\pi\)
							−0.703689	+	0.710508i	\(0.748465\pi\)
\(11\)	−2.34660	+	0.651529i	−0.213327	+	0.0592299i	−0.372548	−	0.928013i	\(-0.621516\pi\)
							0.159221	+	0.987243i	\(0.449102\pi\)
\(13\)	−2.95997	−	5.58310i	−0.227690	−	0.429469i	0.743175	−	0.669097i	\(-0.233319\pi\)
							−0.970865	+	0.239629i	\(0.922974\pi\)
\(17\)	−6.05826	−	13.0947i	−0.356368	−	0.770277i	−0.999988	−	0.00493143i	\(-0.998430\pi\)
							0.643620	−	0.765345i	\(-0.277432\pi\)
\(19\)	28.6449	+	6.30521i	1.50762	+	0.331853i	0.890489	−	0.455005i	\(-0.150362\pi\)
							0.617135	+	0.786858i	\(0.288293\pi\)
\(23\)	−6.93017	+	1.13614i	−0.301312	+	0.0493976i	−0.310541	−	0.950560i	\(-0.600510\pi\)
							0.00922961	+	0.999957i	\(0.497062\pi\)
\(29\)	0.360463	−	1.06982i	0.0124298	−	0.0368902i	−0.941259	−	0.337686i	\(-0.890356\pi\)
							0.953689	+	0.300796i	\(0.0972523\pi\)
\(31\)	36.9353	−	8.13007i	1.19146	−	0.262260i	0.425386	−	0.905012i	\(-0.360138\pi\)
							0.766075	+	0.642751i	\(0.222207\pi\)
\(37\)	−16.2526	−	23.9708i	−0.439259	−	0.647858i	0.542039	−	0.840353i	\(-0.317652\pi\)
							−0.981298	+	0.192495i	\(0.938342\pi\)
\(41\)	−11.9036	−	1.95149i	−0.290331	−	0.0475974i	0.0148564	−	0.999890i	\(-0.495271\pi\)
							−0.305188	+	0.952292i	\(0.598719\pi\)
\(43\)	10.8669	−	39.1391i	0.252719	−	0.910212i	−0.722938	−	0.690913i	\(-0.757209\pi\)
							0.975658	−	0.219300i	\(-0.0703773\pi\)
\(47\)	43.7238	+	17.4212i	0.930294	+	0.370663i	0.785544	−	0.618805i	\(-0.212383\pi\)
							0.144750	+	0.989468i	\(0.453762\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.71.6	yes	1064
3.2	odd	2	inner	177.3.h.a.71.33	yes	1064
59.5	even	29	inner	177.3.h.a.5.33	yes	1064
177.5	odd	58	inner	177.3.h.a.5.6	✓	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.5.6	✓	1064	177.5	odd	58	inner
177.3.h.a.5.33	yes	1064	59.5	even	29	inner
177.3.h.a.71.6	yes	1064	1.1	even	1	trivial
177.3.h.a.71.33	yes	1064	3.2	odd	2	inner