Embedded newform 177.3.h.a.5.6

• Label: 177.3.h.a.5.6
• 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			5.6
Character	\(\chi\)	\(=\)	177.5
Dual form			177.3.h.a.71.6

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

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.5.6	✓	1064	1.1	even	1	trivial
177.3.h.a.5.33	yes	1064	3.2	odd	2	inner
177.3.h.a.71.6	yes	1064	177.71	odd	58	inner
177.3.h.a.71.33	yes	1064	59.12	even	29	inner