Embedded newform 177.4.f.a.101.10

• Label: 177.4.f.a.101.10
• Level: $177$
• Weight: $4$
• Character: 177.101
• Analytic conductor: $10.443$
• Analytic rank: $0$
• Dimension: $1624$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.f (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.4433380710\)
Analytic rank:	\(0\)
Dimension:	\(1624\)
Relative dimension:	\(58\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			101.10
Character	\(\chi\)	\(=\)	177.101
Dual form			177.4.f.a.170.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.36794 + 3.49245i) q^{2} +(-2.95640 + 4.27314i) q^{3} +(-3.62897 - 9.10803i) q^{4} +(8.13982 - 17.5939i) q^{5} +(-7.92315 - 20.4436i) q^{6} +(0.277120 + 0.998095i) q^{7} +(7.43558 + 1.63669i) q^{8} +(-9.51941 - 25.2662i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 21 q^{3} - 278 q^{4} - 29 q^{6} - 42 q^{7} - 25 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{11}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.36794	+	3.49245i	−0.837194	+	1.23477i	0.132680	+	0.991159i	\(0.457642\pi\)
							−0.969873	+	0.243609i	\(0.921669\pi\)
\(3\)	−2.95640	+	4.27314i	−0.568959	+	0.822366i
							\(5\)	8.13982	−	17.5939i	0.728048	−	1.57365i	−0.0883036	−	0.996094i	\(-0.528145\pi\)
0.816351	−	0.577556i	\(-0.195993\pi\)
\(7\)	0.277120	+	0.998095i	0.0149631	+	0.0538921i	0.970655	−	0.240478i	\(-0.0773041\pi\)
							−0.955692	+	0.294370i	\(0.904890\pi\)
\(11\)	20.7585	+	19.6635i	0.568992	+	0.538978i	0.917194	−	0.398442i	\(-0.130449\pi\)
							−0.348201	+	0.937420i	\(0.613207\pi\)
\(13\)	−5.93290	+	54.5521i	−0.126576	+	1.16385i	0.742736	+	0.669585i	\(0.233528\pi\)
							−0.869312	+	0.494264i	\(0.835438\pi\)
\(17\)	−50.0476	−	13.8956i	−0.714019	−	0.198246i	−0.108522	−	0.994094i	\(-0.534612\pi\)
							−0.605497	+	0.795848i	\(0.707026\pi\)
\(19\)	122.466	+	93.0960i	1.47871	+	1.12409i	0.963727	+	0.266891i	\(0.0859964\pi\)
							0.514987	+	0.857198i	\(0.327797\pi\)
\(23\)	31.8225	+	60.0237i	0.288498	+	0.544165i	0.984823	−	0.173561i	\(-0.0555275\pi\)
							−0.696325	+	0.717727i	\(0.745183\pi\)
\(29\)	−58.5316	+	39.6854i	−0.374794	+	0.254117i	−0.734011	−	0.679137i	\(-0.762354\pi\)
							0.359217	+	0.933254i	\(0.383044\pi\)
\(31\)	12.7522	+	16.7752i	0.0738827	+	0.0971911i	0.831550	−	0.555450i	\(-0.187454\pi\)
							−0.757667	+	0.652641i	\(0.773661\pi\)
\(37\)	−19.6822	−	89.4172i	−0.0874523	−	0.397300i	0.912478	−	0.409126i	\(-0.134166\pi\)
							−0.999930	+	0.0118264i	\(0.996235\pi\)
\(41\)	448.119	+	237.577i	1.70694	+	0.904960i	0.974581	+	0.224037i	\(0.0719238\pi\)
							0.732355	+	0.680923i	\(0.238421\pi\)
\(43\)	29.5471	+	31.1924i	0.104788	+	0.110623i	0.776279	−	0.630389i	\(-0.217105\pi\)
							−0.671491	+	0.741012i	\(0.734346\pi\)
\(47\)	148.676	−	68.7846i	0.461416	−	0.213474i	−0.175388	−	0.984499i	\(-0.556118\pi\)
							0.636804	+	0.771026i	\(0.280256\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.f.a.101.10	✓	1624
3.2	odd	2	inner	177.4.f.a.101.49	yes	1624
59.52	odd	58	inner	177.4.f.a.170.49	yes	1624
177.170	even	58	inner	177.4.f.a.170.10	yes	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.f.a.101.10	✓	1624	1.1	even	1	trivial
177.4.f.a.101.49	yes	1624	3.2	odd	2	inner
177.4.f.a.170.10	yes	1624	177.170	even	58	inner
177.4.f.a.170.49	yes	1624	59.52	odd	58	inner