Embedded newform 177.4.f.a.101.49

• Label: 177.4.f.a.101.49
• 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			101.49
Character	\(\chi\)	\(=\)	177.101
Dual form			177.4.f.a.170.49

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.36794 - 3.49245i) q^{2} +(3.40108 + 3.92845i) q^{3} +(-3.62897 - 9.10803i) q^{4} +(-8.13982 + 17.5939i) q^{5} +(21.7735 - 2.57577i) q^{6} +(0.277120 + 0.998095i) q^{7} +(-7.43558 - 1.63669i) q^{8} +(-3.86536 + 26.7219i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 21 q^{3} - 278 q^{4} - 29 q^{6} - 42 q^{7} - 25 q^{9}+O(q^{10}) \)

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	\(-0.542358\pi\)
							0.969873	−	0.243609i	\(-0.0783315\pi\)
\(3\)	3.40108	+	3.92845i	0.654537	+	0.756030i
							\(5\)	−8.13982	+	17.5939i	−0.728048	+	1.57365i	0.0883036	+	0.996094i	\(0.471855\pi\)
−0.816351	+	0.577556i	\(0.804007\pi\)
\(7\)	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	0.348201	−	0.937420i	\(-0.386793\pi\)
							−0.917194	+	0.398442i	\(0.869551\pi\)
\(13\)	−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	0.605497	−	0.795848i	\(-0.292974\pi\)
							0.108522	+	0.994094i	\(0.465388\pi\)
\(19\)	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	0.696325	−	0.717727i	\(-0.254817\pi\)
							−0.984823	+	0.173561i	\(0.944473\pi\)
\(29\)	58.5316	−	39.6854i	0.374794	−	0.254117i	−0.359217	−	0.933254i	\(-0.616956\pi\)
							0.734011	+	0.679137i	\(0.237646\pi\)
\(31\)	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	\(-0.928076\pi\)
							−0.732355	−	0.680923i	\(-0.761579\pi\)
\(43\)	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	−0.636804	−	0.771026i	\(-0.719744\pi\)
							0.175388	+	0.984499i	\(0.443882\pi\)

(See \(a_n\) instead)

Twists

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

    

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