Embedded newform 177.3.g.a.124.7

• Label: 177.3.g.a.124.7
• Level: $177$
• Weight: $3$
• Character: 177.124
• Analytic conductor: $4.823$
• Analytic rank: $0$
• Dimension: $560$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.82290067918\)
Analytic rank:	\(0\)
Dimension:	\(560\)
Relative dimension:	\(20\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			124.7
Character	\(\chi\)	\(=\)	177.124
Dual form			177.3.g.a.10.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.49154 + 0.594282i) q^{2} +(-1.72190 + 0.187268i) q^{3} +(-1.03247 + 0.978012i) q^{4} +(-2.12045 - 2.49638i) q^{5} +(2.45698 - 1.30261i) q^{6} +(6.56905 - 3.95247i) q^{7} +(3.65540 - 7.90102i) q^{8} +(2.92986 - 0.644911i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q + 40 q^{4} + 8 q^{7} - 60 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{51}{58}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.49154	+	0.594282i	−0.745768	+	0.297141i	−0.711896	−	0.702285i	\(-0.752163\pi\)
							−0.0338720	+	0.999426i	\(0.510784\pi\)
\(3\)	−1.72190	+	0.187268i	−0.573966	+	0.0624225i
							\(5\)	−2.12045	−	2.49638i	−0.424089	−	0.499276i	0.508075	−	0.861313i	\(-0.330357\pi\)
−0.932164	+	0.362037i	\(0.882082\pi\)
\(7\)	6.56905	−	3.95247i	0.938436	−	0.564638i	0.0378346	−	0.999284i	\(-0.487954\pi\)
							0.900602	+	0.434646i	\(0.143126\pi\)
\(11\)	−6.49916	+	0.352374i	−0.590833	+	0.0320340i	−0.347132	−	0.937816i	\(-0.612844\pi\)
							−0.243701	+	0.969850i	\(0.578362\pi\)
\(13\)	−4.42943	+	20.1231i	−0.340726	+	1.54793i	0.425258	+	0.905072i	\(0.360183\pi\)
							−0.765984	+	0.642860i	\(0.777748\pi\)
\(17\)	20.4244	+	12.2890i	1.20144	+	0.722880i	0.968355	−	0.249578i	\(-0.0802918\pi\)
							0.233081	+	0.972457i	\(0.425119\pi\)
\(19\)	5.98783	+	21.5662i	0.315149	+	1.13506i	0.936250	+	0.351333i	\(0.114272\pi\)
							−0.621102	+	0.783730i	\(0.713315\pi\)
\(23\)	−2.56095	−	1.73637i	−0.111346	−	0.0754942i	0.504243	−	0.863562i	\(-0.331771\pi\)
							−0.615589	+	0.788067i	\(0.711082\pi\)
\(29\)	5.81698	−	14.5995i	0.200585	−	0.503431i	−0.793932	−	0.608007i	\(-0.791969\pi\)
							0.994517	+	0.104576i	\(0.0333485\pi\)
\(31\)	45.1025	+	12.5226i	1.45492	+	0.403956i	0.902891	−	0.429870i	\(-0.141441\pi\)
							0.552028	+	0.833826i	\(0.313854\pi\)
\(37\)	28.6902	+	62.0128i	0.775410	+	1.67602i	0.736175	+	0.676792i	\(0.236630\pi\)
							0.0392356	+	0.999230i	\(0.487508\pi\)
\(41\)	−4.63693	−	6.83896i	−0.113096	−	0.166804i	0.766928	−	0.641733i	\(-0.221784\pi\)
							−0.880024	+	0.474929i	\(0.842474\pi\)
\(43\)	16.3229	+	0.885000i	0.379602	+	0.0205814i	0.242953	−	0.970038i	\(-0.421884\pi\)
							0.136649	+	0.990620i	\(0.456367\pi\)
\(47\)	−6.71953	−	5.70762i	−0.142969	−	0.121439i	0.573048	−	0.819522i	\(-0.305761\pi\)
							−0.716017	+	0.698083i	\(0.754037\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.g.a.124.7	yes	560
59.10	odd	58	inner	177.3.g.a.10.7	✓	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.g.a.10.7	✓	560	59.10	odd	58	inner
177.3.g.a.124.7	yes	560	1.1	even	1	trivial