Embedded newform 177.4.f.a.2.14

• Label: 177.4.f.a.2.14
• Level: $177$
• Weight: $4$
• Character: 177.2
• 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			2.14
Character	\(\chi\)	\(=\)	177.2
Dual form			177.4.f.a.89.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.191601 + 3.53386i) q^{2} +(-0.887175 + 5.11986i) q^{3} +(-4.49838 - 0.489228i) q^{4} +(2.01755 - 5.98787i) q^{5} +(-17.9229 - 4.11612i) q^{6} +(16.3407 + 24.1007i) q^{7} +(-1.98968 + 12.1365i) q^{8} +(-25.4258 - 9.08442i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 21 q^{3} - 278 q^{4} - 29 q^{6} - 42 q^{7} - 25 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.191601	+	3.53386i	−0.0677410	+	1.24941i	0.745111	+	0.666940i	\(0.232396\pi\)
							−0.812852	+	0.582470i	\(0.802087\pi\)
\(3\)	−0.887175	+	5.11986i	−0.170737	+	0.985317i
							\(5\)	2.01755	−	5.98787i	0.180455	−	0.535571i	−0.818803	−	0.574074i	\(-0.805362\pi\)
0.999259	+	0.0385025i	\(0.0122587\pi\)
\(7\)	16.3407	+	24.1007i	0.882315	+	1.30132i	0.952736	+	0.303800i	\(0.0982556\pi\)
							−0.0704204	+	0.997517i	\(0.522434\pi\)
\(11\)	−40.8859	+	8.99967i	−1.12069	+	0.246682i	−0.736410	−	0.676535i	\(-0.763481\pi\)
							−0.384278	+	0.923217i	\(0.625550\pi\)
\(13\)	8.51733	−	7.23469i	0.181714	−	0.154349i	−0.551913	−	0.833902i	\(-0.686102\pi\)
							0.733627	+	0.679552i	\(0.237826\pi\)
\(17\)	−23.1396	−	15.6890i	−0.330128	−	0.223832i	0.384850	−	0.922979i	\(-0.374253\pi\)
							−0.714978	+	0.699147i	\(0.753563\pi\)
\(19\)	−26.4317	+	49.8555i	−0.319150	+	0.601980i	−0.990236	−	0.139399i	\(-0.955483\pi\)
							0.671087	+	0.741379i	\(0.265828\pi\)
\(23\)	54.6255	−	51.7441i	0.495227	−	0.469103i	−0.398851	−	0.917016i	\(-0.630591\pi\)
							0.894078	+	0.447912i	\(0.147832\pi\)
\(29\)	−29.1882	+	1.58254i	−0.186900	+	0.0101334i	−0.147351	−	0.989084i	\(-0.547075\pi\)
							−0.0395488	+	0.999218i	\(0.512592\pi\)
\(31\)	152.028	−	80.5999i	0.880805	−	0.466973i	0.0343357	−	0.999410i	\(-0.489068\pi\)
							0.846470	+	0.532437i	\(0.178724\pi\)
\(37\)	139.695	−	22.9018i	0.620694	−	0.101758i	0.156774	−	0.987635i	\(-0.449891\pi\)
							0.463920	+	0.885877i	\(0.346442\pi\)
\(41\)	−214.851	+	226.815i	−0.818391	+	0.863965i	−0.992686	−	0.120729i	\(-0.961477\pi\)
							0.174294	+	0.984694i	\(0.444236\pi\)
\(43\)	6.71486	−	30.5059i	0.0238141	−	0.108189i	−0.963214	−	0.268735i	\(-0.913395\pi\)
							0.987028	+	0.160546i	\(0.0513255\pi\)
\(47\)	−242.442	+	81.6882i	−0.752421	+	0.253520i	−0.669272	−	0.743017i	\(-0.733394\pi\)
							−0.0831483	+	0.996537i	\(0.526498\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.f.a.2.14	✓	1624
3.2	odd	2	inner	177.4.f.a.2.45	yes	1624
59.30	odd	58	inner	177.4.f.a.89.45	yes	1624
177.89	even	58	inner	177.4.f.a.89.14	yes	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.f.a.2.14	✓	1624	1.1	even	1	trivial
177.4.f.a.2.45	yes	1624	3.2	odd	2	inner
177.4.f.a.89.14	yes	1624	177.89	even	58	inner
177.4.f.a.89.45	yes	1624	59.30	odd	58	inner