Embedded newform 177.2.f.a.47.14

• Label: 177.2.f.a.47.14
• Level: $177$
• Weight: $2$
• Character: 177.47
• Analytic conductor: $1.413$
• Analytic rank: $0$
• Dimension: $504$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.f (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.41335211578\)
Analytic rank:	\(0\)
Dimension:	\(504\)
Relative dimension:	\(18\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			47.14
Character	\(\chi\)	\(=\)	177.47
Dual form			177.2.f.a.113.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.21980 - 0.410997i) q^{2} +(0.669029 + 1.59762i) q^{3} +(-0.273203 + 0.207683i) q^{4} +(0.772914 - 0.307957i) q^{5} +(1.47270 + 1.67381i) q^{6} +(1.10327 + 0.510426i) q^{7} +(-1.69259 + 2.49638i) q^{8} +(-2.10480 + 2.13771i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{23}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.21980	−	0.410997i	0.862526	−	0.290619i	0.146942	−	0.989145i	\(-0.453057\pi\)
							0.715584	+	0.698526i	\(0.246161\pi\)
\(3\)	0.669029	+	1.59762i	0.386264	+	0.922388i
							\(5\)	0.772914	−	0.307957i	0.345657	−	0.137723i	−0.190847	−	0.981620i	\(-0.561124\pi\)
0.536505	+	0.843897i	\(0.319744\pi\)
\(7\)	1.10327	+	0.510426i	0.416996	+	0.192923i	0.617157	−	0.786840i	\(-0.288284\pi\)
							−0.200161	+	0.979763i	\(0.564146\pi\)
\(11\)	−1.30162	−	4.68803i	−0.392455	−	1.41349i	−0.851160	−	0.524906i	\(-0.824101\pi\)
							0.458706	−	0.888588i	\(-0.348313\pi\)
\(13\)	3.21638	−	1.70522i	0.892064	−	0.472943i	0.0416960	−	0.999130i	\(-0.486724\pi\)
							0.850368	+	0.526188i	\(0.176379\pi\)
\(17\)	−2.22488	−	4.80900i	−0.539613	−	1.16635i	−0.965218	−	0.261445i	\(-0.915801\pi\)
							0.425606	−	0.904909i	\(-0.360061\pi\)
\(19\)	3.68284	+	0.810655i	0.844902	+	0.185977i	0.616269	−	0.787536i	\(-0.288643\pi\)
							0.228633	+	0.973513i	\(0.426574\pi\)
\(23\)	0.277760	+	1.69426i	0.0579169	+	0.353277i	0.999847	+	0.0174696i	\(0.00556102\pi\)
							−0.941930	+	0.335808i	\(0.890991\pi\)
\(29\)	0.917428	−	2.72283i	0.170362	−	0.505617i	−0.828129	−	0.560538i	\(-0.810594\pi\)
							0.998491	+	0.0549214i	\(0.0174908\pi\)
\(31\)	−1.15203	−	5.23370i	−0.206910	−	0.940001i	−0.959543	−	0.281561i	\(-0.909148\pi\)
							0.752634	−	0.658440i	\(-0.228783\pi\)
\(37\)	−6.71605	+	4.55360i	−1.10411	+	0.748607i	−0.970180	−	0.242384i	\(-0.922071\pi\)
							−0.133932	+	0.990991i	\(0.542760\pi\)
\(41\)	−2.78877	−	0.457196i	−0.435533	−	0.0714020i	−0.0599727	−	0.998200i	\(-0.519101\pi\)
							−0.375560	+	0.926798i	\(0.622550\pi\)
\(43\)	6.12814	+	1.70147i	0.934532	+	0.259472i	0.701232	−	0.712933i	\(-0.252634\pi\)
							0.233300	+	0.972405i	\(0.425048\pi\)
\(47\)	−3.65483	+	9.17294i	−0.533112	+	1.33801i	0.377982	+	0.925813i	\(0.376618\pi\)
							−0.911094	+	0.412198i	\(0.864761\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.47.14	yes	504
3.2	odd	2	inner	177.2.f.a.47.5	✓	504
59.54	odd	58	inner	177.2.f.a.113.5	yes	504
177.113	even	58	inner	177.2.f.a.113.14	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.47.5	✓	504	3.2	odd	2	inner
177.2.f.a.47.14	yes	504	1.1	even	1	trivial
177.2.f.a.113.5	yes	504	59.54	odd	58	inner
177.2.f.a.113.14	yes	504	177.113	even	58	inner