Embedded newform 177.2.f.a.113.14

• Label: 177.2.f.a.113.14
• Level: $177$
• Weight: $2$
• Character: 177.113
• 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			113.14
Character	\(\chi\)	\(=\)	177.113
Dual form			177.2.f.a.47.14

$q$-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 \(\left(\mathbb{Z}/177\mathbb{Z}\right)^\times\).

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{35}{58}\right)\)	\(-1\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.47.5	✓	504	59.47	odd	58	inner
177.2.f.a.47.14	yes	504	177.47	even	58	inner
177.2.f.a.113.5	yes	504	3.2	odd	2	inner
177.2.f.a.113.14	yes	504	1.1	even	1	trivial