Embedded newform 177.2.f.a.47.7

• Label: 177.2.f.a.47.7
• 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			47.7
Character	\(\chi\)	\(=\)	177.47
Dual form			177.2.f.a.113.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.875508 + 0.294993i) q^{2} +(0.0745986 + 1.73044i) q^{3} +(-0.912693 + 0.693811i) q^{4} +(3.94962 - 1.57367i) q^{5} +(-0.575780 - 1.49301i) q^{6} +(1.47617 + 0.682948i) q^{7} +(1.63133 - 2.40603i) q^{8} +(-2.98887 + 0.258177i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.875508	+	0.294993i	−0.619077	+	0.208591i	−0.611326	−	0.791379i	\(-0.709364\pi\)
							−0.00775090	+	0.999970i	\(0.502467\pi\)
\(3\)	0.0745986	+	1.73044i	0.0430695	+	0.999072i
							\(5\)	3.94962	−	1.57367i	1.76632	−	0.703767i	0.768356	−	0.640023i	\(-0.221075\pi\)
0.997966	−	0.0637441i	\(-0.0203041\pi\)
\(7\)	1.47617	+	0.682948i	0.557939	+	0.258130i	0.678531	−	0.734572i	\(-0.262617\pi\)
							−0.120592	+	0.992702i	\(0.538479\pi\)
\(11\)	1.09993	+	3.96160i	0.331643	+	1.19447i	0.921956	+	0.387295i	\(0.126590\pi\)
							−0.590313	+	0.807174i	\(0.700996\pi\)
\(13\)	−3.51411	+	1.86306i	−0.974638	+	0.516721i	−0.878032	−	0.478602i	\(-0.841144\pi\)
							−0.0966065	+	0.995323i	\(0.530799\pi\)
\(17\)	0.246036	+	0.531798i	0.0596725	+	0.128980i	0.935118	−	0.354338i	\(-0.115294\pi\)
							−0.875445	+	0.483318i	\(0.839432\pi\)
\(19\)	−2.31422	−	0.509399i	−0.530919	−	0.116864i	−0.0585826	−	0.998283i	\(-0.518658\pi\)
							−0.472336	+	0.881419i	\(0.656589\pi\)
\(23\)	−0.311443	−	1.89972i	−0.0649403	−	0.396118i	−0.999217	−	0.0395690i	\(-0.987402\pi\)
							0.934277	−	0.356549i	\(-0.116047\pi\)
\(29\)	2.65852	−	7.89021i	0.493675	−	1.46518i	−0.354296	−	0.935133i	\(-0.615280\pi\)
							0.847971	−	0.530042i	\(-0.177824\pi\)
\(31\)	0.971279	+	4.41256i	0.174447	+	0.792520i	0.979841	+	0.199780i	\(0.0640228\pi\)
							−0.805394	+	0.592740i	\(0.798046\pi\)
\(37\)	−4.61215	+	3.12711i	−0.758232	+	0.514094i	−0.877973	−	0.478711i	\(-0.841104\pi\)
							0.119740	+	0.992805i	\(0.461794\pi\)
\(41\)	−0.153991	−	0.0252455i	−0.0240493	−	0.00394269i	0.149745	−	0.988725i	\(-0.452155\pi\)
							−0.173795	+	0.984782i	\(0.555603\pi\)
\(43\)	5.87467	+	1.63110i	0.895879	+	0.248740i	0.684802	−	0.728730i	\(-0.259889\pi\)
							0.211078	+	0.977469i	\(0.432303\pi\)
\(47\)	1.64281	−	4.12314i	0.239628	−	0.601422i	−0.759146	−	0.650921i	\(-0.774383\pi\)
							0.998774	+	0.0494989i	\(0.0157624\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.47.7	✓	504
3.2	odd	2	inner	177.2.f.a.47.12	yes	504
59.54	odd	58	inner	177.2.f.a.113.12	yes	504
177.113	even	58	inner	177.2.f.a.113.7	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.47.7	✓	504	1.1	even	1	trivial
177.2.f.a.47.12	yes	504	3.2	odd	2	inner
177.2.f.a.113.7	yes	504	177.113	even	58	inner
177.2.f.a.113.12	yes	504	59.54	odd	58	inner