Embedded newform 177.2.f.a.2.16

• Label: 177.2.f.a.2.16
• Level: $177$
• Weight: $2$
• Character: 177.2
• 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			2.16
Character	\(\chi\)	\(=\)	177.2
Dual form			177.2.f.a.89.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.127399 - 2.34974i) q^{2} +(1.70027 + 0.330268i) q^{3} +(-3.51679 - 0.382473i) q^{4} +(0.844245 - 2.50563i) q^{5} +(0.992658 - 3.95313i) q^{6} +(1.82720 + 2.69492i) q^{7} +(-0.585342 + 3.57043i) q^{8} +(2.78185 + 1.12309i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.127399	−	2.34974i	0.0900850	−	1.66152i	−0.508950	−	0.860796i	\(-0.669966\pi\)
							0.599035	−	0.800723i	\(-0.295551\pi\)
\(3\)	1.70027	+	0.330268i	0.981652	+	0.190680i
							\(5\)	0.844245	−	2.50563i	0.377558	−	1.12055i	−0.575510	−	0.817795i	\(-0.695196\pi\)
0.953068	−	0.302757i	\(-0.0979070\pi\)
\(7\)	1.82720	+	2.69492i	0.690616	+	1.01858i	0.997758	+	0.0669187i	\(0.0213168\pi\)
							−0.307143	+	0.951664i	\(0.599373\pi\)
\(11\)	−3.51705	+	0.774162i	−1.06043	+	0.233419i	−0.710746	−	0.703449i	\(-0.751642\pi\)
							−0.349685	+	0.936867i	\(0.613711\pi\)
\(13\)	−4.57715	+	3.88787i	−1.26947	+	1.07830i	−0.276372	+	0.961051i	\(0.589132\pi\)
							−0.993102	+	0.117250i	\(0.962592\pi\)
\(17\)	0.00712519	+	0.00483100i	0.00172811	+	0.00117169i	0.562051	−	0.827103i	\(-0.310013\pi\)
							−0.560323	+	0.828274i	\(0.689323\pi\)
\(19\)	2.29433	−	4.32756i	0.526355	−	0.992811i	−0.467695	−	0.883890i	\(-0.654916\pi\)
							0.994050	−	0.108921i	\(-0.0347397\pi\)
\(23\)	−3.18354	+	3.01561i	−0.663813	+	0.628797i	−0.943531	−	0.331285i	\(-0.892518\pi\)
							0.279717	+	0.960082i	\(0.409759\pi\)
\(29\)	−2.26562	+	0.122838i	−0.420714	+	0.0228105i	−0.263281	−	0.964719i	\(-0.584805\pi\)
							−0.157433	+	0.987530i	\(0.550322\pi\)
\(31\)	7.25469	−	3.84619i	1.30298	−	0.690797i	0.335094	−	0.942185i	\(-0.391232\pi\)
							0.967886	+	0.251388i	\(0.0808870\pi\)
\(37\)	1.89174	−	0.310135i	0.311000	−	0.0509858i	−0.00426095	−	0.999991i	\(-0.501356\pi\)
							0.315261	+	0.949005i	\(0.397908\pi\)
\(41\)	−4.17864	+	4.41134i	−0.652594	+	0.688935i	−0.965368	−	0.260893i	\(-0.915983\pi\)
							0.312774	+	0.949828i	\(0.398742\pi\)
\(43\)	0.448455	−	2.03735i	0.0683888	−	0.310693i	−0.929992	−	0.367580i	\(-0.880186\pi\)
							0.998381	+	0.0568868i	\(0.0181174\pi\)
\(47\)	10.4169	−	3.50986i	1.51946	−	0.511966i	0.568873	−	0.822425i	\(-0.307380\pi\)
							0.950587	+	0.310460i	\(0.100483\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.2.16	yes	504
3.2	odd	2	inner	177.2.f.a.2.3	✓	504
59.30	odd	58	inner	177.2.f.a.89.3	yes	504
177.89	even	58	inner	177.2.f.a.89.16	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.2.3	✓	504	3.2	odd	2	inner
177.2.f.a.2.16	yes	504	1.1	even	1	trivial
177.2.f.a.89.3	yes	504	59.30	odd	58	inner
177.2.f.a.89.16	yes	504	177.89	even	58	inner