Embedded newform 177.2.f.a.2.3

• Label: 177.2.f.a.2.3
• 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			2.3
Character	\(\chi\)	\(=\)	177.2
Dual form			177.2.f.a.89.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.127399 + 2.34974i) q^{2} +(0.849015 - 1.50969i) q^{3} +(-3.51679 - 0.382473i) q^{4} +(-0.844245 + 2.50563i) q^{5} +(3.43923 + 2.18730i) q^{6} +(1.82720 + 2.69492i) q^{7} +(0.585342 - 3.57043i) q^{8} +(-1.55835 - 2.56350i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

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	\(0.330034\pi\)
							−0.599035	+	0.800723i	\(0.704449\pi\)
\(3\)	0.849015	−	1.50969i	0.490179	−	0.871622i
							\(5\)	−0.844245	+	2.50563i	−0.377558	+	1.12055i	0.575510	+	0.817795i	\(0.304804\pi\)
−0.953068	+	0.302757i	\(0.902093\pi\)
\(7\)	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	0.349685	−	0.936867i	\(-0.386289\pi\)
							0.710746	+	0.703449i	\(0.248358\pi\)
\(13\)	−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	0.560323	−	0.828274i	\(-0.310677\pi\)
							−0.562051	+	0.827103i	\(0.689987\pi\)
\(19\)	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	−0.279717	−	0.960082i	\(-0.590241\pi\)
							0.943531	+	0.331285i	\(0.107482\pi\)
\(29\)	2.26562	−	0.122838i	0.420714	−	0.0228105i	0.157433	−	0.987530i	\(-0.449678\pi\)
							0.263281	+	0.964719i	\(0.415195\pi\)
\(31\)	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	−0.312774	−	0.949828i	\(-0.601258\pi\)
							0.965368	+	0.260893i	\(0.0840169\pi\)
\(43\)	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	−0.950587	−	0.310460i	\(-0.899517\pi\)
							−0.568873	+	0.822425i	\(0.692620\pi\)

(See \(a_n\) instead)

Twists

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

    

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