Embedded newform 177.4.f.a.2.15

• Label: 177.4.f.a.2.15
• Level: $177$
• Weight: $4$
• Character: 177.2
• Analytic conductor: $10.443$
• Analytic rank: $0$
• Dimension: $1624$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.f (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.4433380710\)
Analytic rank:	\(0\)
Dimension:	\(1624\)
Relative dimension:	\(58\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			2.15
Character	\(\chi\)	\(=\)	177.2
Dual form			177.4.f.a.89.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.181075 + 3.33974i) q^{2} +(-4.07453 - 3.22462i) q^{3} +(-3.16798 - 0.344538i) q^{4} +(5.90079 - 17.5129i) q^{5} +(11.5072 - 13.0240i) q^{6} +(-10.9419 - 16.1380i) q^{7} +(-2.60452 + 15.8869i) q^{8} +(6.20361 + 26.2777i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 21 q^{3} - 278 q^{4} - 29 q^{6} - 42 q^{7} - 25 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.181075	+	3.33974i	−0.0640198	+	1.18078i	0.773292	+	0.634050i	\(0.218609\pi\)
							−0.837311	+	0.546726i	\(0.815874\pi\)
\(3\)	−4.07453	−	3.22462i	−0.784144	−	0.620579i
							\(5\)	5.90079	−	17.5129i	0.527783	−	1.56640i	−0.269626	−	0.962965i	\(-0.586900\pi\)
0.797409	−	0.603439i	\(-0.206203\pi\)
\(7\)	−10.9419	−	16.1380i	−0.590805	−	0.871372i	0.408335	−	0.912832i	\(-0.366109\pi\)
							−0.999140	+	0.0414602i	\(0.986799\pi\)
\(11\)	−44.9372	+	9.89142i	−1.23173	+	0.271125i	−0.782705	−	0.622393i	\(-0.786161\pi\)
							−0.449028	+	0.893518i	\(0.648230\pi\)
\(13\)	−47.2303	+	40.1178i	−1.00764	+	0.855899i	−0.989587	−	0.143938i	\(-0.954023\pi\)
							−0.0180550	+	0.999837i	\(0.505747\pi\)
\(17\)	92.6168	+	62.7957i	1.32134	+	0.895894i	0.998717	−	0.0506319i	\(-0.0161235\pi\)
							0.322628	+	0.946526i	\(0.395434\pi\)
\(19\)	−59.2307	+	111.721i	−0.715182	+	1.34898i	0.213075	+	0.977036i	\(0.431652\pi\)
							−0.928257	+	0.371940i	\(0.878693\pi\)
\(23\)	26.4430	−	25.0482i	0.239728	−	0.227083i	−0.558397	−	0.829574i	\(-0.688584\pi\)
							0.798126	+	0.602491i	\(0.205825\pi\)
\(29\)	47.2340	−	2.56095i	0.302453	−	0.0163985i	0.0977121	−	0.995215i	\(-0.468848\pi\)
							0.204741	+	0.978816i	\(0.434365\pi\)
\(31\)	−86.0183	+	45.6040i	−0.498366	+	0.264217i	−0.698576	−	0.715536i	\(-0.746182\pi\)
							0.200210	+	0.979753i	\(0.435838\pi\)
\(37\)	−8.67226	+	1.42174i	−0.0385327	+	0.00631712i	−0.181018	−	0.983480i	\(-0.557939\pi\)
							0.142485	+	0.989797i	\(0.454491\pi\)
\(41\)	159.215	−	168.082i	0.606470	−	0.640243i	−0.348372	−	0.937356i	\(-0.613265\pi\)
							0.954842	+	0.297114i	\(0.0960240\pi\)
\(43\)	1.92073	−	8.72597i	0.00681183	−	0.0309465i	−0.973092	−	0.230417i	\(-0.925991\pi\)
							0.979904	+	0.199470i	\(0.0639221\pi\)
\(47\)	−516.013	+	173.865i	−1.60145	+	0.539592i	−0.971500	−	0.237041i	\(-0.923822\pi\)
							−0.629953	+	0.776633i	\(0.716926\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.f.a.2.15	✓	1624
3.2	odd	2	inner	177.4.f.a.2.44	yes	1624
59.30	odd	58	inner	177.4.f.a.89.44	yes	1624
177.89	even	58	inner	177.4.f.a.89.15	yes	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.f.a.2.15	✓	1624	1.1	even	1	trivial
177.4.f.a.2.44	yes	1624	3.2	odd	2	inner
177.4.f.a.89.15	yes	1624	177.89	even	58	inner
177.4.f.a.89.44	yes	1624	59.30	odd	58	inner