Embedded newform 555.2.bp.a.4.19

• Label: 555.2.bp.a.4.19
• Level: $555$
• Weight: $2$
• Character: 555.4
• Analytic conductor: $4.432$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 555 = 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	555.bp (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.43169731218\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(40\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			4.19
Character	\(\chi\)	\(=\)	555.4
Dual form			555.2.bp.a.139.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0600890 + 0.340781i) q^{2} +(-0.984808 + 0.173648i) q^{3} +(1.76686 + 0.643086i) q^{4} +(1.86567 + 1.23259i) q^{5} -0.346039i q^{6} +(-1.02117 - 1.21698i) q^{7} +(-0.671359 + 1.16283i) q^{8} +(0.939693 - 0.342020i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 12 q^{5}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/555\mathbb{Z}\right)^\times\).

\(n\)	\(76\)	\(112\)	\(371\)
\(\chi(n)\)	\(e\left(\frac{1}{18}\right)\)	\(-1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0600890	+	0.340781i	−0.0424893	+	0.240969i	−0.998654	−	0.0518593i	\(-0.983485\pi\)
							0.956165	+	0.292828i	\(0.0945964\pi\)
\(3\)	−0.984808	+	0.173648i	−0.568579	+	0.100256i
							\(5\)	1.86567	+	1.23259i	0.834352	+	0.551232i
\(7\)	−1.02117	−	1.21698i	−0.385966	−	0.459977i								0.537721	−	0.843123i	\(-0.319285\pi\)
							−0.923688	+	0.383146i	\(0.874841\pi\)
\(11\)	2.55571	−	4.42661i	0.770575	−	1.33467i	−0.166674	−	0.986012i	\(-0.553303\pi\)
							0.937248	−	0.348662i	\(-0.113364\pi\)
\(13\)	5.80142	+	2.11154i	1.60902	+	0.585636i	0.981246	−	0.192757i	\(-0.0617429\pi\)
							0.627777	+	0.778394i	\(0.283965\pi\)
\(17\)	−0.632188	+	0.230097i	−0.153328	+	0.0558068i	−0.417544	−	0.908657i	\(-0.637109\pi\)
							0.264216	+	0.964464i	\(0.414887\pi\)
\(19\)	−5.40111	+	0.952362i	−1.23910	+	0.218487i	−0.754530	−	0.656266i	\(-0.772135\pi\)
							−0.484570	+	0.874752i	\(0.661024\pi\)
\(23\)	−2.72743	−	4.72405i	−0.568708	−	0.985032i	−0.996694	−	0.0812464i	\(-0.974110\pi\)
							0.427986	−	0.903786i	\(-0.359223\pi\)
\(29\)	4.01789	+	2.31973i	0.746104	+	0.430763i	0.824284	−	0.566176i	\(-0.191578\pi\)
							−0.0781807	+	0.996939i	\(0.524911\pi\)
\(31\)			1.46615i			0.263329i	0.991294	+	0.131664i	\(0.0420321\pi\)
							−0.991294	+	0.131664i	\(0.957968\pi\)
\(37\)	2.22310	+	5.66196i	0.365476	+	0.930821i
							\(41\)	−6.25778	−	2.27765i	−0.977301	−	0.355708i	−0.196511	−	0.980502i	\(-0.562961\pi\)
−0.780790	+	0.624793i	\(0.785183\pi\)
\(43\)	4.83128			0.736763			0.368381	−	0.929675i	\(-0.379912\pi\)
							0.368381	+	0.929675i	\(0.379912\pi\)
\(47\)	1.52489	−	0.880393i	0.222427	−	0.128419i	−0.384646	−	0.923064i	\(-0.625677\pi\)
							0.607074	+	0.794645i	\(0.292343\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	555.2.bp.a.4.19	✓	240
5.4	even	2	inner	555.2.bp.a.4.22	yes	240
37.28	even	18	inner	555.2.bp.a.139.22	yes	240
185.139	even	18	inner	555.2.bp.a.139.19	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
555.2.bp.a.4.19	✓	240	1.1	even	1	trivial
555.2.bp.a.4.22	yes	240	5.4	even	2	inner
555.2.bp.a.139.19	yes	240	185.139	even	18	inner
555.2.bp.a.139.22	yes	240	37.28	even	18	inner