Embedded newform 370.2.x.a.49.16

• Label: 370.2.x.a.49.16
• Level: $370$
• Weight: $2$
• Character: 370.49
• Analytic conductor: $2.954$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 370 = 2 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	370.x (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.95446487479\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(18\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			49.16
Character	\(\chi\)	\(=\)	370.49
Dual form			370.2.x.a.219.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.642788 + 0.766044i) q^{2} +(0.958481 - 1.14227i) q^{3} +(-0.173648 + 0.984808i) q^{4} +(-0.774702 - 2.09758i) q^{5} +1.49113 q^{6} +(-1.14359 - 3.14198i) q^{7} +(-0.866025 + 0.500000i) q^{8} +(0.134843 + 0.764731i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/370\mathbb{Z}\right)^\times\).

\(n\)	\(261\)	\(297\)
\(\chi(n)\)	\(e\left(\frac{7}{9}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.642788	+	0.766044i	0.454519	+	0.541675i
							\(3\)	0.958481	−	1.14227i	0.553379	−	0.659491i	−0.414752	−	0.909934i	\(-0.636132\pi\)
0.968131	+	0.250443i	\(0.0805762\pi\)
\(5\)	−0.774702	−	2.09758i	−0.346457	−	0.938066i
							\(7\)	−1.14359	−	3.14198i	−0.432235	−	1.18756i	−0.944438	−	0.328691i	\(-0.893393\pi\)
0.512203	−	0.858865i	\(-0.328830\pi\)
\(11\)	−2.82853	−	4.89915i	−0.852833	−	1.47715i	−0.878641	−	0.477483i	\(-0.841549\pi\)
							0.0258078	−	0.999667i	\(-0.491784\pi\)
\(13\)	7.01795	+	1.23745i	1.94643	+	0.343208i	0.999781	+	0.0209434i	\(0.00666697\pi\)
							0.946650	+	0.322265i	\(0.104444\pi\)
\(17\)	0.135980	−	0.0239769i	0.0329799	−	0.00581525i	−0.157134	−	0.987577i	\(-0.550225\pi\)
							0.190114	+	0.981762i	\(0.439114\pi\)
\(19\)	2.40473	+	2.01781i	0.551682	+	0.462916i	0.875510	−	0.483200i	\(-0.160525\pi\)
							−0.323828	+	0.946116i	\(0.604970\pi\)
\(23\)	−4.10254	−	2.36860i	−0.855438	−	0.493888i	0.00704366	−	0.999975i	\(-0.497758\pi\)
							−0.862482	+	0.506088i	\(0.831091\pi\)
\(29\)	3.66896	+	6.35483i	0.681309	+	1.18006i	0.974581	+	0.224033i	\(0.0719225\pi\)
							−0.293272	+	0.956029i	\(0.594744\pi\)
\(31\)	−2.49268			−0.447699			−0.223849	−	0.974624i	\(-0.571862\pi\)
							−0.223849	+	0.974624i	\(0.571862\pi\)
\(37\)	3.63280	−	4.87881i	0.597229	−	0.802071i
							\(41\)	0.125638	−	0.712527i	0.0196213	−	0.111278i	−0.973424	−	0.229010i	\(-0.926451\pi\)
0.993045	+	0.117732i	\(0.0375623\pi\)
\(43\)			11.3402i			1.72937i	0.502313	+	0.864686i	\(0.332483\pi\)
							−0.502313	+	0.864686i	\(0.667517\pi\)
\(47\)	2.72976	+	1.57603i	0.398176	+	0.229887i	0.685697	−	0.727887i	\(-0.259498\pi\)
							−0.287520	+	0.957774i	\(0.592831\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	370.2.x.a.49.16	yes	108
5.4	even	2	inner	370.2.x.a.49.3	✓	108
37.34	even	9	inner	370.2.x.a.219.3	yes	108
185.34	even	18	inner	370.2.x.a.219.16	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.x.a.49.3	✓	108	5.4	even	2	inner
370.2.x.a.49.16	yes	108	1.1	even	1	trivial
370.2.x.a.219.3	yes	108	37.34	even	9	inner
370.2.x.a.219.16	yes	108	185.34	even	18	inner