Embedded newform 370.2.x.a.49.3

• Label: 370.2.x.a.49.3
• 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			49.3
Character	\(\chi\)	\(=\)	370.49
Dual form			370.2.x.a.219.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.642788 - 0.766044i) q^{2} +(-0.958481 + 1.14227i) q^{3} +(-0.173648 + 0.984808i) q^{4} +(0.754841 + 2.10481i) q^{5} +1.49113 q^{6} +(1.14359 + 3.14198i) q^{7} +(0.866025 - 0.500000i) q^{8} +(0.134843 + 0.764731i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 12 q^{9}+O(q^{10}) \)

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	−0.968131	−	0.250443i	\(-0.919424\pi\)
0.414752	+	0.909934i	\(0.363868\pi\)
\(5\)	0.754841	+	2.10481i	0.337575	+	0.941298i
							\(7\)	1.14359	+	3.14198i	0.432235	+	1.18756i	0.944438	+	0.328691i	\(0.106607\pi\)
−0.512203	+	0.858865i	\(0.671170\pi\)
\(11\)	−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	\(-0.993333\pi\)
							−0.946650	−	0.322265i	\(-0.895556\pi\)
\(17\)	−0.135980	+	0.0239769i	−0.0329799	+	0.00581525i	−0.190114	−	0.981762i	\(-0.560886\pi\)
							0.157134	+	0.987577i	\(0.449775\pi\)
\(19\)	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	0.862482	−	0.506088i	\(-0.168909\pi\)
							−0.00704366	+	0.999975i	\(0.502242\pi\)
\(29\)	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	\(-0.667517\pi\)
							0.502313	−	0.864686i	\(-0.332483\pi\)
\(47\)	−2.72976	−	1.57603i	−0.398176	−	0.229887i	0.287520	−	0.957774i	\(-0.407169\pi\)
							−0.685697	+	0.727887i	\(0.740502\pi\)

(See \(a_n\) instead)

Twists

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

    

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