Embedded newform 370.2.x.a.9.6

• Label: 370.2.x.a.9.6
• Level: $370$
• Weight: $2$
• Character: 370.9
• 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			9.6
Character	\(\chi\)	\(=\)	370.9
Dual form			370.2.x.a.329.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.984808 + 0.173648i) q^{2} +(0.906979 + 0.159925i) q^{3} +(0.939693 - 0.342020i) q^{4} +(-1.36673 - 1.76976i) q^{5} -0.920970 q^{6} +(-1.33907 + 1.59584i) q^{7} +(-0.866025 + 0.500000i) q^{8} +(-2.02204 - 0.735964i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(261\)	\(297\)
\(\chi(n)\)	\(e\left(\frac{4}{9}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.984808	+	0.173648i	−0.696364	+	0.122788i
							\(3\)	0.906979	+	0.159925i	0.523644	+	0.0923327i	0.429221	−	0.903199i	\(-0.358788\pi\)
0.0944234	+	0.995532i	\(0.469899\pi\)
\(5\)	−1.36673	−	1.76976i	−0.611219	−	0.791462i
							\(7\)	−1.33907	+	1.59584i	−0.506120	+	0.603170i	−0.957241	−	0.289293i	\(-0.906580\pi\)
0.451121	+	0.892463i	\(0.351024\pi\)
\(11\)	−2.52305	−	4.37006i	−0.760730	−	1.31762i	−0.942475	−	0.334277i	\(-0.891508\pi\)
							0.181745	−	0.983346i	\(-0.441825\pi\)
\(13\)	−0.845569	−	2.32318i	−0.234519	−	0.644335i	−1.00000		0.000876125i	\(-0.999721\pi\)
							0.765481	−	0.643459i	\(-0.222501\pi\)
\(17\)	−1.32005	+	3.62682i	−0.320160	+	0.879632i	0.670332	+	0.742061i	\(0.266152\pi\)
							−0.990492	+	0.137571i	\(0.956070\pi\)
\(19\)	0.190827	−	1.08224i	0.0437788	−	0.248282i	−0.955063	−	0.296404i	\(-0.904213\pi\)
							0.998841	+	0.0481220i	\(0.0153236\pi\)
\(23\)	−0.664042	−	0.383385i	−0.138462	−	0.0799413i	0.429169	−	0.903224i	\(-0.358807\pi\)
							−0.567631	+	0.823283i	\(0.692140\pi\)
\(29\)	−3.31492	−	5.74160i	−0.615564	−	1.06619i	−0.990285	−	0.139051i	\(-0.955595\pi\)
							0.374721	−	0.927138i	\(-0.377739\pi\)
\(31\)	1.74987			0.314286			0.157143	−	0.987576i	\(-0.449772\pi\)
							0.157143	+	0.987576i	\(0.449772\pi\)
\(37\)	4.42798	+	4.17049i	0.727956	+	0.685624i
							\(41\)	3.96078	−	1.44161i	0.618570	−	0.225141i	−0.0136786	−	0.999906i	\(-0.504354\pi\)
0.632249	+	0.774765i	\(0.282132\pi\)
\(43\)		−	11.4693i		−	1.74905i	−0.484980	−	0.874525i	\(-0.661173\pi\)
							0.484980	−	0.874525i	\(-0.338827\pi\)
\(47\)	−4.95043	−	2.85813i	−0.722094	−	0.416901i	0.0934288	−	0.995626i	\(-0.470217\pi\)
							−0.815523	+	0.578725i	\(0.803551\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	370.2.x.a.9.6	✓	108
5.4	even	2	inner	370.2.x.a.9.13	yes	108
37.33	even	9	inner	370.2.x.a.329.13	yes	108
185.144	even	18	inner	370.2.x.a.329.6	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.x.a.9.6	✓	108	1.1	even	1	trivial
370.2.x.a.9.13	yes	108	5.4	even	2	inner
370.2.x.a.329.6	yes	108	185.144	even	18	inner
370.2.x.a.329.13	yes	108	37.33	even	9	inner