Embedded newform 370.2.v.a.169.7

• Label: 370.2.v.a.169.7
• Level: $370$
• Weight: $2$
• Character: 370.169
• Analytic conductor: $2.954$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.95446487479\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(10\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			169.7
Character	\(\chi\)	\(=\)	370.169
Dual form			370.2.v.a.289.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.939693 - 0.342020i) q^{2} +(0.400420 + 1.10014i) q^{3} +(0.766044 + 0.642788i) q^{4} +(-0.822020 + 2.07949i) q^{5} -1.17075i q^{6} +(3.81475 + 0.672643i) q^{7} +(-0.500000 - 0.866025i) q^{8} +(1.24815 - 1.04732i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 3 q^{5} - 30 q^{8} - 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(261\)	\(297\)
\(\chi(n)\)	\(e\left(\frac{11}{18}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.939693	−	0.342020i	−0.664463	−	0.241845i
							\(3\)	0.400420	+	1.10014i	0.231182	+	0.635169i	0.999991	−	0.00431076i	\(-0.00137216\pi\)
−0.768808	+	0.639479i	\(0.779150\pi\)
\(5\)	−0.822020	+	2.07949i	−0.367618	+	0.929977i
							\(7\)	3.81475	+	0.672643i	1.44184	+	0.254235i	0.839219	−	0.543794i	\(-0.183013\pi\)
0.602620	+	0.798029i	\(0.294124\pi\)
\(11\)	−1.57993	−	2.73651i	−0.476366	−	0.825090i	0.523267	−	0.852169i	\(-0.324713\pi\)
							−0.999633	+	0.0270785i	\(0.991380\pi\)
\(13\)	1.40293	+	1.17720i	0.389102	+	0.326496i	0.816263	−	0.577680i	\(-0.196042\pi\)
							−0.427161	+	0.904176i	\(0.640486\pi\)
\(17\)	−4.46534	+	3.74687i	−1.08300	+	0.908748i	−0.996167	−	0.0874741i	\(-0.972120\pi\)
							−0.0868373	+	0.996223i	\(0.527676\pi\)
\(19\)	2.67996	+	7.36313i	0.614825	+	1.68922i	0.719309	+	0.694690i	\(0.244459\pi\)
							−0.104484	+	0.994527i	\(0.533319\pi\)
\(23\)	−0.0262094	+	0.0453960i	−0.00546504	+	0.00946572i	−0.868745	−	0.495260i	\(-0.835073\pi\)
							0.863280	+	0.504725i	\(0.168406\pi\)
\(29\)	3.42104	−	1.97514i	0.635270	−	0.366774i	−0.147520	−	0.989059i	\(-0.547129\pi\)
							0.782790	+	0.622286i	\(0.213796\pi\)
\(31\)		−	1.88120i		−	0.337874i	−0.985627	−	0.168937i	\(-0.945967\pi\)
							0.985627	−	0.168937i	\(-0.0540334\pi\)
\(37\)	−5.80760	+	1.80882i	−0.954763	+	0.297369i
							\(41\)	2.75128	+	2.30860i	0.429678	+	0.360543i	0.831830	−	0.555030i	\(-0.187293\pi\)
−0.402152	+	0.915573i	\(0.631738\pi\)
\(43\)	−2.80385			−0.427583			−0.213792	−	0.976879i	\(-0.568581\pi\)
							−0.213792	+	0.976879i	\(0.568581\pi\)
\(47\)	−9.42768	−	5.44307i	−1.37517	−	0.793954i	−0.383595	−	0.923502i	\(-0.625314\pi\)
							−0.991573	+	0.129548i	\(0.958647\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	370.2.v.a.169.7	✓	60
5.4	even	2		370.2.v.b.169.4	yes	60
37.30	even	18		370.2.v.b.289.4	yes	60
185.104	even	18	inner	370.2.v.a.289.7	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.v.a.169.7	✓	60	1.1	even	1	trivial
370.2.v.a.289.7	yes	60	185.104	even	18	inner
370.2.v.b.169.4	yes	60	5.4	even	2
370.2.v.b.289.4	yes	60	37.30	even	18