Embedded newform 370.2.bd.a.187.7

• Label: 370.2.bd.a.187.7
• Level: $370$
• Weight: $2$
• Character: 370.187
• Analytic conductor: $2.954$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 370 = 2 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	370.bd (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.95446487479\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(9\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			187.7
Character	\(\chi\)	\(=\)	370.187
Dual form			370.2.bd.a.93.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.642788 + 0.766044i) q^{2} +(0.112127 + 1.28162i) q^{3} +(-0.173648 + 0.984808i) q^{4} +(1.49430 - 1.66345i) q^{5} +(-0.909704 + 0.909704i) q^{6} +(1.94794 + 0.908339i) q^{7} +(-0.866025 + 0.500000i) q^{8} +(1.32445 - 0.233535i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q + 6 q^{3}+O(q^{10}) \)

Character values

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

\(n\)	\(261\)	\(297\)
\(\chi(n)\)	\(e\left(\frac{1}{36}\right)\)	\(e\left(\frac{1}{4}\right)\)

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
							\(3\)	0.112127	+	1.28162i	0.0647367	+	0.739944i	0.957314	+	0.289050i	\(0.0933393\pi\)
−0.892577	+	0.450894i	\(0.851105\pi\)
\(5\)	1.49430	−	1.66345i	0.668271	−	0.743918i
							\(7\)	1.94794	+	0.908339i	0.736252	+	0.343320i	0.754311	−	0.656517i	\(-0.227971\pi\)
−0.0180594	+	0.999837i	\(0.505749\pi\)
\(11\)	−1.69236	+	0.977083i	−0.510265	+	0.294601i	−0.732942	−	0.680291i	\(-0.761854\pi\)
							0.222678	+	0.974892i	\(0.428520\pi\)
\(13\)	0.842036	+	0.148474i	0.233539	+	0.0411792i	0.289193	−	0.957271i	\(-0.406613\pi\)
							−0.0556538	+	0.998450i	\(0.517724\pi\)
\(17\)	−0.444995	−	2.52369i	−0.107927	−	0.612085i	−0.990011	−	0.140992i	\(-0.954971\pi\)
							0.882084	−	0.471093i	\(-0.156140\pi\)
\(19\)	0.481756	+	5.50650i	0.110522	+	1.26328i	0.825610	+	0.564242i	\(0.190831\pi\)
							−0.715087	+	0.699035i	\(0.753613\pi\)
\(23\)	−7.35617	−	4.24708i	−1.53387	−	0.885578i	−0.999178	−	0.0405292i	\(-0.987096\pi\)
							−0.534688	−	0.845049i	\(-0.679571\pi\)
\(29\)	1.74937	+	0.468743i	0.324850	+	0.0870434i	0.417559	−	0.908650i	\(-0.362886\pi\)
							−0.0927085	+	0.995693i	\(0.529552\pi\)
\(31\)	0.659479	+	0.659479i	0.118446	+	0.118446i	0.763845	−	0.645399i	\(-0.223309\pi\)
							−0.645399	+	0.763845i	\(0.723309\pi\)
\(37\)	−4.29578	+	4.30653i	−0.706223	+	0.707990i
							\(41\)	−4.13852	−	0.729733i	−0.646328	−	0.113965i	−0.159132	−	0.987257i	\(-0.550869\pi\)
−0.487197	+	0.873292i	\(0.661981\pi\)
\(43\)		−	8.65527i		−	1.31992i	−0.751303	−	0.659958i	\(-0.770574\pi\)
							0.751303	−	0.659958i	\(-0.229426\pi\)
\(47\)	−3.61275	+	0.968032i	−0.526973	+	0.141202i	−0.512490	−	0.858693i	\(-0.671277\pi\)
							−0.0144830	+	0.999895i	\(0.504610\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	370.2.bd.a.187.7	yes	108
5.3	odd	4		370.2.ba.a.113.3	✓	108
37.19	odd	36		370.2.ba.a.167.3	yes	108
185.93	even	36	inner	370.2.bd.a.93.7	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.ba.a.113.3	✓	108	5.3	odd	4
370.2.ba.a.167.3	yes	108	37.19	odd	36
370.2.bd.a.93.7	yes	108	185.93	even	36	inner
370.2.bd.a.187.7	yes	108	1.1	even	1	trivial