Embedded newform 370.2.bd.a.93.7

• Label: 370.2.bd.a.93.7
• Level: $370$
• Weight: $2$
• Character: 370.93
• 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			93.7
Character	\(\chi\)	\(=\)	370.93
Dual form			370.2.bd.a.187.7

$q$-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 \(\left(\mathbb{Z}/370\mathbb{Z}\right)^\times\).

\(n\)	\(261\)	\(297\)
\(\chi(n)\)	\(e\left(\frac{35}{36}\right)\)	\(e\left(\frac{3}{4}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.ba.a.113.3	✓	108	37.2	odd	36
370.2.ba.a.167.3	yes	108	5.2	odd	4
370.2.bd.a.93.7	yes	108	1.1	even	1	trivial
370.2.bd.a.187.7	yes	108	185.2	even	36	inner