Embedded newform 370.2.bd.a.237.9

• Label: 370.2.bd.a.237.9
• Level: $370$
• Weight: $2$
• Character: 370.237
• 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			237.9
Character	\(\chi\)	\(=\)	370.237
Dual form			370.2.bd.a.153.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.342020 - 0.939693i) q^{2} +(2.88137 + 1.34360i) q^{3} +(-0.766044 - 0.642788i) q^{4} +(-1.84550 + 1.26258i) q^{5} +(2.24806 - 2.24806i) q^{6} +(2.44676 - 1.71324i) q^{7} +(-0.866025 + 0.500000i) q^{8} +(4.56865 + 5.44471i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q + 6 q^{3}+O(q^{10}) \)

Character values

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

\(n\)	\(261\)	\(297\)
\(\chi(n)\)	\(e\left(\frac{13}{36}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.342020	−	0.939693i	0.241845	−	0.664463i
							\(3\)	2.88137	+	1.34360i	1.66356	+	0.775731i	0.999609	+	0.0279538i	\(0.00889914\pi\)
0.663950	+	0.747777i	\(0.268879\pi\)
\(5\)	−1.84550	+	1.26258i	−0.825335	+	0.564644i
							\(7\)	2.44676	−	1.71324i	0.924789	−	0.647544i	−0.0110625	−	0.999939i	\(-0.503521\pi\)
0.935852	+	0.352395i	\(0.114632\pi\)
\(11\)	1.55612	−	0.898424i	0.469186	−	0.270885i	−0.246713	−	0.969089i	\(-0.579350\pi\)
							0.715899	+	0.698204i	\(0.246017\pi\)
\(13\)	−3.34703	+	3.98883i	−0.928299	+	1.10630i	0.0658010	+	0.997833i	\(0.479040\pi\)
							−0.994100	+	0.108470i	\(0.965405\pi\)
\(17\)	4.10923	−	3.44805i	0.996635	−	0.836276i	0.0101201	−	0.999949i	\(-0.496779\pi\)
							0.986515	+	0.163673i	\(0.0523342\pi\)
\(19\)	−5.01176	−	2.33702i	−1.14978	−	0.536150i	−0.248174	−	0.968715i	\(-0.579831\pi\)
							−0.901602	+	0.432566i	\(0.857608\pi\)
\(23\)	−0.0659690	−	0.0380872i	−0.0137555	−	0.00794173i	0.493106	−	0.869969i	\(-0.335861\pi\)
							−0.506862	+	0.862027i	\(0.669195\pi\)
\(29\)	1.25127	+	0.335278i	0.232356	+	0.0622595i	0.373118	−	0.927784i	\(-0.378289\pi\)
							−0.140762	+	0.990043i	\(0.544955\pi\)
\(31\)	−6.24595	−	6.24595i	−1.12181	−	1.12181i	−0.991470	−	0.130335i	\(-0.958395\pi\)
							−0.130335	−	0.991470i	\(-0.541605\pi\)
\(37\)	−3.20280	+	5.17127i	−0.526537	+	0.850152i
							\(41\)	−1.51996	+	1.81142i	−0.237378	+	0.282896i	−0.871561	−	0.490287i	\(-0.836892\pi\)
0.634183	+	0.773183i	\(0.281337\pi\)
\(43\)		−	5.14140i		−	0.784057i	−0.919953	−	0.392028i	\(-0.871773\pi\)
							0.919953	−	0.392028i	\(-0.128227\pi\)
\(47\)	−2.38523	+	0.639120i	−0.347922	+	0.0932253i	−0.428548	−	0.903519i	\(-0.640975\pi\)
							0.0806263	+	0.996744i	\(0.474308\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	370.2.bd.a.237.9	yes	108
5.3	odd	4		370.2.ba.a.163.1	✓	108
37.5	odd	36		370.2.ba.a.227.1	yes	108
185.153	even	36	inner	370.2.bd.a.153.9	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.ba.a.163.1	✓	108	5.3	odd	4
370.2.ba.a.227.1	yes	108	37.5	odd	36
370.2.bd.a.153.9	yes	108	185.153	even	36	inner
370.2.bd.a.237.9	yes	108	1.1	even	1	trivial