Embedded newform 370.2.ba.a.207.1

• Label: 370.2.ba.a.207.1
• Level: $370$
• Weight: $2$
• Character: 370.207
• 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.ba (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			207.1
Character	\(\chi\)	\(=\)	370.207
Dual form			370.2.ba.a.143.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.939693 - 0.342020i) q^{2} +(-2.80744 - 1.30913i) q^{3} +(0.766044 + 0.642788i) q^{4} +(0.849444 - 2.06844i) q^{5} +(2.19038 + 2.19038i) q^{6} +(2.78165 - 1.94773i) q^{7} +(-0.500000 - 0.866025i) q^{8} +(4.23955 + 5.05250i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 6 q^{3} + 6 q^{5} - 54 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(261\)	\(297\)
\(\chi(n)\)	\(e\left(\frac{31}{36}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.939693	−	0.342020i	−0.664463	−	0.241845i
							\(3\)	−2.80744	−	1.30913i	−1.62088	−	0.755828i	−0.621264	−	0.783601i	\(-0.713381\pi\)
−0.999614	+	0.0277732i	\(0.991158\pi\)
\(5\)	0.849444	−	2.06844i	0.379883	−	0.925035i
							\(7\)	2.78165	−	1.94773i	1.05137	−	0.736175i	0.0857949	−	0.996313i	\(-0.472657\pi\)
0.965571	+	0.260138i	\(0.0837681\pi\)
\(11\)	2.76545	−	1.59663i	0.833814	−	0.481403i	−0.0213427	−	0.999772i	\(-0.506794\pi\)
							0.855157	+	0.518369i	\(0.173461\pi\)
\(13\)	3.73489	+	3.13394i	1.03587	+	0.869199i	0.991538	−	0.129817i	\(-0.0414391\pi\)
							0.0443336	+	0.999017i	\(0.485884\pi\)
\(17\)	−3.57133	−	4.25615i	−0.866176	−	1.03227i	−0.999153	−	0.0411491i	\(-0.986898\pi\)
							0.132977	−	0.991119i	\(-0.457546\pi\)
\(19\)	0.735359	−	1.57698i	0.168703	−	0.361785i	−0.803721	−	0.595007i	\(-0.797149\pi\)
							0.972423	+	0.233223i	\(0.0749270\pi\)
\(23\)	3.62767	−	6.28330i	0.756421	−	1.31016i	−0.188244	−	0.982122i	\(-0.560280\pi\)
							0.944665	−	0.328037i	\(-0.106387\pi\)
\(29\)	−2.49247	+	9.30204i	−0.462841	+	1.72735i	0.201110	+	0.979569i	\(0.435545\pi\)
							−0.663950	+	0.747777i	\(0.731121\pi\)
\(31\)	1.89110	−	1.89110i	0.339652	−	0.339652i	−0.516585	−	0.856236i	\(-0.672797\pi\)
							0.856236	+	0.516585i	\(0.172797\pi\)
\(37\)	−4.17321	+	4.42541i	−0.686072	+	0.727533i
							\(41\)	3.28050	−	3.90955i	0.512328	−	0.610569i	−0.446421	−	0.894823i	\(-0.647301\pi\)
0.958749	+	0.284254i	\(0.0917459\pi\)
\(43\)	−5.53097			−0.843465			−0.421732	−	0.906720i	\(-0.638578\pi\)
							−0.421732	+	0.906720i	\(0.638578\pi\)
\(47\)	−5.01501	+	1.34377i	−0.731515	+	0.196009i	−0.605304	−	0.795995i	\(-0.706948\pi\)
							−0.126211	+	0.992003i	\(0.540282\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	370.2.ba.a.207.1	yes	108
5.3	odd	4		370.2.bd.a.133.9	yes	108
37.32	odd	36		370.2.bd.a.217.9	yes	108
185.143	even	36	inner	370.2.ba.a.143.1	✓	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.ba.a.143.1	✓	108	185.143	even	36	inner
370.2.ba.a.207.1	yes	108	1.1	even	1	trivial
370.2.bd.a.133.9	yes	108	5.3	odd	4
370.2.bd.a.217.9	yes	108	37.32	odd	36