Embedded newform 370.2.bd.a.237.3

• Label: 370.2.bd.a.237.3
• 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			237.3
Character	\(\chi\)	\(=\)	370.237
Dual form			370.2.bd.a.153.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.342020 - 0.939693i) q^{2} +(-1.31807 - 0.614627i) q^{3} +(-0.766044 - 0.642788i) q^{4} +(1.81136 + 1.31110i) q^{5} +(-1.02837 + 1.02837i) q^{6} +(1.08886 - 0.762428i) q^{7} +(-0.866025 + 0.500000i) q^{8} +(-0.568816 - 0.677889i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q + 6 q^{3}+O(q^{10}) \)

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
							\(3\)	−1.31807	−	0.614627i	−0.760989	−	0.354855i	0.00307516	−	0.999995i	\(-0.499021\pi\)
−0.764064	+	0.645140i	\(0.776799\pi\)
\(5\)	1.81136	+	1.31110i	0.810065	+	0.586340i
							\(7\)	1.08886	−	0.762428i	0.411550	−	0.288171i	−0.349409	−	0.936970i	\(-0.613618\pi\)
0.760959	+	0.648800i	\(0.224729\pi\)
\(11\)	2.09246	−	1.20808i	0.630900	−	0.364250i	−0.150201	−	0.988656i	\(-0.547992\pi\)
							0.781100	+	0.624405i	\(0.214659\pi\)
\(13\)	2.44954	−	2.91925i	0.679381	−	0.809655i	−0.310647	−	0.950525i	\(-0.600546\pi\)
							0.990028	+	0.140870i	\(0.0449900\pi\)
\(17\)	3.10729	−	2.60733i	0.753628	−	0.632369i	−0.182831	−	0.983144i	\(-0.558526\pi\)
							0.936460	+	0.350775i	\(0.114082\pi\)
\(19\)	−6.56693	−	3.06221i	−1.50656	−	0.702520i	−0.518284	−	0.855209i	\(-0.673429\pi\)
							−0.988274	+	0.152689i	\(0.951207\pi\)
\(23\)	−1.46365	−	0.845040i	−0.305193	−	0.176203i	0.339581	−	0.940577i	\(-0.389715\pi\)
							−0.644773	+	0.764374i	\(0.723048\pi\)
\(29\)	−0.553795	−	0.148389i	−0.102837	−	0.0275551i	0.207033	−	0.978334i	\(-0.433619\pi\)
							−0.309871	+	0.950779i	\(0.600286\pi\)
\(31\)	−1.61472	−	1.61472i	−0.290012	−	0.290012i	0.547073	−	0.837085i	\(-0.315742\pi\)
							−0.837085	+	0.547073i	\(0.815742\pi\)
\(37\)	5.28176	−	3.01711i	0.868317	−	0.496010i
							\(41\)	0.990902	−	1.18091i	0.154753	−	0.184427i	−0.683098	−	0.730327i	\(-0.739368\pi\)
0.837850	+	0.545900i	\(0.183812\pi\)
\(43\)		−	4.07279i		−	0.621095i	−0.950558	−	0.310547i	\(-0.899488\pi\)
							0.950558	−	0.310547i	\(-0.100512\pi\)
\(47\)	3.53553	−	0.947342i	0.515710	−	0.138184i	0.00842837	−	0.999964i	\(-0.497317\pi\)
							0.507281	+	0.861780i	\(0.330650\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	370.2.bd.a.237.3	yes	108
5.3	odd	4		370.2.ba.a.163.7	✓	108
37.5	odd	36		370.2.ba.a.227.7	yes	108
185.153	even	36	inner	370.2.bd.a.153.3	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.ba.a.163.7	✓	108	5.3	odd	4
370.2.ba.a.227.7	yes	108	37.5	odd	36
370.2.bd.a.153.3	yes	108	185.153	even	36	inner
370.2.bd.a.237.3	yes	108	1.1	even	1	trivial