Embedded newform 370.2.bd.a.217.7

• Label: 370.2.bd.a.217.7
• Level: $370$
• Weight: $2$
• Character: 370.217
• 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			217.7
Character	\(\chi\)	\(=\)	370.217
Dual form			370.2.bd.a.133.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.342020 - 0.939693i) q^{2} +(0.695029 + 1.49049i) q^{3} +(-0.766044 + 0.642788i) q^{4} +(-2.05872 - 0.872736i) q^{5} +(1.16289 - 1.16289i) q^{6} +(1.58371 - 2.26177i) q^{7} +(0.866025 + 0.500000i) q^{8} +(0.189856 - 0.226262i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q + 6 q^{3}+O(q^{10}) \)

Character values

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

\(n\)	\(261\)	\(297\)
\(\chi(n)\)	\(e\left(\frac{5}{36}\right)\)	\(e\left(\frac{1}{4}\right)\)

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
							\(3\)	0.695029	+	1.49049i	0.401275	+	0.860537i	0.998298	+	0.0583131i	\(0.0185722\pi\)
−0.597023	+	0.802224i	\(0.703650\pi\)
\(5\)	−2.05872	−	0.872736i	−0.920688	−	0.390300i
							\(7\)	1.58371	−	2.26177i	0.598585	−	0.854868i	−0.399499	−	0.916734i	\(-0.630816\pi\)
0.998085	+	0.0618651i	\(0.0197049\pi\)
\(11\)	4.81498	+	2.77993i	1.45177	+	0.838180i	0.998582	−	0.0532346i	\(-0.0169531\pi\)
							0.453188	+	0.891415i	\(0.350286\pi\)
\(13\)	−0.538253	−	0.641465i	−0.149285	−	0.177910i	0.686220	−	0.727394i	\(-0.259269\pi\)
							−0.835504	+	0.549484i	\(0.814824\pi\)
\(17\)	1.17235	+	0.983718i	0.284337	+	0.238587i	0.773789	−	0.633443i	\(-0.218359\pi\)
							−0.489453	+	0.872030i	\(0.662803\pi\)
\(19\)	−1.19787	−	2.56883i	−0.274810	−	0.589331i	0.719663	−	0.694323i	\(-0.244296\pi\)
							−0.994473	+	0.104992i	\(0.966518\pi\)
\(23\)	5.90081	−	3.40684i	1.23040	−	0.710374i	0.263290	−	0.964717i	\(-0.415192\pi\)
							0.967114	+	0.254342i	\(0.0818589\pi\)
\(29\)	−0.283008	−	1.05620i	−0.0525533	−	0.196132i	0.934658	−	0.355548i	\(-0.115706\pi\)
							−0.987212	+	0.159416i	\(0.949039\pi\)
\(31\)	0.780856	+	0.780856i	0.140246	+	0.140246i	0.773744	−	0.633498i	\(-0.218382\pi\)
							−0.633498	+	0.773744i	\(0.718382\pi\)
\(37\)	−4.20696	−	4.39334i	−0.691621	−	0.722261i
							\(41\)	5.06746	+	6.03916i	0.791404	+	0.943159i	0.999388	−	0.0349732i	\(-0.0111346\pi\)
−0.207984	+	0.978132i	\(0.566690\pi\)
\(43\)		−	5.92174i		−	0.903057i	−0.892257	−	0.451529i	\(-0.850879\pi\)
							0.892257	−	0.451529i	\(-0.149121\pi\)
\(47\)	−1.78083	+	6.64617i	−0.259761	+	0.969443i	0.705618	+	0.708593i	\(0.250670\pi\)
							−0.965379	+	0.260850i	\(0.915997\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	370.2.bd.a.217.7	yes	108
5.3	odd	4		370.2.ba.a.143.3	✓	108
37.22	odd	36		370.2.ba.a.207.3	yes	108
185.133	even	36	inner	370.2.bd.a.133.7	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.ba.a.143.3	✓	108	5.3	odd	4
370.2.ba.a.207.3	yes	108	37.22	odd	36
370.2.bd.a.133.7	yes	108	185.133	even	36	inner
370.2.bd.a.217.7	yes	108	1.1	even	1	trivial