Embedded newform 370.2.bd.b.237.6

• Label: 370.2.bd.b.237.6
• Level: $370$
• Weight: $2$
• Character: 370.237
• Analytic conductor: $2.954$
• Analytic rank: $0$
• Dimension: $120$
• 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:	\(120\)
Relative dimension:	\(10\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			237.6
Character	\(\chi\)	\(=\)	370.237
Dual form			370.2.bd.b.153.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.342020 + 0.939693i) q^{2} +(0.411996 + 0.192117i) q^{3} +(-0.766044 - 0.642788i) q^{4} +(-1.30839 - 1.81332i) q^{5} +(-0.321442 + 0.321442i) q^{6} +(-2.01108 + 1.40817i) q^{7} +(0.866025 - 0.500000i) q^{8} +(-1.79553 - 2.13983i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + 6 q^{3}+O(q^{10}) \)

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
							\(3\)	0.411996	+	0.192117i	0.237866	+	0.110919i	0.537897	−	0.843011i	\(-0.319219\pi\)
−0.300030	+	0.953930i	\(0.596997\pi\)
\(5\)	−1.30839	−	1.81332i	−0.585131	−	0.810939i
							\(7\)	−2.01108	+	1.40817i	−0.760116	+	0.532239i	−0.888163	−	0.459529i	\(-0.848018\pi\)
0.128047	+	0.991768i	\(0.459129\pi\)
\(11\)	0.442572	−	0.255519i	0.133440	−	0.0770418i	−0.431794	−	0.901972i	\(-0.642119\pi\)
							0.565234	+	0.824931i	\(0.308786\pi\)
\(13\)	3.11197	−	3.70870i	0.863105	−	1.02861i	−0.136175	−	0.990685i	\(-0.543481\pi\)
							0.999281	−	0.0379240i	\(-0.0120745\pi\)
\(17\)	4.59927	−	3.85924i	1.11549	−	0.936004i	0.117119	−	0.993118i	\(-0.462634\pi\)
							0.998368	+	0.0571139i	\(0.0181898\pi\)
\(19\)	−6.06697	−	2.82908i	−1.39186	−	0.649034i	−0.425199	−	0.905100i	\(-0.639796\pi\)
							−0.966660	+	0.256065i	\(0.917574\pi\)
\(23\)	−4.73639	−	2.73456i	−0.987606	−	0.570195i	−0.0830485	−	0.996546i	\(-0.526466\pi\)
							−0.904558	+	0.426351i	\(0.859799\pi\)
\(29\)	2.49260	+	0.667890i	0.462864	+	0.124024i	0.482712	−	0.875779i	\(-0.339652\pi\)
							−0.0198483	+	0.999803i	\(0.506318\pi\)
\(31\)	−2.91679	−	2.91679i	−0.523872	−	0.523872i	0.394867	−	0.918738i	\(-0.370791\pi\)
							−0.918738	+	0.394867i	\(0.870791\pi\)
\(37\)	1.19130	−	5.96496i	0.195849	−	0.980634i
							\(41\)	−2.49916	+	2.97838i	−0.390303	+	0.465145i	−0.925038	−	0.379875i	\(-0.875967\pi\)
0.534735	+	0.845020i	\(0.320411\pi\)
\(43\)			8.36237i			1.27525i	0.770347	+	0.637625i	\(0.220083\pi\)
							−0.770347	+	0.637625i	\(0.779917\pi\)
\(47\)	4.61817	−	1.23744i	0.673630	−	0.180498i	0.0942403	−	0.995549i	\(-0.469958\pi\)
							0.579389	+	0.815051i	\(0.303291\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	370.2.bd.b.237.6	yes	120
5.3	odd	4		370.2.ba.b.163.5	✓	120
37.5	odd	36		370.2.ba.b.227.5	yes	120
185.153	even	36	inner	370.2.bd.b.153.6	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.ba.b.163.5	✓	120	5.3	odd	4
370.2.ba.b.227.5	yes	120	37.5	odd	36
370.2.bd.b.153.6	yes	120	185.153	even	36	inner
370.2.bd.b.237.6	yes	120	1.1	even	1	trivial