Embedded newform 37.7.i.a.5.2

• Label: 37.7.i.a.5.2
• Level: $37$
• Weight: $7$
• Character: 37.5
• Analytic conductor: $8.512$
• Analytic rank: $0$
• Dimension: $216$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	37.i (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.51200109393\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(18\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			5.2
Character	\(\chi\)	\(=\)	37.5
Dual form			37.7.i.a.15.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.53836 + 11.8770i) q^{2} +(6.06356 - 16.6595i) q^{3} +(-69.2525 - 82.5319i) q^{4} +(60.5521 - 86.4774i) q^{5} +(164.283 + 164.283i) q^{6} +(72.0577 + 408.660i) q^{7} +(553.647 - 148.349i) q^{8} +(317.674 + 266.560i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 12 q^{2} - 12 q^{3} - 282 q^{4} + 186 q^{5} - 12 q^{6} - 12 q^{7} + 1782 q^{8} - 972 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/37\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{23}{36}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.53836	+	11.8770i	−0.692295	+	1.48463i	0.173815	+	0.984778i	\(0.444390\pi\)
							−0.866110	+	0.499853i	\(0.833387\pi\)
\(3\)	6.06356	−	16.6595i	0.224576	−	0.617018i	−0.775318	−	0.631571i	\(-0.782410\pi\)
							0.999894	+	0.0145530i	\(0.00463253\pi\)
\(5\)	60.5521	−	86.4774i	0.484417	−	0.691819i	−0.500326	−	0.865837i	\(-0.666786\pi\)
							0.984742	+	0.174018i	\(0.0556753\pi\)
\(7\)	72.0577	+	408.660i	0.210081	+	1.19143i	0.889242	+	0.457438i	\(0.151233\pi\)
							−0.679161	+	0.733989i	\(0.737656\pi\)
\(11\)	361.271	+	208.580i	0.271428	+	0.156709i	0.629536	−	0.776971i	\(-0.283245\pi\)
							−0.358108	+	0.933680i	\(0.616578\pi\)
\(13\)	40.2217	+	3.51894i	0.0183076	+	0.00160170i	0.0963058	−	0.995352i	\(-0.469297\pi\)
							−0.0779983	+	0.996953i	\(0.524853\pi\)
\(17\)	483.083	+	5521.67i	0.0983276	+	1.12389i	0.871734	+	0.489979i	\(0.162996\pi\)
							−0.773407	+	0.633910i	\(0.781449\pi\)
\(19\)	2418.83	+	5187.20i	0.352651	+	0.756262i	0.999982	−	0.00594641i	\(-0.00189281\pi\)
							−0.647331	+	0.762209i	\(0.724115\pi\)
\(23\)	−1494.91	+	5579.08i	−0.122866	+	0.458542i	−0.999755	−	0.0221516i	\(-0.992948\pi\)
							0.876889	+	0.480693i	\(0.159615\pi\)
\(29\)	1162.05	+	4336.84i	0.0476466	+	0.177820i	0.985649	−	0.168811i	\(-0.0539926\pi\)
							−0.938002	+	0.346630i	\(0.887326\pi\)
\(31\)	20374.0	−	20374.0i	0.683898	−	0.683898i	−0.276978	−	0.960876i	\(-0.589333\pi\)
							0.960876	+	0.276978i	\(0.0893330\pi\)
\(37\)	−40801.1	+	30016.6i	−0.805502	+	0.592594i
							\(41\)	70012.5	+	83437.7i	1.01584	+	1.21063i	0.977407	+	0.211366i	\(0.0677912\pi\)
0.0384300	+	0.999261i	\(0.487764\pi\)
\(43\)	−81770.2	−	81770.2i	−1.02847	−	1.02847i	−0.999583	−	0.0288824i	\(-0.990805\pi\)
							−0.0288824	−	0.999583i	\(-0.509195\pi\)
\(47\)	5293.68	+	9168.92i	0.0509875	+	0.0883129i	0.890393	−	0.455193i	\(-0.150430\pi\)
							−0.839405	+	0.543506i	\(0.817096\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.7.i.a.5.2	✓	216
37.15	odd	36	inner	37.7.i.a.15.2	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.7.i.a.5.2	✓	216	1.1	even	1	trivial
37.7.i.a.15.2	yes	216	37.15	odd	36	inner