Embedded newform 37.7.g.a.8.7

• Label: 37.7.g.a.8.7
• Level: $37$
• Weight: $7$
• Character: 37.8
• Analytic conductor: $8.512$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	37.g (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.51200109393\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(18\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			8.7
Character	\(\chi\)	\(=\)	37.8
Dual form			37.7.g.a.14.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.02062 - 1.07732i) q^{2} +(-24.6756 - 14.2465i) q^{3} +(-40.4209 - 23.3370i) q^{4} +(-157.005 + 42.0695i) q^{5} +(83.8633 + 83.8633i) q^{6} +(63.0530 - 109.211i) q^{7} +(325.747 + 325.747i) q^{8} +(41.4241 + 71.7487i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 2 q^{2} - 6 q^{3} + 264 q^{4} + 52 q^{5} - 132 q^{6} - 2 q^{7} - 1968 q^{8} + 8094 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.02062	−	1.07732i	−0.502578	−	0.134665i	−0.00138157	−	0.999999i	\(-0.500440\pi\)
							−0.501196	+	0.865334i	\(0.667106\pi\)
\(3\)	−24.6756	−	14.2465i	−0.913912	−	0.527647i	−0.0322241	−	0.999481i	\(-0.510259\pi\)
							−0.881688	+	0.471833i	\(0.843592\pi\)
\(5\)	−157.005	+	42.0695i	−1.25604	+	0.336556i	−0.824668	−	0.565617i	\(-0.808638\pi\)
							−0.431375	+	0.902173i	\(0.641971\pi\)
\(7\)	63.0530	−	109.211i	0.183828	−	0.318400i	−0.759353	−	0.650679i	\(-0.774484\pi\)
							0.943181	+	0.332279i	\(0.107818\pi\)
\(11\)		−	649.879i		−	0.488264i	−0.969742	−	0.244132i	\(-0.921497\pi\)
							0.969742	−	0.244132i	\(-0.0785030\pi\)
\(13\)	806.502	−	216.102i	0.367093	−	0.0983622i	−0.0705571	−	0.997508i	\(-0.522478\pi\)
							0.437650	+	0.899146i	\(0.355811\pi\)
\(17\)	−1411.93	+	5269.40i	−0.287387	+	1.07254i	0.659690	+	0.751538i	\(0.270687\pi\)
							−0.947077	+	0.321005i	\(0.895979\pi\)
\(19\)	3847.09	−	1030.83i	0.560883	−	0.150288i	0.0327708	−	0.999463i	\(-0.489567\pi\)
							0.528112	+	0.849175i	\(0.322900\pi\)
\(23\)	−7687.13	−	7687.13i	−0.631802	−	0.631802i	0.316718	−	0.948520i	\(-0.397419\pi\)
							−0.948520	+	0.316718i	\(0.897419\pi\)
\(29\)	−12265.4	+	12265.4i	−0.502906	+	0.502906i	−0.912340	−	0.409433i	\(-0.865726\pi\)
							0.409433	+	0.912340i	\(0.365726\pi\)
\(31\)	13437.8	−	13437.8i	0.451068	−	0.451068i	−0.444641	−	0.895709i	\(-0.646669\pi\)
							0.895709	+	0.444641i	\(0.146669\pi\)
\(37\)	−39716.9	−	31437.1i	−0.784098	−	0.620637i
							\(41\)	26046.9	+	15038.2i	0.377925	+	0.218195i	0.676915	−	0.736061i	\(-0.263317\pi\)
−0.298990	+	0.954256i	\(0.596650\pi\)
\(43\)	26014.9	+	26014.9i	0.327203	+	0.327203i	0.851522	−	0.524319i	\(-0.175680\pi\)
							−0.524319	+	0.851522i	\(0.675680\pi\)
\(47\)	119880.			1.15466			0.577329	−	0.816511i	\(-0.304095\pi\)
							0.577329	+	0.816511i	\(0.304095\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.7.g.a.8.7	✓	72
37.14	odd	12	inner	37.7.g.a.14.7	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.7.g.a.8.7	✓	72	1.1	even	1	trivial
37.7.g.a.14.7	yes	72	37.14	odd	12	inner