Embedded newform 37.7.i.a.2.5

• Label: 37.7.i.a.2.5
• Level: $37$
• Weight: $7$
• Character: 37.2
• 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			2.5
Character	\(\chi\)	\(=\)	37.2
Dual form			37.7.i.a.19.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.75264 - 0.678269i) q^{2} +(26.0231 - 31.0131i) q^{3} +(-3.38424 - 0.596733i) q^{4} +(-73.2086 + 156.996i) q^{5} +(-222.783 + 222.783i) q^{6} +(-148.093 + 53.9015i) q^{7} +(506.925 + 135.830i) q^{8} +(-158.022 - 896.190i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 12 q^{2} - 12 q^{3} - 282 q^{4} + 186 q^{5} - 12 q^{6} - 12 q^{7} + 1782 q^{8} - 972 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{36}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.75264	−	0.678269i	−0.969081	−	0.0847836i	−0.408389	−	0.912808i	\(-0.633909\pi\)
							−0.560692	+	0.828025i	\(0.689465\pi\)
\(3\)	26.0231	−	31.0131i	0.963818	−	1.14863i	−0.0250268	−	0.999687i	\(-0.507967\pi\)
							0.988845	−	0.148947i	\(-0.0475885\pi\)
\(5\)	−73.2086	+	156.996i	−0.585669	+	1.25597i	0.360640	+	0.932705i	\(0.382558\pi\)
							−0.946308	+	0.323265i	\(0.895219\pi\)
\(7\)	−148.093	+	53.9015i	−0.431759	+	0.157147i	−0.548752	−	0.835985i	\(-0.684897\pi\)
							0.116993	+	0.993133i	\(0.462675\pi\)
\(11\)	−289.694	+	167.255i	−0.217651	+	0.125661i	−0.604862	−	0.796330i	\(-0.706772\pi\)
							0.387211	+	0.921991i	\(0.373439\pi\)
\(13\)	1577.34	+	2252.68i	0.717952	+	1.02534i	0.997865	+	0.0653163i	\(0.0208057\pi\)
							−0.279913	+	0.960026i	\(0.590305\pi\)
\(17\)	5120.95	+	3585.73i	1.04233	+	0.729844i	0.963680	−	0.267058i	\(-0.0860516\pi\)
							0.0786455	+	0.996903i	\(0.474940\pi\)
\(19\)	−2567.92	+	224.664i	−0.374387	+	0.0327546i	−0.272796	−	0.962072i	\(-0.587949\pi\)
							−0.101590	+	0.994826i	\(0.532393\pi\)
\(23\)	2627.21	+	9804.88i	0.215929	+	0.805858i	0.985837	+	0.167704i	\(0.0536352\pi\)
							−0.769908	+	0.638154i	\(0.779698\pi\)
\(29\)	−3648.20	+	13615.3i	−0.149584	+	0.558255i	0.849925	+	0.526904i	\(0.176647\pi\)
							−0.999508	+	0.0313502i	\(0.990019\pi\)
\(31\)	−9357.60	−	9357.60i	−0.314108	−	0.314108i	0.532391	−	0.846499i	\(-0.321294\pi\)
							−0.846499	+	0.532391i	\(0.821294\pi\)
\(37\)	21233.8	+	45987.5i	0.419200	+	0.907894i
							\(41\)	−85160.7	−	15016.1i	−1.23563	−	0.217875i	−0.482587	−	0.875848i	\(-0.660303\pi\)
−0.753041	+	0.657973i	\(0.771414\pi\)
\(43\)	−109259.	+	109259.i	−1.37421	+	1.37421i	−0.520099	+	0.854106i	\(0.674105\pi\)
							−0.854106	+	0.520099i	\(0.825895\pi\)
\(47\)	103235.	−	178808.i	0.994337	−	1.72224i	0.405131	−	0.914259i	\(-0.367226\pi\)
							0.589206	−	0.807983i	\(-0.299441\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.7.i.a.2.5	✓	216
37.19	odd	36	inner	37.7.i.a.19.5	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.7.i.a.2.5	✓	216	1.1	even	1	trivial
37.7.i.a.19.5	yes	216	37.19	odd	36	inner