Embedded newform 37.10.c.a.10.20

• Label: 37.10.c.a.10.20
• Level: $37$
• Weight: $10$
• Character: 37.10
• Analytic conductor: $19.056$
• Analytic rank: $0$
• Dimension: $54$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	37.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.0563259381\)
Analytic rank:	\(0\)
Dimension:	\(54\)
Relative dimension:	\(27\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			10.20
Character	\(\chi\)	\(=\)	37.10
Dual form			37.10.c.a.26.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(11.7918 - 20.4241i) q^{2} +(-110.187 - 190.850i) q^{3} +(-22.0953 - 38.2702i) q^{4} +(-512.092 - 886.969i) q^{5} -5197.24 q^{6} +(-4542.03 - 7867.02i) q^{7} +11032.7 q^{8} +(-14440.9 + 25012.4i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 32 q^{2} + 73 q^{3} - 6486 q^{4} + 2729 q^{5} + 1020 q^{6} + 4231 q^{7} - 7848 q^{8} - 140840 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	11.7918	−	20.4241i	0.521131	−	0.902625i	−0.478567	−	0.878051i	\(-0.658844\pi\)
							0.999698	−	0.0245742i	\(-0.00782301\pi\)
\(3\)	−110.187	−	190.850i	−0.785389	−	1.36033i	−0.928766	−	0.370666i	\(-0.879129\pi\)
							0.143377	−	0.989668i	\(-0.454204\pi\)
\(5\)	−512.092	−	886.969i	−0.366423	−	0.634663i	0.622580	−	0.782556i	\(-0.286084\pi\)
							−0.989003	+	0.147893i	\(0.952751\pi\)
\(7\)	−4542.03	−	7867.02i	−0.715004	−	1.23842i	−0.962958	−	0.269652i	\(-0.913091\pi\)
							0.247954	−	0.968772i	\(-0.420242\pi\)
\(11\)	−39802.9			−0.819687			−0.409843	−	0.912156i	\(-0.634417\pi\)
							−0.409843	+	0.912156i	\(0.634417\pi\)
\(13\)	−2875.66	−	4980.79i	−0.0279250	−	0.0483674i	0.851725	−	0.523989i	\(-0.175557\pi\)
							−0.879650	+	0.475621i	\(0.842223\pi\)
\(17\)	100048.	−	173288.i	0.290528	−	0.503210i	−0.683406	−	0.730038i	\(-0.739502\pi\)
							0.973935	+	0.226828i	\(0.0728356\pi\)
\(19\)	426642.	+	738966.i	0.751057	+	1.30087i	0.947311	+	0.320315i	\(0.103789\pi\)
							−0.196254	+	0.980553i	\(0.562878\pi\)
\(23\)	1.92337e6			1.43313			0.716567	−	0.697519i	\(-0.245713\pi\)
							0.716567	+	0.697519i	\(0.245713\pi\)
\(29\)	1.09256e6			0.286851			0.143425	−	0.989661i	\(-0.454188\pi\)
							0.143425	+	0.989661i	\(0.454188\pi\)
\(31\)	−7.50425e6			−1.45942			−0.729709	−	0.683758i	\(-0.760344\pi\)
							−0.729709	+	0.683758i	\(0.760344\pi\)
\(37\)	−1.05823e7	+	4.23977e6i	−0.928270	+	0.371907i
							\(41\)	−663164.	−	1.14863e6i	−0.0366517	−	0.0634825i	0.847118	−	0.531405i	\(-0.178336\pi\)
−0.883769	+	0.467923i	\(0.845003\pi\)
\(43\)	−4.54500e6			−0.202734			−0.101367	−	0.994849i	\(-0.532322\pi\)
							−0.101367	+	0.994849i	\(0.532322\pi\)
\(47\)	−5.47231e7			−1.63580			−0.817901	−	0.575359i	\(-0.804862\pi\)
							−0.817901	+	0.575359i	\(0.804862\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.c.a.10.20	✓	54
37.26	even	3	inner	37.10.c.a.26.20	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.c.a.10.20	✓	54	1.1	even	1	trivial
37.10.c.a.26.20	yes	54	37.26	even	3	inner